Recent zbMATH articles in MSC 35Rhttps://zbmath.org/atom/cc/35R2021-07-26T21:45:41.944397ZWerkzeugDegenerate fractional differential equations in locally convex spaces with a \(\sigma \)-regular pair of operatorshttps://zbmath.org/1463.342482021-07-26T21:45:41.944397Z"Kostić, Marko"https://zbmath.org/authors/?q=ai:kostic.marko"Fëdorov, Vladimir Evgen'evich"https://zbmath.org/authors/?q=ai:fedorov.v-eSummary: We consider a degenerate fractional order differential equation \(D^\alpha_tLu(t)=Mu(t)\) in a Hausdorff secquentially complete locally convex space is considered. Under the \(p\)-regularity of the operator pair \((L,M)\), we find the phase space of the equation and the family of its resolving operators. We show that the identity image of the latter coincides with the phase space. We prove an unique solvability theorem and obtain the form of the solution to the Cauchy problem for the corresponding inhomogeneous equation. We give an example of application the obtained abstract results to studying the solvability of the initial boundary value problems for the partial differential equations involving entire functions on an unbounded operator in a Banach space, which is a specially constructed Frechét space. It allows us to consider, for instance, a periodic in a spatial variable \(x\) problem for the equation with a shift along \(x\) and with a fractional order derivative with respect to time \(t\).The fundamental solution for the \(m\)-th powers of the sub-Laplacian on the quaternionic Heisenberg grouphttps://zbmath.org/1463.350162021-07-26T21:45:41.944397Z"Wang, Haimeng"https://zbmath.org/authors/?q=ai:wang.haimeng"Zhou, Xuan"https://zbmath.org/authors/?q=ai:zhou.xuan"Zhao, Yujuan"https://zbmath.org/authors/?q=ai:zhao.yujuanSummary: We discuss the fundamental solution for \(m\)-th powers of the sub-Laplacian on the quaternionic Heisenberg group, This result is the extension of the conclusion on the Heisenberg group. We use the representation theory of nilpotent Lie groups of step two to analyze the associated \(m\)-th powers of the sub-Laplacian on the quaternionic Heisenberg group and to construct its fundamental solution.Classical solutions for nonlocal problems on different unbounded domainshttps://zbmath.org/1463.350192021-07-26T21:45:41.944397Z"Wang, Yue"https://zbmath.org/authors/?q=ai:wang.yue.2|wang.yue.1|wang.yue.3"Liang, Jinping"https://zbmath.org/authors/?q=ai:liang.jinping"Suo, Hongmin"https://zbmath.org/authors/?q=ai:suo.hong-min"Lei, Jun"https://zbmath.org/authors/?q=ai:lei.jun"Zhao, Shihai"https://zbmath.org/authors/?q=ai:zhao.shihai"Ye, Hongyan"https://zbmath.org/authors/?q=ai:ye.hongyanSummary: This paper addresses the existence of classical solutions for a kind of nonlocal problems with linear to critical exponents on different unbounded domains, and the expressions of infinitely many classical positive solutions are given based on the idea of constructors. First of all, infinitely many classical solutions are obtained by using the Sobolev's achieving function of the best constant on the whole space with critical growth. Next, we gain the same conclusions on the half-space and the whole space with the help of separation variable methods without the Cartesian coordinate planes. Finally, we prove that the results are suitable for all exponents between linear and critical on the whole space without the Cartesian coordinate planes.Dirichlet boundary value problem in half-strip for fractional differential equation with Bessel operator and Riemann-Liouville partial derivativehttps://zbmath.org/1463.350252021-07-26T21:45:41.944397Z"Khushtova, Fatima Gidovna"https://zbmath.org/authors/?q=ai:khushtova.fatima-gidovnaSummary: In the work we study the Dirichlet boundary value problem in a half-strip for a fractional differential equations with the Bessel operator and the Riemann-Liouville partial derivatives. We formulate the unique solvability theoresm for the considered problem. We find the representations for the solutions in terms of the integral transform with the Wright function in the kernel. The proof of the existence theorem is made on the base of the mentioned integral transform and the modified Bessel function of first kind. The uniqueness of the solutions is shown in the class of the functions satisfying an analogue of Tikhonov equation. In the case, when the considered equations is the fractional order diffusion equation, we show that the obtained solutions coincides with the known solution to the Dirichlet problem for the corresponding equation. We also consider the case when the initial function is power in the spatial variable. In this case the solution to the problem is written out in terms of the Fox \(H\)-function.Oscillation for a class of impulsive fractional partial differential equationshttps://zbmath.org/1463.350332021-07-26T21:45:41.944397Z"Feng, Qian"https://zbmath.org/authors/?q=ai:feng.qian"Ma, Qingxia"https://zbmath.org/authors/?q=ai:ma.qingxia"Liu, Anping"https://zbmath.org/authors/?q=ai:liu.anpingSummary: In this paper, we study the oscillation criteria for a class of impulsive fractional partial differential equations with Neumann boundary condition. By using the properties of the modified Riemann-Liouville fractional partial differential equations and a generalized Riccati technique and the differential inequality methods, some sufficient conditions for the oscillatory behavior of the solution are obtained. As an application, the relevant example is given to illustrate the main conclusions.Oscillation conditions of certain nonlinear impulsive neutral parabolic distributed parameter systemshttps://zbmath.org/1463.350342021-07-26T21:45:41.944397Z"Luo, Liping"https://zbmath.org/authors/?q=ai:luo.liping"Luo, Zhenguo"https://zbmath.org/authors/?q=ai:luo.zhenguo"Zeng, Yunhui"https://zbmath.org/authors/?q=ai:zeng.yunhuiSummary: The oscillation problems for a class of nonlinear impulse parabolic distributed parameter systems with neutral term and higher order Laplace operator are investigated under first boundary value condition. By using the technique of treating neutral term and higher order Laplace operator and integral averaging method, some new sufficient criteria are established for the oscillation of all solutions of such systems. The conclusions fully indicate that the system oscillation is caused by impulse and delay.Oscillation of nonlinear neutral hyperbolic equations with variable delayed influencehttps://zbmath.org/1463.350352021-07-26T21:45:41.944397Z"Luo, Zhenguo"https://zbmath.org/authors/?q=ai:luo.zhenguo"Luo, Liping"https://zbmath.org/authors/?q=ai:luo.liping"Hou, Juan"https://zbmath.org/authors/?q=ai:hou.juanSummary: Some sufficient conditions are established for the oscillation of solutions of a class of neutral hyperbolic equations with nonlinear diffusion term and variable delays under Dirichlet's boundary condition. Our results fully reflect the influence actions of delays in equation oscillation.Lie symmetries solutions for time fractional Cahn-Allen equationhttps://zbmath.org/1463.350372021-07-26T21:45:41.944397Z"Li, Wenting"https://zbmath.org/authors/?q=ai:li.wenting"Liu, Dan"https://zbmath.org/authors/?q=ai:liu.dan"Jiang, Kun"https://zbmath.org/authors/?q=ai:jiang.kunSummary: The researches of time fractional Cahn-Allen equation are done by the Lie group analysis and some Lie symmetries of the equation are obtained. The Lie symmetry analysis is studied for time fractional Cahn-Allen equation that is one of the applications of symmetries. First, we obtain the two order extension of the equation, and then we compute the vector field corresponding to it. Finally, we obtain a reduced equation by using the vector field.A symmetry result for a class of \(p\)-Laplace involving Baouendi-Grushin operators via constrained minimization methodhttps://zbmath.org/1463.350382021-07-26T21:45:41.944397Z"Qian, Hongli"https://zbmath.org/authors/?q=ai:qian.hongli"Huang, Xiaotao"https://zbmath.org/authors/?q=ai:huang.xiaotaoSummary: The purpose of this paper is to investigate a spacial \(p\)-Laplace equation involving Baouendi-Grushin operators. Some existence and symmetry results for positive solutions are obtained by rearrangement of its corresponding constrained minimization. These results are in accordance with those of the classical \(p\)-Laplace equations and the Baouendi-Grushin type sub-Laplacian equations.Positive solutions for a fractional \(p\)-Laplacian Kirchhoff problem with vanishing nonlocal termhttps://zbmath.org/1463.350492021-07-26T21:45:41.944397Z"Han, Tao"https://zbmath.org/authors/?q=ai:han.tao"Wang, Li"https://zbmath.org/authors/?q=ai:wang.li.1|wang.li.4|wang.li.6|wang.li.3|wang.li.2|wang.li.5|wang.li"Jian, Hui"https://zbmath.org/authors/?q=ai:jian.huiSummary: In this paper we study a fractional \(p\)-Laplacian Kirchhoff problem with vanishing nonlocal term in a bounded domain. Under an appropriated condition, we prove a multiplicity result of positive solutions by using truncation techniques and variational methods.Multiple periodic solutions for impulsive differential systems with variable exponenthttps://zbmath.org/1463.350602021-07-26T21:45:41.944397Z"Zhang, Shengui"https://zbmath.org/authors/?q=ai:zhang.shenguiSummary: The existence for periodic solution of impulsive differential systems with variable exponent is studied. When the nonlinearity has a suplinear growth, some results for existence of infinitely many periodic solutions are obtained by using the fountain theorem in critical point theory.Exponential stability of hybrid stochastic heat equation with Markovian switchinghttps://zbmath.org/1463.350882021-07-26T21:45:41.944397Z"Wang, Shan"https://zbmath.org/authors/?q=ai:wang.shan"Wang, Feng"https://zbmath.org/authors/?q=ai:wang.feng.2|wang.feng.4|wang.feng.1|wang.feng.3Summary: In this paper, the hybrid stochastic heat equation with Markovian switching and two independent white noises is studied. By using Markovian switching theory and the properties of heat equation, the explicit expression of the strong solution of equation is obtained, and the sufficient condition for almost surely exponential stability of the solution is given. Finally, some examples are given to illustrate the influence of Markovian switching and white noises on the stability of solutions.Asymptotic solution to nonlinear generalized fractional order thermal wave equationhttps://zbmath.org/1463.350932021-07-26T21:45:41.944397Z"Chen, Huaijun"https://zbmath.org/authors/?q=ai:chen.huaijun"Mo, Jiaqi"https://zbmath.org/authors/?q=ai:mo.jiaqi"Xu, Jianzhong"https://zbmath.org/authors/?q=ai:xu.jianzhongSummary: A class of nonlinear generalized thermal wave equation was considered. Firstly, the solution to reduced thermal wave equation was obtained. Next, the arbitrary order asymptotic solutions to the generalized nonlinear disturbed thermal wave equation initial-boundary value problem were constructed by using the method of functional analysis homotopic mapping. An example was given and the accuracy of its asymptotic solution was obtained. Finally, the physical sense of the solution was briefly stated. The approximate analysis solution makes up for the simple numerical simulation solution deficiency.Asymptotic analysis of vascularized tumor growth model with inhibitorhttps://zbmath.org/1463.350982021-07-26T21:45:41.944397Z"Liu, Jiali"https://zbmath.org/authors/?q=ai:liu.jiali"Wang, Zejia"https://zbmath.org/authors/?q=ai:wang.zejia"Wen, Lishu"https://zbmath.org/authors/?q=ai:wen.lishuSummary: In this paper, we study a free boundary problem of partial differential equation modeling the growth model of vascularized tumor with inhibitors. Firstly, we show the existence and uniqueness of solution for the quasi-steady state problem by the theory of ordinary differential equation. Next, \(\lim\limits_{t\to\infty}{\mathrm{R}}(t) = 0\) is obtained under some assumptions. Finally, the asymptotic behavior of the solution for the considered problem is discussed by the iterative technique.Existence of uniform random attractor for nonautonomous stochastic reaction-diffusion equations on unbounded domainshttps://zbmath.org/1463.351132021-07-26T21:45:41.944397Z"Zhang, Jie"https://zbmath.org/authors/?q=ai:zhang.jie.4|zhang.jie|zhang.jie.5|zhang.jie.3|zhang.jie.1|zhang.jie.2"Li, Xiaojun"https://zbmath.org/authors/?q=ai:li.xiaojunSummary: This paper studies the existence of uniform attractors for a class of nonautonomous stochastic reaction-diffusion equations with white noise on unbounded domains. Firstly, with uniform estimation of the solutions, it is proved that the stochastic dynamical system corresponding to the original equation has a uniformly pullback absorbing set with respect to symbol space. Secondly, by asymptotic tail estimation, it is proved that the solution is uniformly pullback and asymptotically compact. The existence of uniform random attractor of the original system is obtained.Regularity to a Kohn-Laplace equation with bounded coefficients on the Heisenberg Grouphttps://zbmath.org/1463.351482021-07-26T21:45:41.944397Z"Zhang, Junli"https://zbmath.org/authors/?q=ai:zhang.junli"Niu, Pengcheng"https://zbmath.org/authors/?q=ai:niu.pengcheng"Wang, Xiuxiu"https://zbmath.org/authors/?q=ai:wang.xiuxiuSummary: In this paper, we concern the divergence Kohn-Laplace equation \[\sum\limits_{i=1}^n \sum\limits_{j=1}^n (X_j^*({a^{ij}}{X_i}u) + Y_j^*({b^{ij}}{Y_i}u))+Tu = f-\sum\limits_{i=1}^n (X_i^*{f^i} + Y_i^*{g^i})\] with bounded coefficients on the Heisenberg group \(\mathbb{H}^n\), where \({X_1},\cdots, {X_n}, {Y_1}, \cdots, {Y_n}\) and \(T\) are real smooth vector fields defined in a bounded region \(\Omega \subset \mathbb{H}^n\). The local maximum principle of weak solutions to the equation is established. The oscillation properties of the weak solutions are studied and then the Hölder regularity and weak Harnack inequality of the weak solutions are proved.Exact traveling wave solutions for the time fractional nonlinear evolution equation by sub-equation methodhttps://zbmath.org/1463.351562021-07-26T21:45:41.944397Z"Zhao, Xin"https://zbmath.org/authors/?q=ai:zhao.xin"Xia, Shanlei"https://zbmath.org/authors/?q=ai:xia.shanleiSummary: The exact traveling wave solutions of nonlinear evolution equations in mathematical physics, such as time-fractional Burgers equation, mKdV equation, are constructed by a new proposed sub-equation method. Using fractional complex transform skills, the fractional nonlinear evolution differential equation can be easily converted into its equivalent ordinary differential equation form. The results show that fractional complex transform can be efficiently used for solving fractional differential equation, which implies that the method is straight forward and concise.Applying improved Kudryashov method to solve exact solutions of nonlinear equationshttps://zbmath.org/1463.351892021-07-26T21:45:41.944397Z"Feng, Qingjiang"https://zbmath.org/authors/?q=ai:feng.qingjiang"Yang, Juan"https://zbmath.org/authors/?q=ai:yang.juanSummary: By using Kudryashov method, the exact solutions of the \((1+1)\)-dimensional Benjamin-Ono equation, the \((2+1)\)-dimensional AKNS equation and the fractional order biological population model equations are obtained respectively. The practice proves that this method is simple and convenient, it has very important significance for the research of nonlinear fractional evolution equations.Nonhomogeneous systems involving critical or subcritical nonlinearities.https://zbmath.org/1463.352432021-07-26T21:45:41.944397Z"Bhakta, Mousomi"https://zbmath.org/authors/?q=ai:bhakta.mousomi"Chakraborty, Souptik"https://zbmath.org/authors/?q=ai:chakraborty.souptik"Pucci, Patrizia"https://zbmath.org/authors/?q=ai:pucci.patriziaThe authors introduce suitable assumptions under which a system (S) where are considered equations involving nontrivial nonhomogeneous terms and critical or subcritical nonlinearities, admits a positive solution, with different components.
Therefore, is actually solved system (S) when it does not reduce into a single equation.
To the knowledge of the authors the question of finding at least two nontrivial solutions to (S) remains open in the vectorial case.
