Recent zbMATH articles in MSC 26https://zbmath.org/atom/cc/262021-01-08T12:24:00+00:00WerkzeugOn a Hilbert-type inequality defined on the whole plane with mixed kernel.https://zbmath.org/1449.260462021-01-08T12:24:00+00:00"You, Minghui"https://zbmath.org/authors/?q=ai:you.minghui"He, Zhenhua"https://zbmath.org/authors/?q=ai:he.zhenhuaSummary: In the paper, by introducing multiple parameters and using the rational fraction expansion of secant function, we establish a new Hilbert-type integral inequality, which is defined on the whole plane, and with the constant factor related to the even-order derivative of secant function. In addition, the equivalent form of the inequality is considered. Furthermore, the different values of parameters are given and some special results are obtained.Absolute continuity with respect to a subset of an interval.https://zbmath.org/1449.260092021-01-08T12:24:00+00:00"Loukotová, Lucie"https://zbmath.org/authors/?q=ai:loukotova.lucieSummary: The aim of this paper is to introduce a generalization of the classical absolute continuity to a relative case, with respect to a subset \(M\) of an interval \(I\). This generalization is based on adding more requirements to disjoint systems \(\{(a_k, b_k)\}_K\) from the classical definition of absolute continuity -- these systems should be not too far from \(M\) and should be small relative to some covers of \(M\). We discuss basic properties of relative absolutely continuous functions and compare this class with other classes of generalized absolutely continuous functions.Adiabatic invariant for fractional generalized Birkhoffian system with variable order.https://zbmath.org/1449.700172021-01-08T12:24:00+00:00"Xie, Hanxing"https://zbmath.org/authors/?q=ai:xie.hanxing"Song, Chuanjing"https://zbmath.org/authors/?q=ai:song.chuanjing"Zhang, Jia'ning"https://zbmath.org/authors/?q=ai:zhang.jianing"Wu, Xueyan"https://zbmath.org/authors/?q=ai:wu.xueyan"Shen, Jingrong"https://zbmath.org/authors/?q=ai:shen.jingrongSummary: Based on the Caputo fractional order derivative of variable order, we studied the perturbation to symmetry and the adiabatic invariant for the fractional generalized Birkhoffian system. As special cases, we also discussed the adiabatic invariants for the fractional Birkhoffian system with variable order, the fractional generalized Birkhoffian system and the classical generalized Birkhoffian system. Finally, an example was given to illustrate the application of the methods and results.On Jensen's multiplicative inequality for positive convex functions of selfadjoint operators in Hilbert spaces.https://zbmath.org/1449.470352021-01-08T12:24:00+00:00"Dragomir, Silvestru Sever"https://zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: We obtain some multiplicative refinements and reverses of Jensen's inequality for positive convex/concave functions of selfadjoint operators in Hilbert spaces. Natural applications for power and exponential functions are provided.Estimation of unknown functions in a class of nonlinear sum-difference inequalities in three independent variables.https://zbmath.org/1449.260252021-01-08T12:24:00+00:00"Chen, Liqiang"https://zbmath.org/authors/?q=ai:chen.liqiang"Wang, Wusheng"https://zbmath.org/authors/?q=ai:wang.wushengSummary: The discrete form and generalizations form of Gronwall-Bellman type integral inequality are the important tools to study existence, boundedness and uniqueness and other qualitative properties of solutions of the difference equations. This paper studies a class of nonlinear sum-difference inequalities, which include a nonconstant factor outside summarizing symbols and a nonconstant term outside summation term. The upper bound of the unknown function in the sum-difference inequality is estimated explicitly using properties of difference operators, summing techniques, the techniques of change of variable, the method of amplification, the integral mean value theorem, and other analysis techniques, which generalized some known results. Finally an example is given to illustrate that the results can be used to study the qualitative properties of solutions of the difference equations with three independent variables.Estimation of the unknown function of a class of integral inequalities with unknown derivative function.https://zbmath.org/1449.260272021-01-08T12:24:00+00:00"Fan, Lele"https://zbmath.org/authors/?q=ai:fan.lele"Wang, Wusheng"https://zbmath.org/authors/?q=ai:wang.wusheng"Zhong, Hua"https://zbmath.org/authors/?q=ai:zhong.huaSummary: A class of nonlinear integral inequalities is established, which include a nonconstant factor outside integral sign, an unknown function and its derivative function in integrand function. The upper bound of the unknown function in the integro-differential inequality is estimated explicitly using the techniques of variable substitution and amplification, which generalizes some known results. The derived results can be applied in the study of the explicit upper bounds of solutions of a class of integro-differential equations.Existence criteria of positive solutions for a system of Riemann-Liouville type \(p\)-Laplacian fractional order boundary value problems.https://zbmath.org/1449.340212021-01-08T12:24:00+00:00"Krushna, Boddu Muralee Bala"https://zbmath.org/authors/?q=ai:krushna.boddu-muralee-balaSummary: This paper is concerned with determining the eigenvalue intervals of \(\lambda_1\) and \(\lambda_2\) for which there exist positive solutions to a coupled system of Riemann-Liouville type \(p\)-Laplacian fractional order boundary value problems by utilizing a fixed point theorem on a cone under suitable conditions.The study of \(\alpha\)-semi-preinvexity and its applications.https://zbmath.org/1449.260122021-01-08T12:24:00+00:00"Li, Ting"https://zbmath.org/authors/?q=ai:li.ting"Peng, Zaiyun"https://zbmath.org/authors/?q=ai:peng.zaiyun"Shao, Chongyang"https://zbmath.org/authors/?q=ai:shao.chongyang"Wang, Jingjing"https://zbmath.org/authors/?q=ai:wang.jingjingSummary: A class of new generalized convex functions, \(\alpha\)-semi-preinvexity functions, are mainly studied in this paper. Firstly, an example is given to show the existence of \(\alpha\)-semi-preinvexity functions and the relationship with preinvexity functions. Secondly, some properties of the \(\alpha\)-semi-preinvexity functions are discussed. Finally, some optimality results are obtained in nonlinear programming problems without constraint and with inequality constraint, and examples are given for illustration of the corresponding results. The obtained results enrich the study of the generalized convex functions.Chebyshev type inequalities for conformable fractional integrals.https://zbmath.org/1449.260072021-01-08T12:24:00+00:00"Set, Erhan"https://zbmath.org/authors/?q=ai:set.erhan"Akdemir, Ahmet Ocak"https://zbmath.org/authors/?q=ai:akdemir.ahmet-ocak"Mumcu, Ilker"https://zbmath.org/authors/?q=ai:mumcu.ilkerSummary: In this article, firstly some necessary definitions and results involving fractional integrals are given. Secondly, a new identity involving conformable fractional integrals is given. Then, by using this identity, we establish new Chebyshev inequalities for the Chebyshev functional via conformable fractional integral.On the solutions of a Caputo-Katugampola fractional integro-differential inclusion.https://zbmath.org/1449.450132021-01-08T12:24:00+00:00"Cernea, Aurelian"https://zbmath.org/authors/?q=ai:cernea.aurelianSummary: We consider a Cauchy problem associated to an integro-differential inclusion of fractional order defined by Caputo-Katugampola derivative and by a set-valued map with nonconvex values and we prove that the set of selections corresponding to the solutions of the problem considered is a retract of the space of integrable functions on unbounded interval.Characterizing global minimizers of the difference of two positive valued affine increasing and co-radiant functions.https://zbmath.org/1449.903232021-01-08T12:24:00+00:00"Askarizadeh, Khanaman Mohammad"https://zbmath.org/authors/?q=ai:askarizadeh.khanaman-mohammad"Mohebi, Hossein"https://zbmath.org/authors/?q=ai:mohebi.hosseinSummary: Many optimization problems can be reduced to a problem with an increasing and co-radiant objective function by a suitable transformation of variables. Functions, which are increasing and co-radiant, have found many applications in microeconomic analysis. In this paper, the abstract convexity of positive valued affine increasing and co-radiant (ICR) functions are discussed. Moreover, the basic properties of this class of functions such as