In the scalar case, the authors recall that existence of two different solutions are proved by them in very recent papers.Precise Morrey regularity of the weak solutions to a kind of quasilinear systems with discontinuous datahttps://zbmath.org/1463.352472021-07-26T21:45:41.944397Z"Fattorusso, Luisa"https://zbmath.org/authors/?q=ai:fattorusso.luisa"Softova, Lubomira"https://zbmath.org/authors/?q=ai:softova.lubomira-gSummary: We consider the Dirichlet problem for a class of quasilinear elliptic systems in domain with irregular boundary. The principal part satisfies componentwise coercivity condition and the nonlinear terms are Carathéodory maps having Morrey regularity in \(x\) and verifying controlled growth conditions with respect to the other variables. We have obtained boundedness of the weak solution to the problem that permits to apply an iteration procedure in order to find optimal Morrey regularity of its gradient.Multiple solutions for a fractional \(p\)-Laplacian equation with concave nonlinearitieshttps://zbmath.org/1463.352562021-07-26T21:45:41.944397Z"Pei, Ruichang"https://zbmath.org/authors/?q=ai:pei.ruichangSummary: We investigate a fractional \(p\)-Laplacian equation with right-hand-side non-linearity which exhibits \((p - 1)\)-sublinear term of the form \(\lambda {|u|^{q - 2}}, q < p\) (concave term), and a continuous term \(f(x, u)\) which is respectively \((p - 1)\)-superlinear or asymptotically \((p - 1)\)-linear at infinity. Some existence results for multiple nontrivial solutions are established by using variational methods combined with the Morse theory.Existence of infinitely many high energy solutions for fractional Schrödinger-Kirchhoff equationshttps://zbmath.org/1463.352582021-07-26T21:45:41.944397Z"Xu, Jiafa"https://zbmath.org/authors/?q=ai:xu.jiafa"Liu, Lishan"https://zbmath.org/authors/?q=ai:liu.lishan"Jiang, Jiqiang"https://zbmath.org/authors/?q=ai:jiang.jiqiangSummary: We study the following fractional Schrödinger-Kirchhoff equations with sign-changing potential function: \[\left(a+b\iint_{\mathbb{R}^{2N}}\frac{|u(x) - u(y)|^p}{|x-y|^{N+ps}}dxdy\right)^{p-1}(-\Delta)_p^s u + \lambda V(x)|u|^{p-2}u = f(x,u) - \mu g(x)|u|^{q-2}u,\; x \in \mathbb{R}^N,\] where \(s \in (0,1)\), \(p \in [2,\infty)\), \(q \in (1,p)\), \(a, b > 0\), \(\lambda, \mu > 0\) are positive constants. By some appropriate assumptions on \(V, f, g\), we use the fountain theorem to obtain the existence of infinitely many high energy solutions for the above system.Unsolvability conditions for some inequalities and systems with functional parameters and singular coefficients on boundaryhttps://zbmath.org/1463.352682021-07-26T21:45:41.944397Z"Galakhov, Evgeniĭ Igorevich"https://zbmath.org/authors/?q=ai:galakhov.evgeny"Salieva, Ol'ga Alekseevna"https://zbmath.org/authors/?q=ai:salieva.olga-alekseevnaSummary: We consider the problem on nonexistence of positive solutions for some nonlinear elliptic inequalities in a bounded domain. The principal parts of the considered inequalities are \(p(x)\)-Laplacians with variable exponents. The lower terms of the considered inequalities can depend both on the unknown function and its gradient. We assume that the coefficients at the lower terms have singularities at the boundary. To the best of the authors' knowledge, the conditions for nonexistence of solutions to inequalities with variable exponents were not considered before.
We obtain the sufficient conditions for nonexistence of positive solutions in terms of the exponent \(p(x)\), of the singularities order and of parameters in the problem. To prove the obtained conditions, we employ an original modification of the nonlinear capacity method proposed by S. I. Pokhozhaev. The method is based on a special choice of test functions in the generalized formulation of the problem and on algebraic transformations of the obtained expression. This allows us to obtain asymptotically sharp apriori estimates for the solutions leading to a contradiction under a certain choice of the parameters. This implies the nonexistence of the solutions. We generalize the obtained results for the case of nonlinear systems with similar conditions for the operators and coefficients.Existence of solutions for fractional Kirchhoff-type equation with variable exponenthttps://zbmath.org/1463.352742021-07-26T21:45:41.944397Z"Zhang, Shengui"https://zbmath.org/authors/?q=ai:zhang.shenguiSummary: This paper studies a class of Dirichlet boundary value problems for fractional Kirchhoff-type equation with variable exponent. When the nonlinear term is \({p^+}\)-superlinear at infinity, some sufficient conditions for the existence of infinitely many solutions are established by employing the critical point theory, variational methods and the theory of fractional variable exponent space.Picone identities for anisotropic sub-Laplace operator and pseudo \(p\) sub-Laplace operator and its applicationshttps://zbmath.org/1463.352792021-07-26T21:45:41.944397Z"Feng, Tingfu"https://zbmath.org/authors/?q=ai:feng.tingfu"Zhu, Yan"https://zbmath.org/authors/?q=ai:zhu.yan"Mu, Gui"https://zbmath.org/authors/?q=ai:mu.gui"Zhang, Kelei"https://zbmath.org/authors/?q=ai:zhang.keleiSummary: In this paper, anisotropic Laplace operator and pseudo \(p\) sub-Laplace operator in Euclidean space are introduced into Heisenberg group \({\mathbb{H}^n}\), respectively, which are called anisotropic sub-Laplace operator and pseudo \(p\) sub-Laplace operator, respectively. The corresponding Picone identities are not only established, but also the applications of these Picone identities are given. The results in Euclidean space are generalized into Heisenberg group \({\mathbb{H}^n}\).The fractional generalized disturbed thermal wave equationhttps://zbmath.org/1463.352852021-07-26T21:45:41.944397Z"Ouyang, Cheng"https://zbmath.org/authors/?q=ai:ouyang.cheng.1"Wang, Weigang"https://zbmath.org/authors/?q=ai:wang.weigang"Mo, Jiaqi"https://zbmath.org/authors/?q=ai:mo.jiaqiSummary: A class of generalized fractional nonlinear disturbed thermal wave equation is considered. Firstly, the arbitrary order approximate analytic solutions for the initial boundary value problem of the fractional generalized nonlinear disturbed thermal wave equation was constructed by using the singular perturbation method. Then, the uniformly valid asymptotic expansion was proved by using the fixed point theory of the functional analysis. Finally, the physical sense of the solution was stated simply. The approximate analytic solution makes up for the deficiency of the simple numerical method.Optimization method for identifying Robin coefficient in heat equationhttps://zbmath.org/1463.352892021-07-26T21:45:41.944397Z"Wang, Bingxian"https://zbmath.org/authors/?q=ai:wang.bingxian"Du, Haiqing"https://zbmath.org/authors/?q=ai:du.haiqingSummary: The Robin coefficient inverse problem was investigated by transforming it into an optimization problem based on its uniqueness. The regularization penalty of the format \({L^2} + TV\) was introduced in the construction of the objective function. On the basis of discussing the existence of the minimum element of the objective function, a rapidly iterative algorithm was constructed. The numerical simulation results showed the effectiveness of the proposed algorithm.Effect of space dimensions on equilibrium solutions of Cahn-Hilliard and conservative Allen-Cahn equationshttps://zbmath.org/1463.352992021-07-26T21:45:41.944397Z"Lee, Hyun Geun"https://zbmath.org/authors/?q=ai:lee.hyun-geun"Yang, Junxiang"https://zbmath.org/authors/?q=ai:yang.junxiang"Park, Jintae"https://zbmath.org/authors/?q=ai:park.jintae"Kim, Junseok"https://zbmath.org/authors/?q=ai:kim.junseokSummary: In this study, we investigate the effect of space dimensions on the equilibrium solutions of the Cahn-Hilliard (CH) and conservative Allen-Cahn (CAC) equations in one, two, and three dimensions. The CH and CAC equations are fourth-order parabolic partial and second-order integro-partial differential equations, respectively. The former is used to model phase separation in binary mixtures, and the latter is used to model mean curvature flow with conserved mass. Both equations have been used for modeling various interface problems. To study the space dimension effect on both the equations, we consider the equilibrium solution profiles for symmetric, radially symmetric, and spherically symmetric drop shapes. We highlight the different dynamics obtained from the CH and CAC equations. In particular, we find that there is a large difference between the solutions obtained from these equations in three-dimensional space.Temporal decay for the chemotaxis system involving different fractional powers in higher dimensionshttps://zbmath.org/1463.353072021-07-26T21:45:41.944397Z"Wang, Xi"https://zbmath.org/authors/?q=ai:wang.xi"Liu, Zuhan"https://zbmath.org/authors/?q=ai:liu.zuhan"Zhou, Ling"https://zbmath.org/authors/?q=ai:zhou.lingSummary: This paper deals with a fractional Keller-Segel model arising from biology, which involves two parabolic equations with fractional Laplacians and a classical parabolic equation. The global existence result and the optimal temporal decay estimates of the classical solution to the fractional Keller-Segel system are obtained by pure energy method based on the assumption of smallness initial conditions. More precisely, the authors derive the optimal decay rates of the higher-order spatial derivatives of \(u\), \(v\) and \(\nabla \psi\).An inverse problem on determining the right-hand side of fractional equation in weight distributionshttps://zbmath.org/1463.353132021-07-26T21:45:41.944397Z"Lopushansky, A. O."https://zbmath.org/authors/?q=ai:lopushanskyi.a-o"Lopushanska, H. P."https://zbmath.org/authors/?q=ai:lopushanska.galynaThe paper deals with an inverse Cauchy problem for the fractional diffusion equation with distributions in the right-hand sides. The aim is to find a generalized solution of the direct problem and determine an unknown time-dependent factor from the space of weight distributions in the right-hand side of the equation. A unique solvability of the problem is established.Investigation of mathematical models of one-phase Stefan problems with unknown nonlinear coefficientshttps://zbmath.org/1463.353192021-07-26T21:45:41.944397Z"Gol'dman, Nataliya L'vovna"https://zbmath.org/authors/?q=ai:goldman.nataliya-lvovnaSummary: One-phase models of inverse Stefan problems with unknown temperature-dependent convection coefficients are considered. The final observation is considered as an additional information on the solution of the direct Stefan problem. For such inverse problems we justify the corresponding mathematical statements allowing to determine coefficients multiplying the lowest order derivatives in quasilinear parabolic equations in a one-phase domain with an unknown moving boundary. On the basis of the duality principle conditions for the uniqueness of their smooth solution are obtained. The proposed approach allows one to clarity a relationship between the uniqueness property for coefficient inverse Stefan problems and the density property of solutions of the corresponding adjoint problems. It is shown that this density property follows, in turn, from the known inverse uniqueness for linear parabolic equations.Inverse problem for higher order ultraparabolic equation with unknown minor coefficient and right-hand side functionhttps://zbmath.org/1463.353252021-07-26T21:45:41.944397Z"Protsakh, N."https://zbmath.org/authors/?q=ai:protsakh.nataliya-p|protsakh.natalyaSummary: The problem of identifying of the several unknown time dependent parameters in the minor coefficient and in the special type right-hand side function of the semilinear higher order ultraparabolic equation from the additional initial, boundary and integral type overdetermination conditions is considered in this paper. The sufficient conditions of the unique solvability on the interval \([0,T],\) where \(T\) is determined by the coefficients of the equation, are obtained.The wellposedness and energy estimate for wave equations in domains with a space-like boundaryhttps://zbmath.org/1463.353322021-07-26T21:45:41.944397Z"Liu, Lingyang"https://zbmath.org/authors/?q=ai:liu.lingyang"Gao, Hang"https://zbmath.org/authors/?q=ai:gao.hangSummary: This paper is concerned with wave equations defined in domains of \(\mathbb{R}^2\) with an invariable left boundary and a space-like right boundary which means the right endpoint is moving faster than the characteristic. Different from the case where the endpoint moves slower than the characteristic, this problem with ordinary boundary formulations may cause ill-posedness. In this paper, we propose a new kind of boundary condition to make systems well-posed, based on an idea of transposition. The key is to prove wellposedness and a hidden regularity for the corresponding backward system. Moreover, we establish an exponential decay estimate for the energy of homogeneous systems.The problem of determining the matrix kernel of the anisotropic viscoelasticity equations systemhttps://zbmath.org/1463.353372021-07-26T21:45:41.944397Z"Totieva, Zhanna Dmitrievna"https://zbmath.org/authors/?q=ai:totieva.zhanna-dmitrievnaSummary: We consider the problem of determining the matrix kernel \(K(t)=\text{diag}(K_1, K_2, K_3)(t)\), \(t>0\), occurring in the system of integro-differential viscoelasticity equations for anisotropic medium. The direct initial boundary value problem is to determine the displacement vector function \(u(x,t)=(u_1,u_2,u_3)(x,t)\), \(x=(x_1,x_2,x_3) \in R^3\), \(x_3>0\). It is assumed that the coefficients of the system (density and elastic modulus) depend only on the spatial variable \(x_3>0\). The source of perturbation of elastic waves is concentrated on the boundary of \(x_3=0\) and represents the Dirac Delta function (Neumann boundary condition of a special kind). The inverse problem is reduced to the previously studied problems of determining scalar kernels \(K_i(t), i=1,2,3\). As an additional condition, the value of the Fourier transform in \(x_2\) of the function \(u(x,t)\) is given on the surface \(x_3=0\). Theorems of global unique solvability and stability of the solution of the inverse problem are given. The idea of proving global solvability is to apply the contraction mapping principle to a system of nonlinear Volterra integral equations of the second kind in a weighted Banach space.Optimal decay rates of a nonlinear time-delayed viscoelastic wave equation.https://zbmath.org/1463.353492021-07-26T21:45:41.944397Z"Feng, Baowei"https://zbmath.org/authors/?q=ai:feng.baowei"Soufyane, Abdelaziz"https://zbmath.org/authors/?q=ai:soufyane.abdelazizBy use of the Galerkin method and potential well theory the authors prove global existence of weak solution of the initial boundary value problem to the viscoelastic wave equation with time-dependent delay \[u_{tt}(x,t)-\Delta u(x,t)+\int_0^tg(t-s)\Delta u(x,s)\, ds+\mu_1u_t(x,t)+\mu_2u_t(x,t-\tau(t))=|u(x,t)|^{\rho}u(x,t).\] Further, they study the optimal decay rates of energy to the problem and give the assumptions, under which the energy tends to zero as \(t\to\infty.\)Uniqueness of an inverse source non-local problem for fractional order mixed type equationshttps://zbmath.org/1463.353712021-07-26T21:45:41.944397Z"Salakhitdinov, Makhmud S."https://zbmath.org/authors/?q=ai:salakhitdinov.makhmud-s"Karimov, Erkinjon T."https://zbmath.org/authors/?q=ai:karimov.erkinjon-tulkinovichSummary: In the present work, we investigate the uniqueness of a solution to the inverse source problem with non-local conditions for a mixed parabolic-hyperbolic type equation with the Caputo fractional derivative. Solution of the problem we represent as bi-orthogonal series with respect to space variable and will get fractional order differential equations with respect to time-variable. Using boundary and gluing conditions, we deduce system of algebraic equations regarding unknown constants and imposing condition to the determinant of this system, we prove the uniqueness of the considered problem. Moreover, we find some non-trivial solutions to the problem in the case, in which the imposed conditions are not satisfie.Inverse problems for a degenerate mixed parabolic-hyperbolic equation on finding time-depending factors in right hand sideshttps://zbmath.org/1463.353722021-07-26T21:45:41.944397Z"Sidorov, Stanislav Nikolaevich"https://zbmath.org/authors/?q=ai:sidorov.stanislav-nikolaevichSummary: We consider direct and inverse problems on determining time-dependent factors in the right hand sides for a mixed parabolic-hyperbolic equation with a degenerate hyperbolic part in a rectangular area. As a preliminary, we study a direct initial boundary problem for this equation. By the method of spectral analysis we establish the uniqueness criterion for the solution and the solution is constructed as a sum over the system of the eigenfunctions of the corresponding one-dimensional Sturm-Liouville spectral problem. In justifying the convergence of the series, the problem of small denominators arises. Because of this, we prove the estimates for the distance from the zero to the small denominators with a corresponding asymptotics. These estimates allow us to justify the convergence of the constructed series in the class of regular solutions of this equation. On the base of the solution to the direct problem, we formulate and study three inverse problems on finding time-dependent factors in the right hand side only by the parabolic or hyperbolic part of the equation, and also as the factors in the both sides of the equation are unknown. Using the formula of solution to the direct initial boundary problem, the solution of inverse problems is equivalently reduced to the solvability of loaded integral equations. By means of the theory of integral equations, the corresponding theorems of uniqueness and the existence of solutions of the stated inverse problems are proved. At that, the solutions of inverse problems are constructed explicitly, as sums of orthogonal series.On Fredholm property of boundary problemhttps://zbmath.org/1463.353742021-07-26T21:45:41.944397Z"Aliev, N. A."https://zbmath.org/authors/?q=ai:aliev.nihan-a"Bagirov, G. A."https://zbmath.org/authors/?q=ai:bagirov.g-a"Garaeva, N. I."https://zbmath.org/authors/?q=ai:garaeva.n-iSummary: In this paper we consider the boundary problem for a third-order equation of composite type, i.e. in the considered domain the equation has both real and complex characteristics. The presence of real characteristics is the cause of incorrectness of the corresponding boundary problem.Solitons and the generalized Cole-Hopf substitutionshttps://zbmath.org/1463.354232021-07-26T21:45:41.944397Z"Byzykchi, Aleksandr Nikolaevich"https://zbmath.org/authors/?q=ai:byzykchi.aleksandr-nikolaevich"Zhuravlev, Viktor Mikhaĭlovich"https://zbmath.org/authors/?q=ai:zhuravlev.viktor-mikhailovichSummary: This paper examines the relationship between the method of the inverse problem and the method of generalized Cole-Hopf substitutions. The relationship between these methods is established by comparing the method of Darboux transformations and the method of Cole-Hopf substitutions. Concrete examples of using such a relationship are given. Some new examples of the integrable equations are considered.New exact solutions to the fractional Burgers-KdV equationhttps://zbmath.org/1463.354362021-07-26T21:45:41.944397Z"Chen, Zhaohui"https://zbmath.org/authors/?q=ai:chen.zhaohui"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.5"Deng, Shengzhong"https://zbmath.org/authors/?q=ai:deng.shengzhongSummary: The fractional Burgers-KdV equation is explicitly discussed in this paper by the expansion Riccati method. Firstly, the fractional Burgers-KdV equation is converted into an non-fractional ordinary differential equation by fractional complex transformation. Then some exact solutions of the fractional Burgers-KdV equation are obtained through the expansion Riccati method. Also, some specific values are assigned to the variables by choosing one of the exact solutions. Furthermore, some following graphs are drawn with different values of \(\alpha\). The results show that the expansion Riccati method plays an effective role in solving the nonlinear fractional Burgers-KdV equations due to its advantages of simplicity and convenience.New exact solutions to the nonlinear Zoomeron equation with local conformable time-fractional derivativehttps://zbmath.org/1463.354402021-07-26T21:45:41.944397Z"He, Chunlei"https://zbmath.org/authors/?q=ai:he.chunlei"Huang, Shoujun"https://zbmath.org/authors/?q=ai:huang.shoujun"Xia, Chunping"https://zbmath.org/authors/?q=ai:xia.chunping"Xu, Yangyang"https://zbmath.org/authors/?q=ai:xu.yangyangSummary: In this paper, we are concerned with the nonlinear Zoomeron equation with local conformable time-fractional derivative. The concept of local conformable fractional derivative was newly proposed by other researchers. The bifurcation and phase portrait analysis of traveling wave solutions of the nonlinear Zoomeron equation are investigated. Moreover, by utilizing the exp\( (-\phi (\varepsilon))\)-expansion method and the first integral method, we obtain various exact analytical traveling wave solutions to the Zoomeron equation such as solitary wave, breaking wave and periodic wave.Norm inflation for equations of KdV type with fractional dispersion.https://zbmath.org/1463.354412021-07-26T21:45:41.944397Z"Hur, Vera Mikyoung"https://zbmath.org/authors/?q=ai:hur.vera-mikyoungSummary: We demonstrate norm inflation for nonlinear nonlocal equations, which extend the Korteweg-de Vries equation to permit fractional dispersion, in the periodic and non-periodic settings. That is, an initial datum is smooth and arbitrarily small in a Sobolev space but the solution becomes arbitrarily large in the Sobolev space after an arbitrarily short time.New exact solutions of the space-time fractional mKdV-ZK equation by the first integral methodhttps://zbmath.org/1463.354492021-07-26T21:45:41.944397Z"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.5"Chen, Zhaohui"https://zbmath.org/authors/?q=ai:chen.zhaohuiSummary: To provide evidence for dynamic behavior research in physics and better explain some physical phenomena, firstly, space-time and fractional mKdV-ZK equation can be converted into the non-fractional ordinary differential equations by fractional complex transformation. Secondly, the first integral of ordinary differential equations is sought by using the division theorem. Finally, many exact solutions of the original equation are solved by using the first integral. Many exact solutions of the space-time fractional mKdV-ZK equation are obtained. The results show that the first integral method has great effect on solving the space-time fractional mKdV-ZK equation, and it has the advantages of simplicity and convenience.Existence of ground state solutions of Nehari-Pohozaev type for fractional Schrödinger-Poisson systems with a general potentialhttps://zbmath.org/1463.354642021-07-26T21:45:41.944397Z"Liu, Ke"https://zbmath.org/authors/?q=ai:liu.ke"Du, Xinsheng"https://zbmath.org/authors/?q=ai:du.xinshengSummary: In this paper, we consider the existence of ground state solutions to the following fractional Schrödinger-Poisson systems with a general potential \[\begin{cases}(-\Delta)^su + V(x)u + \varphi u = f(u), & \mathrm{in} \mathbb{R}^3,\\(-\Delta)^t\varphi = {u^2},& \mathrm{in} \mathbb{R}^3,\end{cases}\] where \((-\Delta)^s\) and \((-\Delta)^t\) denote the fractional Laplacian, \(0 < s \le t < 1\) and \(2s + 2t > 3\), the potential \(V(x)\) is weakly differentiable and \(f \in C(\mathbb{R, R})\). Under some assumptions on potential \(V(x)\) and \(f(u)\), a nontrivial ground state solution of Nehari-Pohozaev type \((u,\varphi)\) is established through using a subtle approach and global compactness Lemma.Existence of normalized solutions for fractional Schrödinger-Poisson systemhttps://zbmath.org/1463.354692021-07-26T21:45:41.944397Z"Sun, Xia"https://zbmath.org/authors/?q=ai:sun.xia"Teng, Kaimin"https://zbmath.org/authors/?q=ai:teng.kaiminSummary: This paper is devoted to study the existence of the normalized solutions for fractional Schrödinger-Poisson system. We first transform the normalized solutions into minimizers of the constraint minimization problem under the variational framework, then the existence and nonexistence of the minimizers are proved by the concentration-compactness principle.Existence of sign-changing solutions for fractional Schrödinger equationshttps://zbmath.org/1463.354702021-07-26T21:45:41.944397Z"Wang, Jianing"https://zbmath.org/authors/?q=ai:wang.jianing"Liu, Jian"https://zbmath.org/authors/?q=ai:liu.jian.3|liu.jian|liu.jian.4|liu.jian.2|liu.jian.6|liu.jian.1|liu.jian.5Summary: In this paper, we investigate the existence of sign-changing solutions for the fractional Schrödinger equation \[(-\Delta)^\alpha u + V(x)u = f(u),\; x \in {R^3}\] where \(\alpha \in (0, 1)\), \(V(x)\) is smooth function, \(f \in {C^1}(R, R)\). We obtain the existence of the variational solutions of the fractional Schrödinger equation by using the variational method and the approximation principle.Existence, uniqueness and regularity property of solutions to Fokker-Planck type equationshttps://zbmath.org/1463.354772021-07-26T21:45:41.944397Z"Tian, Rongrong"https://zbmath.org/authors/?q=ai:tian.rongrongSummary: In this paper, we study the existence, uniqueness and \(H^k\)-regularity property of solutions to a class of Fokker-Planck type equations with Sobolev coefficients and \(L^2\) initial conditions. Meantime, as a slight extension of \textit{C. Le Bris}' and \textit{P.-L. Lions}' work [``Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients'', Rapport de recherche du CERMICS 349, 52 p. (2007), \url{https://cermics.enpc.fr/cermics-rapports-recherche/2007/CERMICS-2007/CERMICS-349.pdf}; see here [Zbl 1157.35301]], we obtain the existence and uniqueness of weak \(L^p\)-solutions for Fokker-Planck type equations with \(L^p\) initial values.A fractional analysis of Noyes-Field model for the nonlinear Belousov-Zhabotinsky reactionhttps://zbmath.org/1463.354802021-07-26T21:45:41.944397Z"Akinyemi, Lanre"https://zbmath.org/authors/?q=ai:akinyemi.lanreSummary: Nonlinear phenomena play an essential role in various field of natural sciences and engineering. In particular, the nonlinear chemical reactions are observed in various domains, as, for instance, in biological and chemical physics. For this reason, it is important to investigate the solution to this nonlinear phenomenon. This article investigates numerical solutions to a nonlinear oscillatory system called the Belousov-Zhabotinsky with Caputo fractional-time derivative. The simplified Noyes-Field fractional model reads
\[\begin{aligned}
\mathcal{D}_t^\mu p &= \xi_1p_{xx}+\beta \delta w+p-p^2 -\delta pw,\quad 0<\mu \leq 1,\\
\mathcal{D}_t^\mu w &= \xi_2w_{xx}+ \gamma w -\lambda pw,
\end{aligned}\]
where \(\xi_1\) and \(\xi_2\) are the diffusing constants for the concentration \(p\) and \(w\) respectively, \(\gamma\) and \(\beta\) are given constants, \(\lambda \neq 1\) and \(\delta\) are positive parameters. The two iterative techniques used in this work are the fractional reduced differential transform method and q-homotopy analysis transform method. The outcomes using these two methods reveal an efficient numerical solution with high accuracy and minimal computations. Furthermore, to better understand the effect of the fractional order, we present the solution profiles which demonstrate the behavior of the obtained results.On convergence of collocation method for a class of integro-partial differential equationshttps://zbmath.org/1463.354812021-07-26T21:45:41.944397Z"Aliev, R. M."https://zbmath.org/authors/?q=ai:aliev.rafig-m"Gasymova, S. G."https://zbmath.org/authors/?q=ai:gasymova.sara-g"Khalilov, È. G."https://zbmath.org/authors/?q=ai:khalilov.elnur-gasan-oglySummary: In this paper, considered Dirichlet problem for a class of integro-partial differential equations in the rectangle \(D(a\le x\le b;\ c\le y\le d)\). For this problem investigated the convergence of the collocation method.Regularized asymptotics of solutions to integro-differential partial differential equations with rapidly varying kernelshttps://zbmath.org/1463.354822021-07-26T21:45:41.944397Z"Bobodzhanov, Abdukhafiz Abdurasulovich"https://zbmath.org/authors/?q=ai:bobodzhanov.abdukhafiz-a"Safonov, Valeriĭ Fedorovich"https://zbmath.org/authors/?q=ai:safonov.valerii-fSummary: We generalize the Lomov's regularization method for partial differential equations with integral operators, whose kernel contains a rapidly varying exponential factor. We study the case when the upper limit of the integral operator coincides with the differentiation variable. For such problems we develop an algorithm for constructing regularized asymptotics. In contrast to the work by Imanaliev M. I., where for analogous problems with slowly varying kernel only the passage to the limit studied as the small parameter tended to zero, here we construct an asymptotic solution of any order (with respect to the parameter). We note that the Lomov's regularization method was used mainly for ordinary singularly perturbed integro-differential equations (see detailed bibliography at the end of the article). In one of the authors' papers the case of a partial differential equation with slowly varying kernel was considered. The development of this method for partial differential equations with rapidly changing kernel was not made before. The type of the upper limit of an integral operator in such equations generates two fundamentally different situations. The most difficult situation is when the upper limit of the integration operator does not coincide with the differentiation variable. As studies have shown, in this case, the integral operator can have characteristic values, and for the construction of the asymptotics, more strict conditions on the initial data of the problem are required. It is clear that these difficulties also arise in the study of an integro-differential system with a rapidly changing kernels, therefore in this paper the case of the dependence of the upper limit of an integral operator on the variable \(x\) is deliberately avoided. In addition, it is assumed that the same regularity is observed in a rapidly decreasing kernel exponent integral operator. Any deviations from these (seemingly insignificant) limitations greatly complicate the problem from the point of view of constructing its asymptotic solution. We expect that in our further works in this direction we will succeed to weak these restrictions.Investigation of solutions to the fractional integro-differential equations of Bratu-type using Legendre wavelets methodhttps://zbmath.org/1463.354832021-07-26T21:45:41.944397Z"Felahat, M."https://zbmath.org/authors/?q=ai:felahat.m"Kadkhoda, N."https://zbmath.org/authors/?q=ai:kadkhoda.nematollah"Fečkan, M."https://zbmath.org/authors/?q=ai:feckan.michalSummary: In this study, Legendre wavelets has been applied to solve the fractional integro-differential equations of Bratu-type. In this method, Legendre wavelet operational matrix and numerical integration techniques have been used. Finally, this method is used for solving some examples to illustrate the simplicity of the suggested method.Analysis of scaling transformation group and self-similar solutions of a class of population balance equationhttps://zbmath.org/1463.354842021-07-26T21:45:41.944397Z"Lin, Fubiao"https://zbmath.org/authors/?q=ai:lin.fubiao"Zhang, Qianhong"https://zbmath.org/authors/?q=ai:zhang.qianhongSummary: Firstly, a class of population balance equation involving growth process, aggregation process and breakage process was investigated and analyzed. Secondly, the method of scaling transformation group was applied to this population balance equation. Thirdly, the admitted scaling group, self-similar solutions and reduced equations of this population balance equation were obtained. Finally, these obtained results showed that the method of scaling transformation group could be used not only for partial differential equations but also for integro-differential equations.Analysis for a delayed viral infection model with diffusion and general incidence functionhttps://zbmath.org/1463.354852021-07-26T21:45:41.944397Z"An, Zhaofeng"https://zbmath.org/authors/?q=ai:an.zhaofeng"Zhang, Suxia"https://zbmath.org/authors/?q=ai:zhang.suxiaSummary: In this paper, we propose a virus compartmental model with general incidence function that incorporates diffusion and time delays. A detailed analysis of dynamic behaviors is conducted, including defining the basic reproductive number \({\mathcal{R}_0}\), discussing the existence of equilibria and proving the stability by constructing Lyapunov functional. The results show that if \({\mathcal{R}_0} < 1\), then the infection-free equilibrium is globally asymptotically stable. Conversely, if \({\mathcal{R}_0} > 1\), the infection-free equilibrium is unstable and the endemic equilibrium is globally asymptotically stable under certain conditions. Meanwhile, taking Beddington-DeAngelis incidence function as a typical example, numerical simulations are provided to illustrate the main theoretical results.A survey on new methods for partial functional differential equations and applicationshttps://zbmath.org/1463.354862021-07-26T21:45:41.944397Z"Ezzinbi, Khalil"https://zbmath.org/authors/?q=ai:ezzinbi.khalilSummary: This work is a survey of many papers dealing with new methods to study partial functional differential equations. We propose a new reduction method of the complexity of partial functional differential equations and its applications. Since, any partial functional differential equation is well-posed in a infinite dimensional space, this presents many difficulties to study the qualitative analysis of the solutions. Here, we propose to reduce the dimension from infinite to finite. We suppose that the undelayed part is not necessarily densely defined and satisfies the Hille-Yosida condition. The delayed part is continuous. We prove the dynamic of solutions are obtained through an ordinary differential equations that is well-posed in a finite dimensional space. The powerty of this results is used to show the existence of almost automorphic solutions for partial functional differential equations. For illustration, we provide an application to the Lotka-Volterra model with diffusion and delay.A hierarchical size-structured population model with delayhttps://zbmath.org/1463.354872021-07-26T21:45:41.944397Z"Han, Mengjie"https://zbmath.org/authors/?q=ai:han.mengjie"He, Zerong"https://zbmath.org/authors/?q=ai:he.zerong"Zhou, Nan"https://zbmath.org/authors/?q=ai:zhou.nanSummary: We analyze a class of hierarchical size-structured population model, which is of nonlinear partial functional integro-differential equation with a delay incorporated into the size boundary condition. By means of frozen coefficients, fixed point principle and a partition of the horizon, the well-posedness of the