support set, subdifferential and maximal elements of support set are characterized. Finally, as an application, necessary and sufficient conditions for the global minimum of the difference of two strictly positive valued affine ICR functions are presented.Numerical treatment for a class of partial integro-differential equations with a weakly singular kernel using Chebyshev wavelets.https://zbmath.org/1449.653382021-01-08T12:24:00+00:00"Xu, Xiaoyong"https://zbmath.org/authors/?q=ai:xu.xiaoyong"Zhou, Fengying"https://zbmath.org/authors/?q=ai:zhou.fengying"Xie, Yu"https://zbmath.org/authors/?q=ai:xie.yuSummary: In this paper, a numerical method based on fourth kind Chebyshev wavelet collocation method is applied for solving a class of partial integro-differential equations (PIDEs) with a weakly singular kernel under three types of boundary conditions. Fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of shifted Chebyshev polynomials of the fourth kind. By implementing fractional integral formula and two-dimensional fourth kind Chebyhev wavelets together with collocation method, PIDEs with a weakly singular kernel are converted into system of algebraic equation. The convergence analysis of two-dimensional fourth kind Chebyhev wavelets is investigated. Some numerical examples are included for demonstrating the efficiency of the proposed method.Quasi-invariant and attractive sets of inertial neural networks with time-varying and infinite distributed delays.https://zbmath.org/1449.342632021-01-08T12:24:00+00:00"Tang, Qian"https://zbmath.org/authors/?q=ai:tang.qian"Jian, Jigui"https://zbmath.org/authors/?q=ai:jian.jiguiSummary: This paper aims at analyzing the quasi-invariant and attractive sets for a class of inertial neural networks with time-varying and infinite distributed delays. By utilizing the properties of nonnegative matrices, a new bidirectional-like delay integral inequality is developed. Some sufficient conditions are obtained for the existence of quasi-invariant and attractive sets of the discussed system according to the bidirectional-like integral inequality. Besides, the framework of the quasi-invariant and attractive sets for the concerned system is provided. Finally, one example is analyzed to illustrate our results.Exponential time differencing methods for the time-space-fractional Schrödinger equation.https://zbmath.org/1449.651892021-01-08T12:24:00+00:00"Liang, Xiao"https://zbmath.org/authors/?q=ai:liang.xiao"Bhatt, Harish"https://zbmath.org/authors/?q=ai:bhatt.harish-pSummary: In this paper, exponential time differencing schemes with Padé approximation to the Mittag-Leffler function are proposed for the time-space-fractional nonlinear Schrödinger equations. Ways of increasing the efficiency of the proposed schemes are discussed. Numerical experiments are performed on the time-space-fractional nonlinear Schrödinger equations with various parameters. The accuracy, efficiency, and reliability of the proposed method are illustrated by numerical results.Existence and uniqueness of solutions for a class of integro-differential equation.https://zbmath.org/1449.450142021-01-08T12:24:00+00:00"Cheng, Rong"https://zbmath.org/authors/?q=ai:cheng.rong"Ye, Guoju"https://zbmath.org/authors/?q=ai:ye.guoju"Liu, Wei"https://zbmath.org/authors/?q=ai:liu.wei.7"Zhao, Dafang"https://zbmath.org/authors/?q=ai:zhao.dafangSummary: Using Schauder's fixed point theorem and Banach contraction mapping principle, we discussed a class of integro-differential equations with Kurzweil-Henstock-Stieltjes integral, and proved the existence and uniqueness of their solutions.Equivalent condition for obtaining the best constant factor of Hilbert-type multiple series inequalities with homogeneous kernel and applications.https://zbmath.org/1449.260312021-01-08T12:24:00+00:00"Hong, Yong"https://zbmath.org/authors/?q=ai:hong.yongSummary: By using the technique of real analysis and the method of weight function, the author discussed Hilbert-type multiple series inequalities with homogeneous kernel, obtained the necessary and sufficient condition for obtaining the best constant factor, and gave its applications.Compact finite-difference method for 2D time-fractional convection-diffusion equation of groundwater pollution