model solutions is established.Controllability results for second order impulsive stochastic functional differential systems with state-dependent delayhttps://zbmath.org/1463.354882021-07-26T21:45:41.944397Z"Parthasarathy, C."https://zbmath.org/authors/?q=ai:parthasarathy.chinnasamy"Mallika Arjunan, M."https://zbmath.org/authors/?q=ai:mallika-arjunan.mSummary: In this paper, we study the controllability results of second-order impulsive stochastic differential and neutral differential systems with state-dependent delay. Sufficient conditions for controllability of a class of second-order stochastic differential systems are formulated then the results are obtained by using the theory of strongly continuous cosine families and Sadovskii fixed point theorem. An example is provided to illustrate the theory.Homotopy perturbation method combined with Z transform to solve some nonlinear fractional differential equationshttps://zbmath.org/1463.354892021-07-26T21:45:41.944397Z"Riabi, Lakhdar"https://zbmath.org/authors/?q=ai:riabi.lakhdar"Belghaba, Kacem"https://zbmath.org/authors/?q=ai:belghaba.kacem"Hamdi Cherif, Mountassir"https://zbmath.org/authors/?q=ai:hamdi-cherif.mountassir"Ziane, Djelloul"https://zbmath.org/authors/?q=ai:ziane.djelloulSummary: The idea proposed in this work is to extend the Z transform method to resolve the nonlinear fractional partial differential equations by combining them with the so-called homotopy perturbation method (HPM). We apply this technique to solve some nonlinear fractional equations as: nonlinear time-fractional Fokker-Planck equation, the cubic nonlinear time-fractional Schrödinger equation and the nonlinear time-fractional KdV equation. The fractional derivative is described in the Caputo sense. The results show that this is the appropriate method to solve some models of nonlinear partial differential equations with time-fractional derivative.On qualitative behavior of multiple solutions of quasilinear parabolic functional equationshttps://zbmath.org/1463.354902021-07-26T21:45:41.944397Z"Simon, László"https://zbmath.org/authors/?q=ai:simon.laszloSummary: We shall consider weak solutions of initial-boundary value problems for semilinear and nonlinear parabolic differential equations for \(t\in (0,\infty)\) with certain nonlocal terms. We shall prove theorems on the number of solutions and certain qualitative properties of the solutions. These statements are based on arguments for fixed points of some real functions and operators, respectively, and theorems on the existence, uniqueness and qualitative properties of the solutions of partial differential equations (without functional terms).Controlled forced fractional vibrating systemhttps://zbmath.org/1463.354912021-07-26T21:45:41.944397Z"Agila, Adel"https://zbmath.org/authors/?q=ai:agila.adel"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-iSummary: The reliability of dynamic systems is enhanced by vibration control. Many types of controllers are used to control the dynamic systems' vibrations. The integer and fractional PID controllers are used to control the fractional and integer dynamic systems. Different techniques are utilized to model the controlled systems. In this study, the discrete integer proportional integral derivative (PID) controller is used to control a forced damped variable-order fractional oscillatory systems. The objectives of this study are the analysis of controlled fractional system responses, and the investigation of controller gains' effects on system response characteristics. The Caputo-Fabrizio fractional derivative is used to model the system fractional dissipating force. The system responses are approximated by numerical and time discretization techniques. In order to verify the feasibility and effectiveness of the introduced methods, the fractional system response and integer system response are compared at fractional order close to one. The controlled responses of the fractional system are obtained for different fractional derivative order values. The results demonstrate same effects of PID gains on the fractional and integer oscillatory system responses' metrics. However, the system responses are varying based on the fractional derivative order values. The study shows that the integer response and the fractional responses have same behaviors and different instantaneous values.Mathematical modeling of salt condition of soils with fractal structurehttps://zbmath.org/1463.354922021-07-26T21:45:41.944397Z"Bedanokova, S. Yu."https://zbmath.org/authors/?q=ai:bedanokova.s-yuSummary: We obtain and study the linear mathematical models salt condition in soils with fractal structure. The models are based on the loaded differential equations of fractal order.Existence and multiplicity of solutions for a fractional \(p\)-Laplacian problem of Kirchhoff type via Krasnoselskii's genus.https://zbmath.org/1463.354932021-07-26T21:45:41.944397Z"Benhamida, Ghania"https://zbmath.org/authors/?q=ai:benhamida.ghania"Moussaoui, Toufik"https://zbmath.org/authors/?q=ai:moussaoui.toufikSummary: We use the genus theory to prove the existence and multiplicity of solutions for the fractional \(p\)-Kirchhoff problem \[ \begin{cases}\displaystyle -\left[ M\left( \int_Q\frac{| u(x)-u(y)|^p}{| x-y|^{N+ps}}dx\, dy\right)\right]^{p-1}(-\Delta)_p^su=\lambda h(x,u)\quad\text{in}\;\Omega,\\ u=0\quad\text{on}\;\mathbb{R}^N\setminus\Omega,\end{cases} \] where \(\Omega\) is an open bounded smooth domain of \(\mathbb{R}^N\), \(p>1\), \(N>ps\) with \(s\in(0,1)\) fixed, \(Q=\mathbb{R}^{2N}\setminus(C\Omega\times C\Omega)\), \(\lambda >0\) is a numerical parameter, \(M\) and \(h\) are continuous functions.Solutions of some linear fractional partial differential equations in mathematical physicshttps://zbmath.org/1463.354942021-07-26T21:45:41.944397Z"Dhunde, Ranjit R."https://zbmath.org/authors/?q=ai:dhunde.ranjit-r"Waghmare, G. L."https://zbmath.org/authors/?q=ai:waghmare.g-lSummary: In this article, we use double Laplace transform method to find solution of general linear fractional partial differential equation in terms of Mittag-Leffler function subject to the initial and boundary conditions. The efficiency of the method is illustrated by considering fractional wave and diffusion equations, Klein-Gordon equation, Burger's equation, Fokker-Planck equation, KdV equation, and KdV-Burger's equation of mathematical physics.Generalized heat equation under conform derivativehttps://zbmath.org/1463.354952021-07-26T21:45:41.944397Z"Elomari, M'hamed"https://zbmath.org/authors/?q=ai:elomari.mhamed"Melliani, Said"https://zbmath.org/authors/?q=ai:melliani.said"Saadia Chadli, Lalla"https://zbmath.org/authors/?q=ai:chadli.lalla-saadiaSummary: In the present work, we establish the existence and uniqueness result of the linear heat equation with Conform derivative in Colombeau generalized algebra. We using for the first time the notion of a generalized conformable semigroup and the purpose of introducing Conform derivative is regularizing it in Colombeau.Multiple solutions of fractional Laplacian equationshttps://zbmath.org/1463.354962021-07-26T21:45:41.944397Z"Fu, Yongqiang"https://zbmath.org/authors/?q=ai:fu.yongqiangSummary: The existence of three nontrivial solutions for fractional Laplace equations with super-linear growth at infinity and saddle structure near zero is established. The approach is based on a combination of bifurcation theory and minimax methods.Group classification and symmetry reduction of three-dimensional nonlinear anomalous diffusion equationhttps://zbmath.org/1463.354972021-07-26T21:45:41.944397Z"Gazizov, Rafail Kavyevich"https://zbmath.org/authors/?q=ai:gazizov.rafail-kavevich"Kasatkin, Alekseĭ Aleksandrovich"https://zbmath.org/authors/?q=ai:kasatkin.aleksei-aleksandrovich"Lukashchuk, Stanislav Yur'evich"https://zbmath.org/authors/?q=ai:lukashchuk.stanislav-yurevichSummary: The work is devoted to studying symmetry properties of a nonlinear anomalous diffusion equation involving a Riemann-Liouville fractional derivative with respect to the time. We resolve a problem on group classification with respect to the diffusion coefficient treated as a function of the unknown. We show that for an arbitrary function, the equation admits a seven-dimensional Lie algebra of infinitesimal operators corresponding to the groups of translations, rotations and dilations. In contrast to the symmetries of the equations with integer order derivatives, the translation in time is not admitted. Moreover, the coefficients of the group of dilations are different. If the coefficient is power, the admissible algebra is extended to a eight-dimensional one by an additional operator generating the group of dilatations. For two specific values of the exponent in the power, the algebra can be further extended to a nine-dimensional one or to a eleven-dimensional one and at that, additional admissible operators correspond to various projective transformations. For the obtained Lie algebras of symmetries with dimensions from seven to nine, we construct optimal systems of subalgebras and provide ansatzes for corresponding invariant solutions of various ranks. We also provide general forms of invariant solutions convenient for the symmetry reduction as the fractional Riemann-Liouville derivative is present. We make a symmetry reduction on subalgebras allowing one to find invariant solutions of rank one. We provide corresponding reduced ordinary fractional differential equations.On the solution of fractional Burgers' equation and its optimal control problemhttps://zbmath.org/1463.354982021-07-26T21:45:41.944397Z"Georgiev, Svetlin G."https://zbmath.org/authors/?q=ai:georgiev.svetlin-georgiev"Mohammadizadeh, Fatemeh"https://zbmath.org/authors/?q=ai:mohammadizadeh.fatemeh"Tehrani, Hojjat A."https://zbmath.org/authors/?q=ai:tehrani.hojjat-ahsani"Noori, Skandari M. H."https://zbmath.org/authors/?q=ai:noori.skandari-m-hSummary: The main aim of this study is to prove that a class of fractional Burgers' equations has a unique solution under some special conditions. Then it is demonstrated that an optimal control problem for this class of fractional Burgers' equations has at least one optimal solution.Fractional eigenvalue problems on \(\mathbb{R}^N\)https://zbmath.org/1463.354992021-07-26T21:45:41.944397Z"Grecu, Andrei"https://zbmath.org/authors/?q=ai:grecu.andreiSummary: Let \(N\geq 2\) be an integer. For each real number \(s\in(0,1)\) we denote by \((-\Delta)^s\) the corresponding fractional Laplace operator. First, we investigate the eigenvalue problem \((-\Delta)^s u=\lambda V(x)u\) on \(\mathbb{R}^N\), where \(V:\mathbb{R}^N\rightarrow\mathbb{R}\) is a given function. Under suitable conditions imposed on \(V\) we show the existence of an unbounded, increasing sequence of positive eigenvalues. Next, we perturb the above eigenvalue problem with a fractional \((t,p)\)-Laplace operator, when \(t\in(0,1)\) and \(p\in (1,\infty)\) are such that \(t<s\) and \(s-N/2=t-N/p\). We show that when the function \(V\) is nonnegative on \(\mathbb{R}^N\), the set of eigenvalues of the perturbed eigenvalue problem is exactly the unbounded interval \((\lambda_1,\infty)\), where \(\lambda_1\) stands for the first eigenvalue of the initial eigenvalue problem.Critical well-posedness and scattering results for fractional Hartree-type equations.https://zbmath.org/1463.355002021-07-26T21:45:41.944397Z"Herr, Sebastian"https://zbmath.org/authors/?q=ai:herr.sebastian"Yang, Changhun"https://zbmath.org/authors/?q=ai:yang.changhunSummary: Scattering for the mass-critical fractional Schrödinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is established. For this, we prove a bilinear estimate for free solutions and extend it to perturbations of bounded quadratic variation. This result is shown to be sharp by proving the discontinuity of the flow map in the super-critical range.Analytical investigation for modified Riemann-Liouville fractional equal-width equation types based on \((G'/G)\)-expansion techniquehttps://zbmath.org/1463.355012021-07-26T21:45:41.944397Z"Karaman, Bahar"https://zbmath.org/authors/?q=ai:karaman.baharSummary: The present investigation studies exact solutions of modified Riemann-Liouville fractional Equal-Width (MRLFEW) equation types with the help of the \((G'/G)\)-expansion method. Firstly, the MRLFEW equation is converted into an ordinary differential equation via fractional complex transform. Then, the proposed method has applied this equation to construct the exact solutions.Numerical solution of two dimensional reaction-diffusion equation using operational matrix method based on Genocchi polynomial. I: Genocchi polynomial and operational matrixhttps://zbmath.org/1463.355022021-07-26T21:45:41.944397Z"Kumar, Sachin"https://zbmath.org/authors/?q=ai:kumar.sachin"Pandey, Prashant"https://zbmath.org/authors/?q=ai:pandey.prashant-k"Das, Subir"https://zbmath.org/authors/?q=ai:das.subir-k"Craciun, E.-M."https://zbmath.org/authors/?q=ai:craciun.eduard-mariusSummary: In this article, the aim is to find the solution of fractional order non-linear reaction-diffusion equation using collocation method through deriving the operational matrix of fractional derivative. For this purpose the required definitions of fractional order derivatives, Genocchi polynomial and properties of Kronecker product of matrices used for the approximation of arbitrary functions are discussed.Exact solutions for a class of time fractional coupled Boussinesq-Burger equations in the invariant subspacehttps://zbmath.org/1463.355032021-07-26T21:45:41.944397Z"Li, Linfang"https://zbmath.org/authors/?q=ai:li.linfang"Shu, Ji"https://zbmath.org/authors/?q=ai:shu.ji"Wen, Huixia"https://zbmath.org/authors/?q=ai:wen.huixiaSummary: In this paper, the invariant subspace method is applied to study fractional coupled nonlinear partial differential equations and construct exact solutions for the time fractional Boussinesq-Burger equations. In the sense of variable transformations, the invariant subspace of the equations is given by invariant conditions. Then the equations are reduced to ordinary differential equations in the invariant subspace, and their exact solutions are obtained by solving ordinary differential equations.Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a source termhttps://zbmath.org/1463.355042021-07-26T21:45:41.944397Z"Lukashchuk, Stannislav Yur'evich"https://zbmath.org/authors/?q=ai:lukashchuk.stannislav-yurevichSummary: We consider a problem on constructing invariant solutions to a nonlinear fractional differential equations of anomalous diffusion with a source. On the base of an earlier made group classification of the considered equation, for each case in the classification we construct the optimal systems of one-dimensional subalgebras of Lie algebras of infinitesimal operators of the point transformations group admitted by the equation. For each one-dimensional subalgebra of each optimal system we find the corresponding form of the invariant solution and made the symmetry reduction to an ordinary differential equation. We prove that there are three different types of the reduction equations (factor equations): a second order ordinary differential equation integrated by quadratures and two ordinary nonlinear fractional differential equations. For particular cases of the latter we find exact solutions.Group classification, invariant solutions and conservation laws of nonlinear orthotropic two-dimensional filtration equation with the Riemann-Liouville time-fractional derivativehttps://zbmath.org/1463.355052021-07-26T21:45:41.944397Z"Lukashchuk, Veronika Olegovna"https://zbmath.org/authors/?q=ai:lukashchuk.veronika-olegovna"Lukashchuk, Stanislav Yur'evich"https://zbmath.org/authors/?q=ai:lukashchuk.stanislav-yurevichSummary: A nonlinear two-dimensional orthotropic filtration equation with the Riemann-Liouville time-fractional derivative is considered. It is proved that this equation can admits only linear autonomous groups of point transformations. The Lie point symmetry group classification problem for the equation in question is solved with respect to coefficients of piezoconductivity. These coefficients are assumed to be functions of the square of the pressure gradient absolute value. It is proved that if the order of fractional differentiation is less than one then the considered equation with arbitrary coefficients admits a four-parameter group of point transformations in orthotropic case, and a five-parameter group in isotropic case. For the power-law piezoconductivity, the group admitted by the equation is five-parametric in orthotropic case, and six-parametric in isotropic case. Also, a special case of power function of piezoconductivity is determined for which there is an additional extension of admitted groups by the projective transformation. There is no an analogue of this case for the integer-order filtration equation. It is also shown that if the order of fractional differentiation \(\alpha \in (1,2)\) then dimensions of admitted groups are incremented by one for all cases since an additional translation symmetry exists. This symmetry is corresponded to an additional particular solution of the fractional filtration equation under consideration.
Using the group classification results for orthotropic case, the representations of group-invariant solutions are obtained for two-dimensional subalgebras from optimal systems of symmetry subalgebras. Examples of reduced equations obtained by the symmetry reduction technique are given, and some exact solutions of these equations are presented.
It is proved that the considered time-fractional filtration equation is nonlinearly self-adjoint and therefore the corresponding conservation laws can be constructed. The components of obtained conserved vectors are given in an explicit form.Fractional \(\mathrm{exp}(-\phi(\xi))\)-expansion method and its application to space-time nonlinear fractional equationshttps://zbmath.org/1463.355062021-07-26T21:45:41.944397Z"Moussa, Alaeddin Amin"https://zbmath.org/authors/?q=ai:moussa.alaeddin-amin"Alhakim, Abdulaziz"https://zbmath.org/authors/?q=ai:alhakim.abdulazizSummary: In this paper, we mainly suggest a new method that depends on the fractional derivative proposed by Katugampola for solving nonlinear fractional partial differential equations. Using this method, we obtained numerous useful and surprising solutions for the space-time fractional nonlinear Whitham-Broer-Kaup equations and space-time fractional generalized nonlinear Hirota-Satsuma coupled KdV equations. The solutions obtained varied between hyperbolic, trigonometric, and rational functions, and we hope those interested in the real-life applications of the previous two equations will find this approach useful.Exact solutions of the conformable space-time chiral nonlinear Schrödinger's equationshttps://zbmath.org/1463.355072021-07-26T21:45:41.944397Z"Sirisubtawee, Sekson"https://zbmath.org/authors/?q=ai:sirisubtawee.sekson"Khansai, Nattawut"https://zbmath.org/authors/?q=ai:khansai.nattawut"Koonprasert, Sanoe"https://zbmath.org/authors/?q=ai:koonprasert.sanoe"Torvattanabun, Montri"https://zbmath.org/authors/?q=ai:torvattanabun.montriSummary: In this paper, the modified simple equation method is used to obtain exact solutions of the space-time (1+1) and (2+1)-dimensional chiral nonlinear Schrödinger's equations in the sense of the conformable derivative. As a consequence, these obtained solutions with their constraint conditions can be useful to explain some physical phenomena such as dark or singular soliton solutions. Graphical representations of selected solutions are illustrated using a range of fractional orders. The performance of the method is concise, effective and reliable for solving nonlinear partial differential equations (NPDEs) including the NPDEs with conformable derivatives.Existence of radial sign-changing solution for a fractional autonomous Kirchhoff equationhttps://zbmath.org/1463.355082021-07-26T21:45:41.944397Z"Zhang, Dandan"https://zbmath.org/authors/?q=ai:zhang.dandan"Ding, Ling"https://zbmath.org/authors/?q=ai:ding.lingSummary: The existence of sign-changing solution for a fractional autonomous Kirchhoff equation is considered in the whole space. We prove that this equation is equivalent to a fractional autonomous Schrödinger system under appropriate conditions. Then, by using the existence of radial sign-changing solution of the fractional autonomous Schrödinger equation, the existence of solutions for the fractional autonomous Schrödinger system is proved. The existence of radial sign-changing solution for the fractional autonomous Kirchhoff equation is obtained as well.Maximal regularity of parameter dependent differential-operator equations on the halflinehttps://zbmath.org/1463.355092021-07-26T21:45:41.944397Z"Musaev, Hummet K."https://zbmath.org/authors/?q=ai:musaev.hummet-kSummary: This paper focuses on boundary value problems for differential-operator equations in half line. The equations and boundary conditions contain certain small and spectral parameters. The uniform \(L_p\)-separability is obtained. Here the explicit formula for the solution is given and behavior of solution is established with small parameter. It used to obtain singular perturbation result for the convolution differential-operator equation.Regularization method for an ill-posed Cauchy problem of nonlinear elliptic equationhttps://zbmath.org/1463.355102021-07-26T21:45:41.944397Z"Zhang, Hongwu"https://zbmath.org/authors/?q=ai:zhang.hongwu"Zhang, Xiaoju"https://zbmath.org/authors/?q=ai:zhang.xiaojuSummary: This paper considers an ill-posed Cauchy problem of nonlinear elliptic equation. By using a regularization method to overcome the ill-posedness, we obtain the existence, uniqueness, stability and convergence result of the regularization solution. An iterative scheme is constructed to calculate the regularization solution, which is an extension on the related research results of existing literature in the aspect of regularization theory and algorithm for Cauchy problem of elliptic equation.Determination of parameters in telegraph equationhttps://zbmath.org/1463.355112021-07-26T21:45:41.944397Z"Kozhanov, Aleksandr Ivanovich"https://zbmath.org/authors/?q=ai:kozhanov.aleksandr-ivanovich"Safiullova, Regina Rafailovna"https://zbmath.org/authors/?q=ai:safiullova.regina-rafailovnaSummary: We study the solvability of the inverse problems on finding a solution \(u(x,t)\) and an unknown coefficient \(c\) for a telegraph equation
\[u_{tt}-\Delta u +cu=f(x,t).\]
We prove the theorems on the existence of the regular solutions. The feature of the problems is a presence of new overdetermination conditions for the considered class of equations.Heat potentials method in the treatment of one-dimensional free boundary problems applied in cryomedicinehttps://zbmath.org/1463.355122021-07-26T21:45:41.944397Z"Kudayeva, Fatimat K."https://zbmath.org/authors/?q=ai:kudayeva.fatimat-k"Kaigermazov, Arslan A."https://zbmath.org/authors/?q=ai:kaigermazov.arslan-a"Edgulova, Elizaveta K."https://zbmath.org/authors/?q=ai:edgulova.elizaveta-k"Tkhabisimova, Mariya M."https://zbmath.org/authors/?q=ai:tkhabisimova.mariya-m"Bechelova, Aminat R."https://zbmath.org/authors/?q=ai:bechelova.aminat-rSummary: Free boundary problems are considered to be the most difficult and the least researched in the field of mathematical physics. The present article is concerned with the research of the following issue: treatment of one-dimensional free boundary problems. The treated problem contains a nonlinear evolutionary equation, which occurs within the context of mathematical modeling of cryosurgery problems. In the course of the research, an integral expression has been obtained. The obtained integral expression presents a general solution to the non-homogeneous evolutionary equation which contains the functions that represent simple-layer and double-layer heat potential density. In order to determine the free boundary and the density of potential a system of nonlinear, the second kind of Fredholm integral equations was obtained within the framework of the given work. The treated problem has been reduced to the system of integral equations. In order to reduce the problem to the integral equation system, a method of heat potentials has been used. In the obtained system of integral equations instead of \(K(\xi, x, \tau - t)\) in case of Dirichlet or Neumann conditions the corresponding Greens functions \(G(\xi, x, \tau - t)\) or \(N(\xi, x, \tau - t)\) have been applied. Herewith the integral expression contains fewer densities, but the selection of arbitrary functions is reserved. The article contains a number of results in terms of building a mathematical model of cooling and freezing processes of biological tissue, as well as their effective solution development.The free boundary problem modeling the growth of tumor cord with angiogenesishttps://zbmath.org/1463.355132021-07-26T21:45:41.944397Z"Zhang, Qin"https://zbmath.org/authors/?q=ai:zhang.qin"Wang, Zejia"https://zbmath.org/authors/?q=ai:wang.zejiaSummary: In this paper, the free boundary problem modeling the growth of tumor cord with angiogenesis is studied. Assuming that the tumor grows along the outside of blood vessel, the domain of tumor considered has two boundary, the inside boundary is fixed and the outside is free. For this problem, It is proved that there is a radially stationary solution to the problem. If the angiogenesis function \(\alpha (t)\) is uniformly bounded, then the free boundary \(R (t)\) is uniformly bounded. If \(\lim\limits_{t \to \infty} \alpha (t) = 0\), the free boundary will shrink to the inner boundary, that is, the tumor disappears.Survival analysis of a stochastic single-population system with dispersal in driven by the toxin of polluted environmenthttps://zbmath.org/1463.355142021-07-26T21:45:41.944397Z"Dai, Xiangjun"https://zbmath.org/authors/?q=ai:dai.xiangjun"Mao, Zhi"https://zbmath.org/authors/?q=ai:mao.zhiSummary: In this paper, a stochastic single population system with dispersal in driven by the toxin of polluted environment is studied. By constructing a suitable Lyapunov function, the existence and uniqueness of global positive solutions are analyzed, and the stochastic ultimate boundedness of solutions is discussed. Finally, sufficient conditions for the stochastic persistence, persistence in the mean and extinction of population are obtained.Some research on Navier-Stokes-Voight equationhttps://zbmath.org/1463.355152021-07-26T21:45:41.944397Z"Sun, Chengfeng"https://zbmath.org/authors/?q=ai:sun.chengfeng"Liu, Xingchen"https://zbmath.org/authors/?q=ai:liu.xingchen"Jiao, Xiaoyu"https://zbmath.org/authors/?q=ai:jiao.xiaoyuSummary: In this paper, we studied the well-posedness of three dimensional stochastic Navier-Stokes-Voight equation and the existence of invariant measure. While compared with the determination Navier-Stokes-Voight equation, stochastic Navier-Stokes-Voight equation with the term of multiplicative noise was suitable for more general turbulence. It had a broad application in meteorology, geophysics and biology.Approximation of finite dimensional reduction for a class of stochastic partial differential equationshttps://zbmath.org/1463.355162021-07-26T21:45:41.944397Z"Yang, Min"https://zbmath.org/authors/?q=ai:yang.min"Chen, Guanggan"https://zbmath.org/authors/?q=ai:chen.guanggan"Li, Qin"https://zbmath.org/authors/?q=ai:li.qinSummary: This paper studies a class of stochastic partial differential equations with Stratonovich multiplicative noise. The solution of the equation is reduced to a finite dimensional random invariant manifold, and a new class of simplified stochastic evolution equations is used to approximate the original system. It is shown that the finite dimensional reduction of the new approximation system converges to the finite dimensional reduction of the original system.Matrix fractional systemshttps://zbmath.org/1463.370172021-07-26T21:45:41.944397Z"Machado, J. A. Tenreiro"https://zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiroSummary: This paper addresses the matrix representation of dynamical systems in the perspective of fractional calculus. Fractional elements and fractional systems are interpreted under the light of the classical Cole-Cole, Davidson-Cole, and Havriliak-Negami heuristic models. Numerical simulations for an electrical circuit enlighten the results for matrix based models and high fractional orders. The conclusions clarify the distinction between fractional elements and fractional systems.Approximate nonclassical symmetries for the time-fractional KdV equations with the small parameterhttps://zbmath.org/1463.370362021-07-26T21:45:41.944397Z"Najafi, Ramin"https://zbmath.org/authors/?q=ai:najafi.raminSummary: In this paper, the Lie symmetry analysis is presented for the time-fractional KdV equation with the Riemann-Liouville derivative. We introduce a generalized approximate nonclassical method that is applied to differential equations with fractional order. In the sense of this symmetry, the vector fields of fractional KdV equation are obtained. The similarity reduction corresponding to the symmetries of the equation is constructed.Inverse spectral results for AKNS systems with partial information on the potentialshttps://zbmath.org/1463.370442021-07-26T21:45:41.944397Z"Sun, Yixin"https://zbmath.org/authors/?q=ai:sun.yixin"Wei, Guangsheng"https://zbmath.org/authors/?q=ai:wei.guangshengSummary: We consider the AKNS systems on the interval \([0,1]\) and prove that two potentials already known on \([a, 1]\left({a \in \left({0, \frac{1}{2}} \right]} \right)\) and their difference in \({L^p}\) are equal if the number of their common eigenvalues is sufficiently large. The result here is to write down explicitly this number in terms of \(p\) (and \(a\)) showing the role of \(p\).Robustness of random attractors for a stochastic reaction-diffusion systemhttps://zbmath.org/1463.370542021-07-26T21:45:41.944397Z"You, Yuncheng"https://zbmath.org/authors/?q=ai:you.yunchengSummary: Asymptotic pullback dynamics of a typical stochastic reaction-diffusion system, the reversible Schnackenberg equations, with multiplicative white noise is investigated. The robustness of random attractor with respect to the reverse reaction rate as it tends to zero is proved through the uniform pullback absorbing property and the uniform convergence of reversible to non-reversible cocycles. This result means that, even if the reverse reactions would be neglected, the dynamics of this class of stochastic reversible reaction-diffusion systems can still be captured by the random attractor of the non-reversible stochastic raction-diffusion system in a long run.Besov maximal regularity for a class of degenerate integro-differential equations with infinite delay in Banach spaceshttps://zbmath.org/1463.450482021-07-26T21:45:41.944397Z"Aparicio, Rafael"https://zbmath.org/authors/?q=ai:aparicio.rafael"Keyantuo, Valentin"https://zbmath.org/authors/?q=ai:keyantuo.valentinThe authors consider the equation
\[
(Mu')'(t) -\Lambda u'(t)- \frac d{dt} \int_{-\infty}^tc(t-s)u(s)\, ds \\
= \gamma u(t) + Au(t) + \int_{-\infty}^t b(t-s)Bu(s)\, ds + f(t),
\]
where \(A\), \(B\), \(\Lambda\), and \(M\) are closed linear operators in a Banach space \(X\) satisfying \(D(A)\cap D(B) \subset D(\Lambda)\cap D(M)\), \(b\), \(c\in L^1(\mathbb R_+)\), \(f\) is an \(X\)-valued function defined on \(\mathbb R\), and \(\gamma\) is a constant. They study the question when the equation is well-posed in the Besov space \(B^{s}_{pq}(\mathbb R)\), i.e., when there for each \(f\in B^{s}_{pq}(\mathbb R)\) is a unique strong solution \(u\in B^{s}_{pq}(\mathbb R)\) to the equation that depends continuously on \(f\). In most of the results an important condition is that \(\{BN(\xi)\,:\, \xi\in \mathbb R\}\), \(\{\xi N(\xi)\,:\, \xi\in \mathbb R\}\), \(\{\xi \Lambda N(\xi)\,:\, \xi\in \mathbb R\}\), and \(\{\xi^2MN(\xi)\,:\, \xi\in \mathbb R\}\) are Fourier multipliers where \(N(\xi)= [ \xi^2M+A + L_b(\xi) B + i\xi \Lambda + i\xi L_c(\xi)I +
\gamma I]^{-1}\) and \(L_b(\xi)=\int_0^\infty e^{-i\xi t}b(t)\,dt\). In some cases results on maximal regularity are obtained. A number of examples are studied as well.Optimal control problem and maximum principle for fractional order cooperative systems.https://zbmath.org/1463.490052021-07-26T21:45:41.944397Z"Bahaa, G. M."https://zbmath.org/authors/?q=ai:bahaa.gaber-mohamedSummary: In this paper, by using the classical control theory, the optimal control problem for fractional order cooperative system governed by Schrödinger operator is considered. The fractional time derivative is considered in a Riemann-Liouville and Caputo senses. The maximum principle for this system is discussed. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.Asymptotics of the mild solution of a wave equation in three-dimensional space driven by a general stochastic measurehttps://zbmath.org/1463.600842021-07-26T21:45:41.944397Z"Bodnarchuk, I. M."https://zbmath.org/authors/?q=ai:bodnarchuk.i-mSummary: We study the Cauchy problem for a wave equation in three-dimensional space driven by a general stochastic measure. Under some assumptions, we prove that the mild solution tends to zero almost surely as the absolute value of the spatial variable tends to infinity.Exact solutions for the Wick-type stochastic time-fractional KdV equationshttps://zbmath.org/1463.600852021-07-26T21:45:41.944397Z"Ghany, Hossam A."https://zbmath.org/authors/?q=ai:ghany.hossam-a"Hyder, Abd-Allah"https://zbmath.org/authors/?q=ai:hyder.abd-allahSummary: Our aim in this paper is to explore white noise functional solutions for the variable coefficients Wick-type stochastic time-fractional KdV equations. Using the modified fractional sub-equation method, we can find out new exact solutions for the time-fractional KdV equations. Subsequently, the Hermite transform and the inverse Hermite transform are employed to find white noise functional solutions for the variable coefficients Wick-type stochastic time-fractional KdV equations.On regularized Hermitian splitting iteration methods for solving discretized almost-isotropic spatial fractional diffusion equations.https://zbmath.org/1463.650392021-07-26T21:45:41.944397Z"Bai, Zhong-Zhi"https://zbmath.org/authors/?q=ai:bai.zhongzhi"Lu, Kang-Ya"https://zbmath.org/authors/?q=ai:lu.kang-yaThe paper is concerned with numerical solutions of one-dimensional almost-isotropic spatial fractional diffusion equations, particularly focused on the situation that the diffusion coefficients are almost equal. The authors propose a class of regularized Hermitian splitting (RHS) iteration methods for solving the associated discrete linear system, and under mild conditions, they prove the asymptotic convergence property and derive a tight upper bound on the contraction factor in the Euclidean norm. Numerical experiments and comparisons with other methods are presented.Diagonal and Toeplitz splitting iteration methods for diagonal-plus-Toeplitz linear systems from spatial fractional diffusion equations.https://zbmath.org/1463.650402021-07-26T21:45:41.944397Z"Bai, Zhong-Zhi"https://zbmath.org/authors/?q=ai:bai.zhongzhi"Lu, Kang-Ya"https://zbmath.org/authors/?q=ai:lu.kang-ya"Pan, Jian-Yu"https://zbmath.org/authors/?q=ai:pan.jianyuSummary: The finite difference discretization of the spatial fractional diffusion equations gives discretized linear systems whose coefficient matrices have a diagonal-plus-Toeplitz structure. For solving these diagonal-plus-Toeplitz linear systems, we construct a class of diagonal and Toeplitz splitting iteration methods and establish its unconditional convergence theory. In particular, we derive a sharp upper bound about its asymptotic convergence rate and deduct the optimal value of its iteration parameter. The diagonal and Toeplitz splitting iteration method naturally leads to a diagonal and circulant splitting preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1, especially when the discretization step-size \(h\) is small. Numerical results exhibit that the diagonal and circulant splitting preconditioner can significantly improve the convergence properties of GMRES and BiCGSTAB, and these preconditioned Krylov subspace iteration methods outperform the conjugate gradient method preconditioned by the approximate inverse circulant-plus-diagonal preconditioner proposed recently by \textit{M. K. Ng} and \textit{J.