problems.https://zbmath.org/1449.651912021-01-08T12:24:00+00:00"Li, Lingyu"https://zbmath.org/authors/?q=ai:li.lingyu"Jiang, Ziwen"https://zbmath.org/authors/?q=ai:jiang.ziwen"Yin, Zhe"https://zbmath.org/authors/?q=ai:yin.zheSummary: In this work, we provide a compact finite-difference scheme (CFDS) of 2D time-fractional convection-diffusion equation (TF-CDE) for solving fluid dynamics problem, especially groundwater pollution. The successful predication of the pollutants concentration in groundwater will greatly benefit the protection of water resources for provide the fast and intuitive decision-makings in response to sudden water pollution events. Here, we creatively use the dimensionality reduction technology (DRT) to rewrite the original 2D problem as two equations, and we handle each one as a 1D problem. Particularly, the spatial derivative is approximated by fourth-order compact finite-difference method (CFDM) and time-fractional derivative is approximated by \(L_1\) interpolation of Caputo fractional derivative. Based on the approximations, we obtain the CFDS with fourth-order in spatial and \((2-\alpha )\)-order in temporal by adding two 1D results. In addition, the unique solvability, unconditional stability, and convergence order \(\mathcal{O}(\tau^{2-\alpha} +h_1^4+h_2^4)\) of the proposed scheme are studied. Finally, several numerical examples are carried out to support the theoretical results and demonstrate the effectiveness of the CFDS based DRT strategy. Obviously, the method developed in 2D TF-CDE of groundwater pollution problem can be easily extended for the other complex problems.Noether symmetries for fractional generalized Birkhoffian systems in terms of Caputo derivatives.https://zbmath.org/1449.700252021-01-08T12:24:00+00:00"Zhou, Ying"https://zbmath.org/authors/?q=ai:zhou.ying"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.9|zhang.yi|zhang.yi.1|zhang.yi.3|zhang.yi.2|zhang.yi.5|zhang.yi.11|zhang.yi.4|zhang.yi.12|zhang.yi.8|zhang.yi.7|zhang.yi.10Summary: Noether's theorems for a fractional generalized Birkhoffian system in terms of Caputo derivatives are studied. Firstly, the generalized Pfaff-Birkhoff principle based on Caputo fractional derivatives is established, the fractional generalized Birkhoffian equations are derived. Then, the fractional Noether symmetry and the fractional conserved quantity under special infinitesimal transformations without transforming the time are studied. Noether's theorem for the fractional generalized Birkhoffian system is established. Once more, the fractional Noether symmetry and the fractional conserved quantity under general infinitesimal transformations with transforming the time are studied, Noether's theorem for the fractional generalized Birkhoffian system is established. The proof is given by using the time reparametric method. Finally, an example is given to illustrate its application.Some integral inequalities of Simpson type for \(s\)-logarithmically convex functions.https://zbmath.org/1449.260242021-01-08T12:24:00+00:00"Chang, Xiuling"https://zbmath.org/authors/?q=ai:chang.xiulingSummary: In the paper, some new integral inequalities of Simpson type were established for the differentiable \(s\)-logarithmic convex functions. Several inequalities to the means of some positive numbers were given as their applications.Fractional Tikhonov method of a non-characteristic Cauchy problem for a parabolic equation.https://zbmath.org/1449.652272021-01-08T12:24:00+00:00"Chen, Yawen"https://zbmath.org/authors/?q=ai:chen.yawen"Xiong, Xiangtuan"https://zbmath.org/authors/?q=ai:xiong.xiangtuanSummary: The ill-posed non-characteristic Cauchy problem for a parabolic equation is considered. A fractional Tikhonov regularization method is applied to solve the the problem. Some stability error estimates under a-priori and a-posteriori choice rules are given.The uniqueness of solution for initial value problems for fractional differential equation involving the Caputo-Fabrizio derivative.https://zbmath.org/1449.340402021-01-08T12:24:00+00:00"Zhang, Shuqin"https://zbmath.org/authors/?q=ai:zhang.shuqin"Hu, Lei"https://zbmath.org/authors/?q=ai:hu.lei"Sun, Sujing"https://zbmath.org/authors/?q=ai:sun.sujingSummary: In this paper, we study some results about the expression of solutions to some linear differential equations for the