-Y. Pan} [SIAM J. Sci. Comput. 32, No. 3, 1442--1464 (2010; Zbl 1222.65028)]. Moreover, unlike this preconditioned conjugate gradient method, the preconditioned GMRES and BiCGSTAB methods show \(h\)-independent convergence behavior even for the spatial fractional diffusion equations of discontinuous or big-jump coefficients.Fast evaluation of linear combinations of Caputo fractional derivatives and its applications to multi-term time-fractional sub-diffusion equationshttps://zbmath.org/1463.652212021-07-26T21:45:41.944397Z"Gao, Guanghua"https://zbmath.org/authors/?q=ai:gao.guanghua"Yang, Qian"https://zbmath.org/authors/?q=ai:yang.qianSummary: In the present work, linear combinations of Caputo fractional derivatives are fast evaluated based on the efficient sum-of-exponentials (SOE) approximation for kernels in Caputo fractional derivatives with an absolute error \(\epsilon\), which is a further work of the existing results in literatures. Both the storage needs and computational amount are significantly reduced compared with the direct algorithm. Applications of the proposed fast algorithm are illustrated by solving a second-order multi-term time-fractional sub-diffusion problem. The unconditional stability and convergence of the fast difference scheme are proved. The CPU time is largely reduced while the accuracy is kept, especially for the cases of large temporal level, which is displayed by numerical experiments.The implicit midpoint method for Riesz tempered fractional advection-diffusion equationhttps://zbmath.org/1463.652222021-07-26T21:45:41.944397Z"Guan, Wenhui"https://zbmath.org/authors/?q=ai:guan.wenhui"Cao, Xuenian"https://zbmath.org/authors/?q=ai:cao.xuenianSummary: In this paper, an implicit midpoint method is applied to discretize the first order time partial derivative, the modified second-order Lubich tempered difference operator is used to approximate Riesz space tempered fractional partial derivative, and the central difference formula is utilized to discrete the advection term, a numerical scheme is constructed for solving Riesz tempered fractional advection-diffusion equation. The stability and convergence of the numerical scheme is established, and the convergence order of numerical scheme can reach two order accuracy on temporal and spatial directions respectively. The numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the schemes.High order finite difference/spectral methods to a water wave model with nonlocal viscosityhttps://zbmath.org/1463.652232021-07-26T21:45:41.944397Z"Hasan, Mohammad Tanzil"https://zbmath.org/authors/?q=ai:hasan.mohammad-tanzil"Xu, Chuanju"https://zbmath.org/authors/?q=ai:xu.chuanjuSummary: In this paper, an efficient numerical scheme is proposed for solving the water wave model with nonlocal viscous term that describes the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions is investigated.Computational study for a class of time-dependent singularly perturbed parabolic partial differential equation through tension splinehttps://zbmath.org/1463.652272021-07-26T21:45:41.944397Z"Kumar, P. Murali Mohan"https://zbmath.org/authors/?q=ai:murali-mohan-kumar.p"Ravi Kanth, A. S. V."https://zbmath.org/authors/?q=ai:ravi-kanth.a-s-vSummary: This article contributes a numerical technique for a class of singularly perturbed time delayed parabolic partial differential equation. A priori results of maximum principle, stability and bounds are discussed. The continuous problem is semi-discretized by the Crank-Nicolson based scheme in the temporal direction and then discretized by the tension spline scheme on non-uniform Shishkin mesh. Error estimation for the discretized problem is derived. To validate the theoretical findings, the numerical outcomes for linear and nonlinear problems are tested.Error analysis of compact implicit-explicit BDF method for nonlinear partial integral differential equationshttps://zbmath.org/1463.652282021-07-26T21:45:41.944397Z"Lan, Haifeng"https://zbmath.org/authors/?q=ai:lan.haifeng"Xiao, Feiyan"https://zbmath.org/authors/?q=ai:xiao.feiyan"Zhang, Gengen"https://zbmath.org/authors/?q=ai:zhang.gengen"Zhu, Rui"https://zbmath.org/authors/?q=ai:zhu.ruiSummary: In this paper, the compact implicit-explicit BDF method is proposed to solve nonlinear partial integral differential equations, i.e. the equation is discretized by the implicit-explicit BDF method in time and compact finite difference approximations in space. Then, the global convergence of the scheme is proved rigorously with convergence order \(O({\tau^2} + {h^4})\). Finally, numerical examples are presented to verify the accuracy and validity of the numerical scheme.Modeling and computing of fractional convection equationhttps://zbmath.org/1463.652292021-07-26T21:45:41.944397Z"Li, Changpin"https://zbmath.org/authors/?q=ai:li.changpin"Yi, Qian"https://zbmath.org/authors/?q=ai:yi.qianSummary: In this paper, we derive the fractional convection (or advection) equations (FCEs) (or FAEs) to model anomalous convection processes. Through using a continuous time random walk (CTRW) with power-law jump length distributions, we formulate the FCEs depicted by Riesz derivatives with order in \((0,1)\). The numerical methods for fractional convection operators characterized by Riesz derivatives with order lying in \((0,1)\) are constructed too. Then the numerical approximations to FCEs are studied in detail. By adopting the implicit Crank-Nicolson method and the explicit Lax-Wendroff method in time, and the second-order numerical method to the Riesz derivative in space, we, respectively, obtain the unconditionally stable numerical scheme and the conditionally stable numerical one for the FCE with second-order convergence both in time and in space. The accuracy and efficiency of the derived methods are verified by numerical tests. The transport performance characterized by the derived fractional convection equation is also displayed through numerical simulations.An efficient second-order convergent scheme for one-side space fractional diffusion equations with variable coefficientshttps://zbmath.org/1463.652332021-07-26T21:45:41.944397Z"Lin, Xue-Lei"https://zbmath.org/authors/?q=ai:lin.xuelei"Lyu, Pin"https://zbmath.org/authors/?q=ai:lyu.pin"Ng, Michael K."https://zbmath.org/authors/?q=ai:ng.michael-k"Sun, Hai-Wei"https://zbmath.org/authors/?q=ai:sun.haiwei"Vong, Seakweng"https://zbmath.org/authors/?q=ai:vong.seakwengSummary: In this paper, a second-order finite-difference scheme is investigated for time-dependent space fractional diffusion equations with variable coefficients. In the presented scheme, the Crank-Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald-Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the variable coefficients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes, so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efficiency of the proposed scheme.An approximation to the solution of time fractional modified Burgers' equation using extended cubic B-spline methodhttps://zbmath.org/1463.652372021-07-26T21:45:41.944397Z"Majeed, Abdul"https://zbmath.org/authors/?q=ai:majeed.abdul"Kamran, Mohsin"https://zbmath.org/authors/?q=ai:kamran.mohsin"Rafique, Muhammad"https://zbmath.org/authors/?q=ai:rafique.muhammad-aasim|rafique.muhammad-awaisSummary: This paper aims to investigate numerical solution of time fractional modified Burgers' equation via Caputo fractional derivative. Extended cubic B-spline collocation scheme which reduces the nonlinear equation to a system of linear equation in the matrix form has been used for this investigation. The nonlinear part in fractional partial differential equation has been linearized by modified form of the existing method. The validity of proposed scheme has been examined on three test problems and effect of viscosity \(\nu\) and \(\alpha \in [0,1]\) variation displayed in 2D and 3D graphics. Moreover, the working of proposed scheme has also been explained through algorithm and stability of proposed scheme has been analyzed by von Neumann scheme and has proved to be unconditionally stable. To quantify the accuracy of suggested scheme, error norms have been computed.A high-order accuracy method for solving the fractional diffusion equationshttps://zbmath.org/1463.652412021-07-26T21:45:41.944397Z"Ran, Maohua"https://zbmath.org/authors/?q=ai:ran.maohua"Zhang, Chengjian"https://zbmath.org/authors/?q=ai:zhang.chengjianSummary: In this paper, an efficient numerical method for solving the general fractional diffusion equations with Riesz fractional derivative is proposed by combining the fractional compact difference operator and the boundary value methods. In order to efficiently solve the generated linear large-scale system, the generalized minimal residual (GMRES) algorithm is applied. For accelerating the convergence rate of the iterative, the Strang-type, Chan-type and P-type preconditioners are introduced. The suggested method can reach higher order accuracy both in space and in time than the existing methods. When the used boundary value method is \(A_{{k_1}, {k_2}}\)-stable, it is proven that Strang-type preconditioner is invertible and the spectra of preconditioned matrix is clustered around 1. It implies that the iterative solution is convergent rapidly. Numerical experiments with the absorbing boundary condition and the generalized Dirichlet type further verify the efficiency.Convergence analysis of the space fractional-order diffusion equation based on the compact finite difference schemehttps://zbmath.org/1463.652442021-07-26T21:45:41.944397Z"Safdari, H."https://zbmath.org/authors/?q=ai:safdari.hamid"Mesgarani, H."https://zbmath.org/authors/?q=ai:mesgarani.hamid"Javidi, M."https://zbmath.org/authors/?q=ai:javidi.mohammad"Aghdam, Y. Esmaeelzade"https://zbmath.org/authors/?q=ai:aghdam.y-esmaeelzadeSummary: This paper develops a numerical method for approximating the space fractional diffusion equation in Caputo derivative sense. In this discretization process, firstly, the compact finite difference with convergence order \(\mathcal{O}(\delta \tau^2)\) is used to obtain the semi-discrete in time derivative. Afterward, the spatial fractional derivative is discretized by using the Chebyshev collocation method of the third-kind. This collocation scheme is based on the operational matrix. In addition, time-discrete stability and convergence are theoretically proved in detail. We solve two examples by the proposed method and the obtained results are compared with other numerical methods. The numerical results show that our method is much more accurate than existing methods.Mean square convergent three and five points finite difference scheme for stochastic parabolic partial differential equationshttps://zbmath.org/1463.652472021-07-26T21:45:41.944397Z"Sohaly, Mohammed. A."https://zbmath.org/authors/?q=ai:sohaly.mohammed-aSummary: In this paper, we focus on the use of two finite difference schemes in order to approximate the solution of stochastic parabolic partial differential equations problem. The conditions of the mean square convergence of the numerical solution are studied.A robust numerical method for a two-parameter singularly perturbed time delay parabolic problemhttps://zbmath.org/1463.652492021-07-26T21:45:41.944397Z"Sumit"https://zbmath.org/authors/?q=ai:sumit."Kumar, Sunil"https://zbmath.org/authors/?q=ai:kumar.sunil"Kuldeep"https://zbmath.org/authors/?q=ai:kuldeep.gajraj|kuldeep.b"Kumar, Mukesh"https://zbmath.org/authors/?q=ai:kumar.mukeshSummary: In this article, we consider a class of singularly perturbed two-parameter parabolic partial differential equations with time delay on a rectangular domain. The solution bounds are derived by asymptotic analysis of the problem. We construct a numerical method using a hybrid monotone finite difference scheme on a rectangular mesh which is a product of uniform mesh in time and a layer-adapted Shishkin mesh in space. The error analysis is given for the proposed numerical method using truncation error and barrier function approach, and it is shown to be almost second- and first-order convergent in space and time variables, respectively, independent of both the perturbation parameters. At the end, we present some numerical results in support of the theory.A linearized second-order difference scheme for the nonlinear time-fractional fourth-order reaction-diffusion equationhttps://zbmath.org/1463.652502021-07-26T21:45:41.944397Z"Sun, Hong"https://zbmath.org/authors/?q=ai:sun.hong"Sun, Zhizhong"https://zbmath.org/authors/?q=ai:sun.zhizhong"Du, Rui"https://zbmath.org/authors/?q=ai:du.ruiSummary: This paper presents a second-order linearized finite difference scheme for the nonlinear time-fractional fourth-order reaction-diffusion equation. The temporal Caputo derivative is approximated by \(L2 - {1_\sigma}\) formula with the approximation order of \(O (\tau^{3 - \alpha})\). The unconditional stability and convergence of the proposed scheme are proved by the discrete energy method. The scheme can achieve the global second-order numerical accuracy both in space and time. Three numerical examples are given to verify the numerical accuracy and efficiency of the difference scheme.Crank-Nicolson finite difference method for parabolic optimal control problemhttps://zbmath.org/1463.652552021-07-26T21:45:41.944397Z"Yang, Caijie"https://zbmath.org/authors/?q=ai:yang.caijie"Sun, Tongjun"https://zbmath.org/authors/?q=ai:sun.tongjunSummary: A one dimensional parabolic optimal control problem with Neumann boundary condition is considered. The co-state equations and optimality conditions are presented and the optimality system is obtained. By applying a ghost-point based central difference approximation to the boundary condition, Crank-Nicolson finite difference discrete schemes are established for the optimality system. The maximum norm error estimates are proved to be of second-order convergence in both time and space for the state, co-state and control variables. Finally, a numerical example is presented. In order to avoid solving large coupled algebraic equations, the iterative method is used. The numerical results validate the theoretical conclusion.A class of efficient difference methods for the double-term time fractional sub-diffusion equationhttps://zbmath.org/1463.652562021-07-26T21:45:41.944397Z"Yang, Xiaozhong"https://zbmath.org/authors/?q=ai:yang.xiaozhong"Shao, Jing"https://zbmath.org/authors/?q=ai:shao.jing"Sun, Shuzhen"https://zbmath.org/authors/?q=ai:sun.shuzhenSummary: Anomalous diffusion is not only an important physical issue, but also a practical issue commonly involved in engineering. For the double-term time fractional sub-diffusion equation, this paper proposes explicit-implicit (E-I) difference method and implicit-explicit (I-E) difference method in combination with classical explicit and classical implicit difference methods. The unique solvability and the unconditional stability and convergence for the numerical solutions of E-I and I-E schemes are well illustrated. Besides that, both theoretical analysis and numerical experiments show that E-I and I-E difference methods are unconditionally stable, with 2-order spatial accuracy and \(2 - \alpha\) order temporal accuracy. Compared with the classical implicit difference method, E-I and E-I difference methods are more time-saving when the computational accuracy is consistent, which shows that E-I and E-I difference methods proposed in this paper are efficient and feasible to solve the double-term time fractional sub-diffusion equation.A fast and high accuracy numerical simulation for a fractional Black-Scholes model on two assetshttps://zbmath.org/1463.652572021-07-26T21:45:41.944397Z"Zhang, Hongmei"https://zbmath.org/authors/?q=ai:zhang.hongmei"Liu, Fawang"https://zbmath.org/authors/?q=ai:liu.fawang"Chen, Shanzhen"https://zbmath.org/authors/?q=ai:chen.shanzhen"Shen, Ming"https://zbmath.org/authors/?q=ai:shen.mingSummary: In this paper, a two dimensional (2D) fractional Black-Scholes (FBS) model on two assets following independent geometric Lévy processes is solved numerically. A high order convergent implicit difference scheme is constructed and detailed numerical analysis is established. The fractional derivative is a quasidifferential operator, whose nonlocal nature yields a dense lower Hessenberg block coefficient matrix. In order to speed up calculation and save storage space, a fast bi-conjugate gradient stabilized (FBi-CGSTAB) method is proposed to solve the resultant linear system. Finally, one example with a known exact solution is provided to assess the effectiveness and efficiency of the presented fast numerical technique. The pricing of a European Call-on-Min option is shown in the other example, in which the influence of fractional derivative order and volatility on the 2D FBS model is revealed by comparing with the classical 2D B-S model.Difference schemes for partial differential equations of fractional orderhttps://zbmath.org/1463.652702021-07-26T21:45:41.944397Z"Bazzaev, Aleksandr Kazbekovich"https://zbmath.org/authors/?q=ai:bazzaev.aleksandr-kazbekovich"Tsopanov, Igor' Dzastemirovich"https://zbmath.org/authors/?q=ai:tsopanov.igor-dzastemirovichSummary: Nowadays, fractional differential equations arise while describing physical systems with such properties as power nonlocality, long-term memory and fractal property. The order of the fractional derivative is determined by the dimension of the fractal. Fractional mathematical calculus in the theory of fractals and physical systems with memory and non-locality becomes as important as classical analysis in continuum mechanics.