Caputo-Fabrizio fractional derivative. Furthermore, by the Banach contraction principle, the unique existence of the solution to an initial value problem for nonlinear differential equation involving the Caputo-Fabrizio fractional derivative is obtained.A numerical solution of the pricing model of Asian options under sub-fractional jump-diffusion process.https://zbmath.org/1449.651832021-01-08T12:24:00+00:00"Hu, Pan"https://zbmath.org/authors/?q=ai:hu.panSummary: Under the assumption of the sub-fractional Ho-Lee stochastic interest rate model, this research firstly uses the delta hedging principle and establishes the partial differential equation of geometric average Asian options under the sub-fractional jump-diffusion process with transaction costs and dividends. Secondly, the pricing model is simplified to the Cauchy problem by using the variable substitution. Finally, a numerical solution of the pricing model is given by using the finite difference method and the composite trapezoid method. An example is also given to verify the effectiveness of the algorithm design.A split-step predictor-corrector method for space-fractional reaction-diffusion equations with nonhomogeneous boundary conditions.https://zbmath.org/1449.651862021-01-08T12:24:00+00:00"Kazmi, Kamran"https://zbmath.org/authors/?q=ai:kazmi.kamran"Khaliq, Abdul"https://zbmath.org/authors/?q=ai:khaliq.abdul-q-mSummary: A split-step second-order predictor-corrector method for space-fractional reaction-diffusion equations with nonhomogeneous boundary conditions is presented and analyzed for the stability and convergence. The matrix transfer technique is used for spatial discretization of the problem. The method is shown to be unconditionally stable and second-order convergent. Numerical experiments are performed to confirm the stability and second-order convergence of the method. The split-step predictor-corrector method is also compared with an IMEX predictor-corrector method which is found to incur oscillatory behavior for some time steps. Our method is seen to produce reliable and oscillation-free results for any time step when implemented on numerical examples with nonsmooth initial data. We also present a priori reliability constraint for the IMEX predictor-corrector method to avoid unwanted oscillations and show its validity numerically.Numerical methods for solving space fractional partial differential equations using Hadamard finite-part integral approach.https://zbmath.org/1449.652042021-01-08T12:24:00+00:00"Wang, Yanyong"https://zbmath.org/authors/?q=ai:wang.yanyong"Yan, Yubin"https://zbmath.org/authors/?q=ai:yan.yubin"Hu, Ye"https://zbmath.org/authors/?q=ai:hu.yeSummary: We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case. The approximation of the space fractional Riemann-Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order \(O(h^{3-\alpha})\), where \(h\) is the space step size and \(\alpha \in (1, 2)\) is the order of Riemann-Liouville fractional derivative. Based on this scheme, we introduce a shifted finite difference method for solving space fractional partial differential equations. We obtained the error estimates with the convergence orders \(O(\tau +h^{3-\alpha}+ h^\beta)\), where \(\tau\) is the time step size and \(\beta >0\) is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation. Unlike the numerical methods for solving space fractional partial differential equations constructed using the standard shifted Grünwald-Letnikov formula or higher order Lubich's methods which require the solution of the equation to satisfy the homogeneous Dirichlet boundary condition to get the first-order convergence, the numerical method for solving the space fractional partial differential equation constructed using the Hadamard finite-part integral approach does not require the solution of the equation to satisfy the Dirichlet homogeneous boundary condition. Numerical results show that the experimentally determined convergence order obtained using the Hadamard finite-part integral approach for solving the space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained using the numerical methods constructed with the standard shifted Grünwald-Letnikov formula or Lubich's higher order approximation schemes.Preface to the focused issue on fractional derivatives