In this paper we consider higher order difference schemes of approximation for differential equations with fractional-order derivatives with respect to both spatial and time variables. Using the maximum principle, we obtain apriori estimates and prove the stability and the uniform convergence of difference schemes.The Fourier method for the Cauchy problem of the Helmholtz equation with inhomogeneous Dirichlet datahttps://zbmath.org/1463.652712021-07-26T21:45:41.944397Z"Ren, Liting"https://zbmath.org/authors/?q=ai:ren.liting"Xiong, Xiangtuan"https://zbmath.org/authors/?q=ai:xiong.xiangtuanSummary: The Cauchy problem for the Helmholtz equation with inhomogeneous Dirichlet data is a severely ill-posed problem, because its solution does not depend continuously on the data. The regularized approximate solution of the problem in an infinite ``strip'' domain is obtained by a Fourier regularization method. Then, some convergence error estimations with the asymptotic Hölder type error for Fourier regularization method can be proved by using an a priori regularization parameter choice rule and an a posterior regularization parameter choice rule.The simultaneous determination of initial value and source term in the heat conduction problem by mollification methodhttps://zbmath.org/1463.652722021-07-26T21:45:41.944397Z"Wen, Jin"https://zbmath.org/authors/?q=ai:wen.jin"Ren, Xuejuan"https://zbmath.org/authors/?q=ai:ren.xuejuanSummary: The inverse problem of determining a heat source and initial temperature simultaneously in a parabolic equation is investigated. Since it is an ill-posed problem, the mollification method is used to solve it and an optimal error estimate from the a priori parameter choice rule is obtained. Finally, two examples show that the proposed method is feasible and effective.The multilevel augmentation method for solving time fractional subdiffusion equationhttps://zbmath.org/1463.652902021-07-26T21:45:41.944397Z"Chen, Jian"https://zbmath.org/authors/?q=ai:chen.jian"Zeng, Taishan"https://zbmath.org/authors/?q=ai:zeng.taishanSummary: Based on \(L1\) formula and the multiscale Galerkin method, a fully-discrete scheme is proposed for solving time fractional subdiffusion equations with \(\alpha\) order Caputo fractional derivative. The existence and uniqueness of the solution of the fully-discrete scheme are proved, and the optimal convergence order \(O({h^r} + {\tau^{2-\alpha}})\) is also deduced, where \(r\) is the order of piecewise polynomials. A multilevel augmentation method (MAM) is developed to solve the linear systems resulting from the fully-discrete scheme at each time step, and MAM preserves the optimal convergence order. A numerical experiment is presented at last to show the validity of the theoretical analysis.High accuracy analysis of the bilinear element for the time-fractional diffusion equationshttps://zbmath.org/1463.652952021-07-26T21:45:41.944397Z"Fan, Minzhi"https://zbmath.org/authors/?q=ai:fan.minzhi"Wang, Fenling"https://zbmath.org/authors/?q=ai:wang.fenling"Zhao, Yanmin"https://zbmath.org/authors/?q=ai:zhao.yanmin"Shi, Yanhua"https://zbmath.org/authors/?q=ai:shi.yanhua"Zhang, Yadong"https://zbmath.org/authors/?q=ai:zhang.yadongSummary: High accuracy analysis of the finite element is proposed for two-dimensional time-fractional diffusion equations with Caputo fractional derivative. Firstly, based on bilinear element and \(L1\) scheme, a fully-discrete approximate scheme is established and the unconditional stability analysis in \({H^1}\) norm is investigated. Secondly, with the help of Riesz projection and skill of the fractional derivative, the optimal order error estimate in \({L^2}\) norm is obtained, and by use of high accuracy result between the interpolation operator and Riesz projection and the interpolated postprocessing technique, the superclose property and the superconvergence estimate are derived. It is worth mentioning that the above superclose and superconvergence results will not be derived by the interpolation operator and Riesz projection alone. Finally, numerical results are provided to confirm the validity of our theoretical analysis.Local discontinuous Galerkin scheme for space fractional Allen-Cahn equationhttps://zbmath.org/1463.653012021-07-26T21:45:41.944397Z"Li, Can"https://zbmath.org/authors/?q=ai:li.can|li.can.1"Liu, Shuming"https://zbmath.org/authors/?q=ai:liu.shumingSummary: This paper is concerned with the efficient numerical solution for a space fractional Allen-Cahn (AC) equation. Based on the features of the fractional derivative, we design and analyze a semi-discrete local discontinuous Galerkin (LDG) scheme for the initial-boundary problem of the space fractional AC equation. We prove the optimal convergence rates of the semi-discrete LDG approximation for smooth solutions. Finally, we test the accuracy and efficiency of the designed numerical scheme on a uniform grid by three examples. Numerical simulations show that the space fractional AC equation displays abundant dynamical behaviors.A new fully discrete weak Galerkin finite element method for parabolic integro-differential equationhttps://zbmath.org/1463.653032021-07-26T21:45:41.944397Z"Liu, Xuanyu"https://zbmath.org/authors/?q=ai:liu.xuanyu"Luo, Kun"https://zbmath.org/authors/?q=ai:luo.kun"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.2|wang.hao.11|wang.hao.7|wang.hao.10|wang.hao.1|wang.hao.3|wang.hao.4|wang.hao.6|wang.hao.12|wang.hao.9|wang.hao.5|wang.hao.13Summary: In this paper, we study a fully discrete weak Galerkin finite element method for solving parabolic integro-differential equations based on polygons/polyhedrons mesh of any shape. The method contains three variables: \({u_0}\), \({u_b}\) and \(\nabla u\), where \({u_0}\) is the part of the exact solution \(u\) in the interior of elements, \({u_b}\) is the trace of \(u\) on the mesh interface, and \(\nabla u\) is the gradient of \(u\) in the interior of elements. The method uses discontinuous piecewise polynomials of degrees \(k\), \(k-1\) and \(k-1\) (\(k \ge 1\)) to approximate \({u_0}\), \({u_b}\) and \(\nabla u\) respectively. The time derivative is discretized by the Crack-Nicolson difference scheme. We prove the existence and uniqueness of the solution of the fully discrete scheme of the method. Corresponding error estimates are derived. Numerical experiments are provided to verify the theoretical results.Superconvergence analysis of quadratic triangular element for multi-term time-fractional diffusion equationshttps://zbmath.org/1463.653052021-07-26T21:45:41.944397Z"Niu, Yuqi"https://zbmath.org/authors/?q=ai:niu.yuqi"Wang, Pingli"https://zbmath.org/authors/?q=ai:wang.pingli"Wang, Fenling"https://zbmath.org/authors/?q=ai:wang.fenlingSummary: Based on the quadratic triangular finite element and time L1 approximate scheme, a fully-discrete scheme is established for multi-term time-fractional diffusion equation with Caputo derivative. Firstly, the high precision result of the quadratic triangular element is proved by using the integral identity technique under the uniform grid. Then, the spatial superclose result and temporal optimal error estimate are obtained by using the fractional derivative technique and the relationship between interpolation and projection. Furthermore, the superconvergence analysis is given through the interpolated postprocessing technique.A new approach of high accuracy analysis of fully discrete finite element method for distributed order fractional wave equationshttps://zbmath.org/1463.653072021-07-26T21:45:41.944397Z"Ren, Jincheng"https://zbmath.org/authors/?q=ai:ren.jincheng"Shi, Dongyang"https://zbmath.org/authors/?q=ai:shi.dongyangSummary: In this paper, a new approach of numerical fully discrete scheme based on the finite element approximation for the distributed order time fractional diffusion equations is developed and high accuracy error analysis is provided. Firstly, based on the \(L1\) formula for the approximation of the time distributed order fractional derivative, the fully discrete finite element scheme is derived and the unconditional stability of the scheme is obtained. Secondly, by use of the superclose estimate between the Ritz projection operator \({R_h}\) and interpolation operator \({I_h}\) of the bilinear element and the interpolated post-processing technique, the superclose and superconvergence results for the fully discrete scheme are obtained, which can't be deduced by the interpolation or Ritz projection alone. Furthermore, the proposed method is applied to the equations with variable coefficient and the unconditional stability and superconvergent estimates are also proved. Finally, some popular finite elements are investigated.Auxiliary equations approach for the stochastic unsteady Navier-Stokes equations with additive random noisehttps://zbmath.org/1463.653172021-07-26T21:45:41.944397Z"Zhao, Wenju"https://zbmath.org/authors/?q=ai:zhao.wenju"Gunzburger, Max"https://zbmath.org/authors/?q=ai:gunzburger.max-dSummary: This paper presents a Martingale regularization method for the stochastic Navier-Stokes equations with additive noise. The original system is split into two equivalent parts, the linear stochastic Stokes equations with Martingale solution and the stochastic modified Navier-Stokes equations with relatively-higher regularities. Meanwhile, a fractional Laplace operator is introduced to regularize the noise term. The stability and convergence of numerical scheme for the pathwise modified Navier-Stokes equations are proved. The comparisons of non-regularized and regularized noises for the Navier-Stokes system are numerically presented to further demonstrate the efficiency of our numerical scheme.The average vector field method for fractional Schrödinger equationhttps://zbmath.org/1463.653242021-07-26T21:45:41.944397Z"Kong, Jiameng"https://zbmath.org/authors/?q=ai:kong.jiameng"Sun, Jianqiang"https://zbmath.org/authors/?q=ai:sun.jianqiang"Liu, Ying"https://zbmath.org/authors/?q=ai:liu.ying.5|liu.ying.1|liu.ying.4|liu.ying.3|liu.ying.6|liu.ying.2|liu.yingSummary: The Hamiltonian energy preserving scheme of fractional Schrödinger equation is constructed by the second-order average vector field method and Fourier spectral method. The evolution behavior of the equation is numerically simulated with the new scheme. The numerical results show that the new scheme of fractional Schrödinger equation has second-order accuracy and can preserve the energy and mass conservation property.A new spectral method using nonstandard singular basis functions for time-fractional differential equationshttps://zbmath.org/1463.653252021-07-26T21:45:41.944397Z"Liu, Wenjie"https://zbmath.org/authors/?q=ai:liu.wen-jie"Wang, Li-Lian"https://zbmath.org/authors/?q=ai:wang.lilian"Xiang, Shuhuang"https://zbmath.org/authors/?q=ai:xiang.shuhuangSummary: In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches, the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.Approximate solution of the multi-term time fractional diffusion and diffusion-wave equationshttps://zbmath.org/1463.653272021-07-26T21:45:41.944397Z"Rashidinia, Jalil"https://zbmath.org/authors/?q=ai:rashidinia.jalil"Mohmedi, Elham"https://zbmath.org/authors/?q=ai:mohmedi.elhamSummary: We develop a numerical scheme for finding the approximate solution for one- and two-dimensional multi-term time fractional diffusion and diffusion-wave equations considering smooth and nonsmooth solutions. The concept of multi-term time fractional derivatives is conventionally defined in the Caputo view point. In the current research, the convergence analysis of Legendre collocation spectral method was carried out. Spectral collocation method is consequently tested on several benchmark examples, to verify the accuracy and to confirm effectiveness of proposed method. The main advantage of the method is that only a small number of shifted Legendre polynomials are required to obtain accurate and efficient results. The numerical results are provided to demonstrate the reliability of our method and also to compare with other previously reported methods in the literature survey.Efficient and energy stable scheme for an anisotropic phase-field dendritic crystal growth model using the scalar auxiliary variable (SAV) approachhttps://zbmath.org/1463.653292021-07-26T21:45:41.944397Z"Yang, Xiaofeng"https://zbmath.org/authors/?q=ai:yang.xiaofengSummary: The phase-field dendritic crystal growth model is a highly nonlinear system that couples the anisotropic Allen-Cahn type equation and the heat equation. By combining the recently developed SAV (scalar auxiliary variable) method with the linear stabilization approach, as well as a special decoupling technique, we arrive at a totally decoupled, linear, and unconditionally energy stable scheme for solving the dendritic model. We prove its unconditional energy stability rigorously and present various numerical simulations to demonstrate the stability and accuracy.Numerical investigation on nonlinear ion-acoustic waves in a nonextensive plasma: fractional modelhttps://zbmath.org/1463.653312021-07-26T21:45:41.944397Z"Zhao, Meimei"https://zbmath.org/authors/?q=ai:zhao.meimeiSummary: The paper is intended to study the nonlinear behavior of planar ion-acoustic waves in an unmagnetized plasma consisting of positive ions, non-extensive electrons, positrons, and stationary negatively charged dust grains. In the basic fluid equations, a modified Riemann-Liouville fractional derivative is adopted as the time derivative for the first time. Combined with the reductive perturbation method, a Korteweg-de Vries (KdV) equation is derived for the motions of ion-acoustic waves. The Chebyshev-Legendre-Galerkin (CLG) pseudospectral method is applied to solve this KdV equation numerically for analyzing how the plasma parameters influence the soliton structures of ion-acoustic waves. It is concluded that increasing the order of fractional derivatives can increase the amplitude of solitary waves, which enables a better understanding of nonlinear wave phenomena in both astrophysical environments and laboratory plasma experiments.Numerical simulation of variation regulation of free surface wave for multiple sourceshttps://zbmath.org/1463.653342021-07-26T21:45:41.944397Z"Zhai, Shuaitao"https://zbmath.org/authors/?q=ai:zhai.shuaitao"Luo, Zhiqiang"https://zbmath.org/authors/?q=ai:luo.zhiqiang"Han, Yonghao"https://zbmath.org/authors/?q=ai:han.yonghao"Lv, Yibin"https://zbmath.org/authors/?q=ai:lv.yibinSummary: Based on the numerical method of desingular dissipative Green's function integral, a computational model of free surface wave elevation under uniform flow passing double-point sources is established in this paper. The variation of free surface wave elevation and the profile of the wave contour are presented with the different dissipation parameters, Froude number and different distances between the double sources. The numerical simulation results are verified and compared with the previous numerical results by the desingularity dissipative Green's function and double sources, and the numerical results agree well with the previous papers.Numerical method of a nonlinear hierarchical age-structured population modelhttps://zbmath.org/1463.653362021-07-26T21:45:41.944397Z"He, Zerong"https://zbmath.org/authors/?q=ai:he.zerong"Zhang, Zhiqiang"https://zbmath.org/authors/?q=ai:zhang.zhiqiang"Qiu, Zheyong"https://zbmath.org/authors/?q=ai:qiu.zheyongSummary: Based upon the assumption that the young individuals are more competitive than the old ones within a species, a class of nonlinear hierarchical age-structured population model is established in a form of IBVP of integro-partial differential equations. We propose an algorithm for the solutions to the model, analyze its convergence, and make some numerical experiments.Approximate solutions of fractional wave equations using variational iteration method and Laplace transformhttps://zbmath.org/1463.653382021-07-26T21:45:41.944397Z"Liu, Yanqin"https://zbmath.org/authors/?q=ai:liu.yanqin"Yin, Xiuling"https://zbmath.org/authors/?q=ai:yin.xiuling"Zhao, Linlin"https://zbmath.org/authors/?q=ai:zhao.linlinSummary: A relatively novel modification of the variational iteration method, by means of the Laplace transform, is applied to obtain an approximate solution of fractional wake-like equations with variable coefficients. The fractional derivatives described in this paper are in the Caputo sense. It is observed that the approach is a reliable tool to analytically investigating wave models with fractional derivatives and can be implemented to other fractional models.Local fractional Laplace homotopy analysis method for solving non-differentiable wave equations on Cantor setshttps://zbmath.org/1463.653392021-07-26T21:45:41.944397Z"Maitama, Shehu"https://zbmath.org/authors/?q=ai:maitama.shehu"Zhao, Weidong"https://zbmath.org/authors/?q=ai:zhao.weidongSummary: In this paper, we introduce a semi-analytical method called the local fractional Laplace homotopy analysis method (LFLHAM) for solving wave equations with local fractional derivatives. The LFLHAM is based on the homotopy analysis method and the local fractional Laplace transform method, respectively. The proposed analytical method was a modification of the homotopy analysis method and converged rapidly within a few iterations. The nonzero convergence-control parameter was used to adjust the convergence of the series solutions. Three examples of non-differentiable wave equations were provided to demonstrate the efficiency and the high accuracy of the proposed technique. The results obtained were completely in agreement with the results in the existing methods and their qualitative and quantitative comparison of the results.Boundary value problems for degenerate and degenerate fractional order differential equations with non-local linear source and difference methods for their numerical implementationhttps://zbmath.org/1463.653412021-07-26T21:45:41.944397Z"Beshtokov, Murat Khamidbievich"https://zbmath.org/authors/?q=ai:beshtokov.murat-khamidbievichSummary: In the paper we study non-local boundary value problems for differential and partial differential equations of fractional order with a non-local linear source being mathematical models of the transfer of water and salts in soils with fractal organization. Apart of the Cartesian case, in the paper we consider one-dimensional cases with cylindrical and spherical symmetry. By the method of energy inequalities, we obtain apriori estimates of solutions to nonlocal boundary value problems in differential form. We construct difference schemes and for these schemes, we prove analogues of apriori estimates in the difference form and provide estimates for errors assuming a sufficient smoothness of solutions to the equations. By the obtained apriori estimates, we get the uniqueness and stability of the solution with respect to the the initial data and the right par, as well as the convergence of the solution of the difference problem to the solution of the corresponding differential problem with the rate of \(O(h^2+\tau^2)\).Erratum to: ``Boundary value problems for degenerate and degenerate fractional order differential equations with non-local linear source and difference methods for their numerical implementation''https://zbmath.org/1463.653422021-07-26T21:45:41.944397Z"Beshtokov, Murat Khamidbievich"https://zbmath.org/authors/?q=ai:beshtokov.murat-khamidbievichErratum to the author's paper [ibid. 11, No. 2, 36--55 (2019; Zbl 1463.65341)].A simple semi-implicit scheme for partial differential equations with obstacle constraintshttps://zbmath.org/1463.653442021-07-26T21:45:41.944397Z"Liu, Hao"https://zbmath.org/authors/?q=ai:liu.hao|liu.hao.1|liu.hao.2"Leung, Shingyu"https://zbmath.org/authors/?q=ai:leung.shingyuSummary: We develop a simple and efficient numerical scheme to solve a class of obstacle problems encountered in various applications. Mathematically, obstacle problems are usually formulated using nonlinear partial differential equations (PDE). To construct a computationally efficient scheme, we introduce a time derivative term and convert the PDE into a time-dependent problem. But due to its nonlinearity, the time step is in general chosen to satisfy a very restrictive stability condition. To relax such a time step constraint when solving a time dependent evolution equation, we decompose the nonlinear obstacle constraint in the PDE into a linear part and a nonlinear part and apply the semi-implicit technique. We take the linear part implicitly while treating the nonlinear part explicitly. Our method can be easily applied to solve the fractional obstacle problem and min curvature flow problem. The article analyzes the convergence of our proposed algorithm. Numerical experiments are given to demonstrate the efficiency of our algorithm.An efficient symmetric finite volume element method for second-order variable coefficient parabolic integro-differential equationshttps://zbmath.org/1463.653482021-07-26T21:45:41.944397Z"Gan, Xiaoting"https://zbmath.org/authors/?q=ai:gan.xiaoting"Xu, Dengguo"https://zbmath.org/authors/?q=ai:xu.dengguoSummary: This paper is devoted to develop a symmetric finite volume element (FVE) method to solve second-order variable coefficient parabolic integro-differential equations, arising in modeling of nonlocal reactive flows in porous media. Based on barycenter dual mesh, one semi-discrete and two fully discrete backward Euler and Crank-Nicolson symmetric FVE schemes are presented. Then, the optimal order error estimates in \(L^2\)-norm are derived for the semi-discrete and two fully discrete schemes. Numerical experiments are performed to examine the convergence rate and verify the effectiveness and usefulness of the new numerical schemes.An optimal filtering method for the sideways fractional heat equationhttps://zbmath.org/1463.653512021-07-26T21:45:41.944397Z"Xiong, Xiangtuan"https://zbmath.org/authors/?q=ai:xiong.xiangtuan"Bai, Enpeng"https://zbmath.org/authors/?q=ai:bai.enpengSummary: The Cauchy problem for the sideways fractional heat equation is discussed in the quarter plane. An approximate solution of the problem is given using the optimal filtering regularization method, and the Hölder type error estimates are obtained by using Fourier transforms.Error estimates and superconvergence of a high-accuracy difference scheme for a parabolic inverse problem with unknown boundary conditionshttps://zbmath.org/1463.653522021-07-26T21:45:41.944397Z"Zhou, Liping"https://zbmath.org/authors/?q=ai:zhou.liping"Shu, Shi"https://zbmath.org/authors/?q=ai:shu.shi"Yu, Haiyuan"https://zbmath.org/authors/?q=ai:yu.haiyuanSummary: In this work, we firstly construct an implicit Euler difference scheme for a one-dimensional parabolic inverse problem with a unknown time-dependent function in the boundary conditions. Then we initially prove that this scheme can reach the asymptotic optimal error estimate in the maximum norm. Next, we present some approximation formulas for the solution derivative and the unknown boundary function and prove that they have superconvergence properties. In the end, numerical experiment demonstrates the theoretical results.Discontinuous Galerkin methods for fractional elliptic problemshttps://zbmath.org/1463.653592021-07-26T21:45:41.944397Z"Aboelenen, Tarek"https://zbmath.org/authors/?q=ai:aboelenen.tarekSummary: The aim of this paper is to provide a mathematical framework for studying different versions of discontinuous Galerkin (DG) approaches for solving 2D Riemann-Liouville fractional elliptic problems on a finite domain. The boundedness and stability analysis of the primal bilinear form are provided. A priori error estimate under energy norm and optimal error estimate under \(L^2\) norm are obtained for DG methods of the different formulations. Finally, the performed numerical examples confirm the optimal convergence order of the different formulations.The implication of local thin plate splines for solving nonlinear mixed integro-differential equations based on the Galerkin schemehttps://zbmath.org/1463.653602021-07-26T21:45:41.944397Z"Assari, Pouria"https://zbmath.org/authors/?q=ai:assari.pouria"Asadi, Mehregan Fatemeh"https://zbmath.org/authors/?q=ai:asadi.mehregan-fatemeh"Dehghan, Mehdi"https://zbmath.org/authors/?q=ai:dehghan.mehdiSummary: In this article, we investigate the construction of a computational method for solving nonlinear mixed Volterra-Fredholm integro-differential equations of the second kind. The method firstly converts these types of integro-differential equations to a class of nonlinear integral equations and then utilizes the locally supported thin plate splines as a basis in the discrete Galerkin method to estimate the solution. The local thin plate splines are known as a type of the free shape parameter radial basis functions constructed on a small set of nodes in the support domain of any node which establish a stable technique to approximate an unknown function. The presented method in comparison with the method based on the globally supported thin plate splines for solving integral equations is well-conditioned and uses much less computer memory. Moreover, the algorithm of the presented approach is attractive and easy to implement on computers. The numerical method developed in the current paper does not require any cell structures, so it is meshless. Finally, numerical examples are considered to demonstrate the validity and efficiency of the new method.A balanced oversampling finite element method for elliptic problems with observational boundary datahttps://zbmath.org/1463.653642021-07-26T21:45:41.944397Z"Chen, Zhiming"https://zbmath.org/authors/?q=ai:chen.zhiming"Tuo, Rui"https://zbmath.org/authors/?q=ai:tuo.rui"Zhang, Wenlong"https://zbmath.org/authors/?q=ai:zhang.wenlongSummary: In this paper we propose a finite element method for solving elliptic equations with observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier and requires balanced oversampling of the measurements of the boundary data to control the random noises. We show the convergence of the random finite element error in expectation and, when the noise is sub-Gaussian, in the Orlicz \({\psi_2}\)-norm which implies the probability that the finite element error estimates are violated decays exponentially. Numerical examples are included.Superconvergence analysis of the polynomial preserving recovery for elliptic problems with Robin boundary conditionshttps://zbmath.org/1463.653662021-07-26T21:45:41.944397Z"Du, Yu"https://zbmath.org/authors/?q=ai:du.yu"Wu, Haijun"https://zbmath.org/authors/?q=ai:wu.haijun"Zhang, Zhimin"https://zbmath.org/authors/?q=ai:zhang.zhiminSummary: We analyze the superconvergence property of the linear finite element method based on the polynomial preserving recovery (PPR) for Robin boundary elliptic problems on triangulations. First, we improve the convergence rate between the finite element solution and the linear interpolation under the \({H^1}\)-norm by introducing a class of meshes satisfying the condition \((\alpha, \sigma, \mu)\). Then we prove the superconvergence of the recovered gradients post-processed by PPR and define an asymptotically exact a posteriori error estimator. Finally, numerical tests are provided to verify the theoretical findings.Finite difference/\(H^1\)-Galerkin MFE procedure for a fractional water wave modelhttps://zbmath.org/1463.653782021-07-26T21:45:41.944397Z"Wang, Jin-Feng"https://zbmath.org/authors/?q=ai:wang.jinfeng"Zhang, Min"https://zbmath.org/authors/?q=ai:zhang.min.6|zhang.min.4|zhang.min.7|zhang.min.3|zhang.min.2|zhang.min.5|zhang.min.1|zhang.min"Li, Hong"https://zbmath.org/authors/?q=ai:li.hong"Liu, Yang"https://zbmath.org/authors/?q=ai:liu.yang.11Summary: In this article, an \(H^1\)-Galerkin mixed finite element (MFE) method for solving the time fractional water wave model is presented. First-order backward Euler difference method and \(L1\) formula are applied to approximate integer derivative and Caputo fractional derivative with order \(1/2\), respectively, and \(H^1\)-Galerkin mixed finite element method is used to approximate the spatial direction. The analysis of stability for fully discrete mixed finite element scheme is made and the optimal space-time orders of convergence for two unknown variables in both \(H^1\)-norm and \(L^2\)-norm are derived. Further, some computing results for a priori analysis and numerical figures based on four changed parameters in the studied problem are given to illustrate the effectiveness of the current method.Well-posedness and finite element approximations for elliptic SPDEs with Gaussian noiseshttps://zbmath.org/1463.653922021-07-26T21:45:41.944397Z"Cao, Yanzhao"https://zbmath.org/authors/?q=ai:cao.yanzhao"Hong, Jialin"https://zbmath.org/authors/?q=ai:hong.jialin"Liu, Zhihui"https://zbmath.org/authors/?q=ai:liu.zhihuiSummary: The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises. The noise term is approximated through the spectral projection of the covariance operator, which is not required to be commutative with the Laplacian operator. Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises, the well-posedness of the SPDE is established under certain covariance operator-dependent conditions. These SPDEs with projected noises are then numerically approximated with the finite element method. A general error estimate framework is established for the finite element approximations. Based on this framework, optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained. It is shown that with the proposed approach, convergence order of white noise driven SPDEs is improved by half for one-dimensional problems, and by an infinitesimal factor for higher-dimensional problems.Numerical analysis of linear and nonlinear time-fractional subdiffusion equationshttps://zbmath.org/1463.653982021-07-26T21:45:41.944397Z"Yang, Yubo"https://zbmath.org/authors/?q=ai:yang.yubo"Zeng, Fanhai"https://zbmath.org/authors/?q=ai:zeng.fanhaiSummary: In this paper, a new type of the discrete fractional Grönwall inequality is developed, which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdiffusion equation. Based on the temporal-spatial error splitting argument technique, the discrete fractional Grönwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdiffusion equation.Operational matrices based on hybrid functions for solving general nonlinear two-dimensional fractional integro-differential equationshttps://zbmath.org/1463.654282021-07-26T21:45:41.944397Z"Maleknejad, Khosrow"https://zbmath.org/authors/?q=ai:maleknejad.khosrow"Rashidinia, Jalil"https://zbmath.org/authors/?q=ai:rashidinia.jalil"Eftekhari, Tahereh"https://zbmath.org/authors/?q=ai:eftekhari.taherehSummary: In the present paper, an attempt was made to develop a numerical method for solving a general form of two-dimensional nonlinear fractional integro-differential equations using operational matrices. Our approach is based on the hybrid of two-dimensional block-pulse functions and two-variable shifted Legendre polynomials. Error bound and convergence analysis of the proposed method are discussed. We prove that the order of convergence of our method is \(O( \frac{1}{2^{2M-1}N^MM!})\). The presented method is tested by seven test problems to demonstrate the accuracy and computational efficiency of the proposed method and to compare our results with other well-known methods. The comparison highlighted that the proposed method exhibits superior performance than the existing methods, even using a few numbers of bases.The least square support machine method for solving the inverse heat conduction problem with a source parameterhttps://zbmath.org/1463.654472021-07-26T21:45:41.944397Z"Wu, Chunmei"https://zbmath.org/authors/?q=ai:wu.chunmei"Wu, Ziku"https://zbmath.org/authors/?q=ai:wu.zikuSummary: A function approximate method is proposed based on the least square support vector machine, which is employed to solve the one-dimensional inverse heat conduction problem with a source parameter. The trial solution of this method is a combination of kernel functions, which satisfies the initial conditions. It turns the parameters estimation problem into a quadratic programming under the constraints of boundary conditions and the differential equation. Numerical example shows the effectiveness of the method.An approximate similarity solution for spatial fractional boundary-layer flow over an infinite vertical platehttps://zbmath.org/1463.760232021-07-26T21:45:41.944397Z"Mohammadein, Ali S."https://zbmath.org/authors/?q=ai:mohammadein.ali-s"El-Amin, Mohamed F."https://zbmath.org/authors/?q=ai:el-amin.mohamed-fathy"Ali, Hegagi M."https://zbmath.org/authors/?q=ai:ali.hegagi-mohamedSummary: In this paper, an approximate similarity solution of a fractional laminar boundary layer of viscous fluid flow over an infinite vertical plate has been presented. We introduce new fractional similarity transformations to convert the partial differential equation into a similarity ordinary differential equation with a fractional context. The viscous term of the flow momentum equation is written based on the Caputo fractional derivative. The proposed analytical solution has been developed using fractional power-series technique, and the convergence of the solution has been examined. The velocity distributions with the similarity variable are plotted against different values of the fractional order to illustrate the effects of fractional argument on the velocity behavior.Derivation of Lie groups for some higher order stochastic differential equationshttps://zbmath.org/1463.760392021-07-26T21:45:41.944397Z"Sakulrang, Sasikarn"https://zbmath.org/authors/?q=ai:sakulrang.sasikarn"Sungnul, Surattana"https://zbmath.org/authors/?q=ai:sungnul.surattana"Srihirun, Boonlert"https://zbmath.org/authors/?q=ai:srihirun.boonlert"Moore, Elvin J."https://zbmath.org/authors/?q=ai:moore.elvin-jSummary: In this paper, we give examples of the construction of Lie groups for some examples of second and third order stochastic differential equations of physical interest. Determining equations are derived for admitted Lie groups of transformations that transform Brownian motion terms into Brownian motion terms. The examples include both fiber-preserving and non-fiber-preserving transformations. It is proved by direct transformation that the derived Lie groups transform the original stochastic equations into stochastic equations with the same solutions.An efficient computational method for local fractional transport equation occurring in fractal porous mediahttps://zbmath.org/1463.760502021-07-26T21:45:41.944397Z"Singh, Jagdev"https://zbmath.org/authors/?q=ai:singh.jagdev"Kumar, Devendra"https://zbmath.org/authors/?q=ai:kumar.devendra.3"Kumar, Sunil"https://zbmath.org/authors/?q=ai:kumar.sunilSummary: The present article deals with the local fractional linear transport equations (LFLTE) in fractal porous media. LFLTE play a key role in different scientific problems such as aeronomy, superconductor, semiconductors, turbulence, gas mixture, plasma and biology. A numerical scheme namely \(q\)-local fractional homotopy analysis transform method \((q\)-LFHATM) is applied to get the solution of LFLTE. The results obtained by using of \(q\)-LFHATM show that the proposed scheme is very suitable and easy to perform with high accuracy.Unsteady magnetohydrodynamic flow of second grade fluid due to impulsive motion of platehttps://zbmath.org/1463.760612021-07-26T21:45:41.944397Z"Khan, Amir"https://zbmath.org/authors/?q=ai:khan.amir-sultan"Zaman, Gul"https://zbmath.org/authors/?q=ai:zaman.gulSummary: New analytic solutions for unsteady magnetohydrodynamic (MHD) flows of a generalized second-grade fluid have been derived. The generalized second-grade fluid saturates the porous space. Fractional derivative is used in the governing equation. The analytical expressions for velocity and shear stress fields have been obtained by using Laplace transform technique for the fractional calculus. The obtained solutions are expressed in series form in terms of Fox H-functions. The corresponding solutions for ordinary second-grade fluid passing through a porous space are obtained as special cases of general solutions. Moreover, several figures are sketched for the pertinent parameters to analyze the characteristics of velocity field and shear stress.Research on the application of adaptive genetic algorithm in the MEI method of two-dimensional electromagnetic wave scatteringhttps://zbmath.org/1463.780092021-07-26T21:45:41.944397Z"Jia, Hanqin"https://zbmath.org/authors/?q=ai:jia.hanqin"Jia, Zupeng"https://zbmath.org/authors/?q=ai:jia.zupengSummary: MEI (Measured Equation of Invariance) method is an effective numerical method for boundary truncation, which has been widely used in the field of computational electromagnetics. The ill-conditioning of MEI equation is a problem worthy of attention. In this paper, the finite element method is used to solve Helmholtz equation related to two-dimensional electromagnetic wave scattering, and the application of adaptive genetic algorithm to the solution of MEI equation is focused on. The results show that the application of adaptive genetic algorithm to the solution of MEI equation is effective.New wave solutions of an integrable dispersive wave equation with a fractional time derivative arising in ocean engineering modelshttps://zbmath.org/1463.860022021-07-26T21:45:41.944397Z"Tozar, Ali"https://zbmath.org/authors/?q=ai:tozar.ali"Kurt, Ali"https://zbmath.org/authors/?q=ai:kurt.ali"Tasbozan, Orkum"https://zbmath.org/authors/?q=ai:tasbozan.orkumSummary: The fractional Camassa-Holm equation is generally used as a powerful tool in computer simulations of water waves in shallow water, coastal and harbor models. In this paper, new wave solutions of this equation are obtained by using a new extended direct algebraic method. Thirty-six completely new solutions are obtained and are graphically represented. These solutions may motivate future research on the topic.An inverse finance problem for estimating volatility in American option pricing under jump-diffusion dynamicshttps://zbmath.org/1463.912012021-07-26T21:45:41.944397Z"Neisy, Abdolsadeh"https://zbmath.org/authors/?q=ai:neisy.abdolsadeh"Bidarvand, Mandana"https://zbmath.org/authors/?q=ai:bidarvand.mandanaSummary: This study attempts to estimate the volatility of the American options pricing model under jump-diffusion underlying asset model. Therefore, the problem is formulated then inverted, and afterward, direct finance problems are defined. Then, it is demonstrated that the price of this type of options satisfies a free boundary Partial Integral Differential Equation (PIDE). The inverse method for estimating the volatility and the American options price is also described in three phases: first, transformation of the direct problem to a non-linear initial and boundary value problem. Second, finding the solution by using the method of lines and the fourth-order Runge-Kutta method. Third, presenting a minimization function with Tikhonov regularization.Solving Black-Scholes equations using fractional generalized homotopy analysis methodhttps://zbmath.org/1463.912022021-07-26T21:45:41.944397Z"Saratha, S. R."https://zbmath.org/authors/?q=ai:saratha.s-r"Sai Sundara Krishnan, G."https://zbmath.org/authors/?q=ai:saisundarakrishnan.g"Bagyalakshmi, M."https://zbmath.org/authors/?q=ai:bagyalakshmi.morachan"Lim, Chee Peng"https://zbmath.org/authors/?q=ai:lim.chee-pengSummary: This paper aims to solve the Black-Scholes (B-S) model for the European options pricing problem using a hybrid method called fractional generalized homotopy analysis method (FGHAM). The convergence region of the B-S model solutions are clearly identified using \(h\)-curve and the closed form series solutions are produced using FGHAM. To verify the convergence of the proposed series solutions, sequence of errors are obtained by estimating the deviation between the exact solution and the series solution, which is increased in number of terms in the series. The convergence of sequence of errors is verified using the convergence criteria and the results are graphically illustrated. Moreover, the FGHAM approach has overcome the difficulties of applying multiple integration and differentiation procedures while obtaining the solution using well-established methods such as homotopy analysis method and homotopy perturbation method. The computational efficiency of the proposed method is analyzed using a comparative study. The advantage of the proposed method is shown with a numerical example using the comparative study between FGHAM and Monte Carlo simulation. Using the numerical example, analytical expression for the implied volatility is derived and the non-local behavior is studied for the various values of the fractional parameter. The results of FGHAM are statistically validated with the exact solution and the other existing computational methods.On the pricing of multi-asset options under jump-diffusion processes using meshfree moving least-squares approximationhttps://zbmath.org/1463.912032021-07-26T21:45:41.944397Z"Shirzadi, Mohammad"https://zbmath.org/authors/?q=ai:shirzadi.mohammad"Dehghan, Mehdi"https://zbmath.org/authors/?q=ai:dehghan.mehdi"Foroush Bastani, Ali"https://zbmath.org/authors/?q=ai:foroush-bastani.aliSummary: The moving least-squares (MLS) approximation is a powerful numerical scheme widely used in the meshfree literature to construct local multivariate polynomial basis functions for expanding the solution of a given differential or integral equation. For partial integro-differential equations arising from the valuation of multi-asset options written on correlated Lévy-driven assets, we propose here an MLS-based collocation scheme in conjunction with implicit-explicit (IMEX) temporal discretization to numerically solve the problem. We apply the method to price both European and American options and compute the option hedge parameters. In the case of American options, we use an operator splitting approach to solve the linear complementarity formulation of the problem. Our numerical experiments show the efficiency of the proposed scheme in comparison with some competing approaches, specially finite difference methods.Control design for partial stabilization of nonlinear mechanical systems with random disturbanceshttps://zbmath.org/1463.932142021-07-26T21:45:41.944397Z"Zuyev, A. L."https://zbmath.org/authors/?q=ai:zuev.a-l|zuyev.alexander"Vasylieva, I. G."https://zbmath.org/authors/?q=ai:vasylieva.i-gSummary: The problem of partial stabilization of nonlinear control systems described by the Itô's stochastic differential equations is considered. For these systems, we propose a constructive control design method, which provides the partial asymptotic stability in probability of the trivial solution of the closed-loop system with respect to a part of state variables. Mechanical examples are presented to illustrate the efficiency of the proposed controllers.