and general nonlocal models.https://zbmath.org/1449.000022021-01-08T12:24:00+00:00"Du, Qiang (ed.)"https://zbmath.org/authors/?q=ai:du.qiang"Hesthaven, Jan S. (ed.)"https://zbmath.org/authors/?q=ai:hesthaven.jan-s"Li, Changpin (ed.)"https://zbmath.org/authors/?q=ai:li.changpin"Shu, Chi-Wang (ed.)"https://zbmath.org/authors/?q=ai:shu.chi-wang"Tang, Tao (ed.)"https://zbmath.org/authors/?q=ai:tang.taoFrom the text: The present focused issue entitled ``fractional derivatives and general nonlocal models'' in Communications on Applied Mathematics and Computation (CAMC) covers fractional derivatives problems and nonlocal problems.Numerical simulation of two-dimensional tempered fractional diffusion equation.https://zbmath.org/1449.651742021-01-08T12:24:00+00:00"Chen, Jinghua"https://zbmath.org/authors/?q=ai:chen.jinghua"Chen, Xuejuan"https://zbmath.org/authors/?q=ai:chen.xuejuan"Zhang, Hongmei"https://zbmath.org/authors/?q=ai:zhang.hongmeiSummary: This paper discusses a two-dimensional tempered fractional diffusion equation, in which the tempered fractional derivative is the extension of fractional derivative. The scheme can model the transition from super-diffusion early time to diffusive late-time behavior. We apply the alternating directions implicit approach and the Crank-Nicolson (C-N) algorithm to establish our numerical discretization scheme. Furthermore, we obtain the second-order accurate difference method by a Richardson extrapolation. The stability and the convergence of the numerical scheme are proven via the technique of matrix analysis. A numerical example is given to demonstrate the efficiency of the designed schemes.The implicit midpoint method for two-side space fractional diffusion equation with a nonlinear source term.https://zbmath.org/1449.651822021-01-08T12:24:00+00:00"Hu, Dongdong"https://zbmath.org/authors/?q=ai:hu.dongdong"Cao, Xuenian"https://zbmath.org/authors/?q=ai:cao.xuenian"Jiang, Huiling"https://zbmath.org/authors/?q=ai:jiang.huilingSummary: In this paper, the numerical scheme was constructed for solving the space fractional diffusion equation with a nonlinear source term where the implicit midpoint method was applied to discretize the first order time partial derivative, and the quasi-compact difference operator was utilized to approximate Riemann-Liouville space fractional partial derivative. Stability and convergence analysis of this numerical method were given. Numerical experiments show that the numerical method is effective.Hadamard type fractional integral inequalities with parameter for harmonically quasi-convex functions.https://zbmath.org/1449.260412021-01-08T12:24:00+00:00"Sun, Wenbing"https://zbmath.org/authors/?q=ai:sun.wenbingSummary: First, the author established a Riemann-Liouville fractional integral identity with parameter. Then, according to the identity, some Hermite-Hadamard type fractional integral inequalities with parameter for harmonically quasi-convex functions involving a Riemann-Liouville fractional integral were established by using a power mean inequality and Hölder inequality.Fractional-order Legendre functions for solving fractional-order differential equations.https://zbmath.org/1449.330122021-01-08T12:24:00+00:00"Kazem, S."https://zbmath.org/authors/?q=ai:kazem.saeed"Abbasbandy, S."https://zbmath.org/authors/?q=ai:abbasbandy.saeid"Kumar, Sunil"https://zbmath.org/authors/?q=ai:kumar.sunilSummary: In this article, a general formulation for the fractional-order Legendre functions (FLFs) is constructed to obtain the solution of the fractional-order differential equations. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, an efficient and reliable technique for the solution of them is important, too. For the concept of fractional derivative we will adopt Caputo's definition by using Riemann-Liouville fractional integral operator. Our main aim is to generalize the new orthogonal functions based on Legendre polynomials to the fractional calculus. Also a general formulation for FLFs fractional derivatives and product operational matrices is driven. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.A new discussion on inequality of arithmetic and geometric means.https://zbmath.org/1449.260442021-01-08T12:24:00+00:00"Wang, Gaizhen"https://zbmath.org/authors/?q=ai:wang.gaizhen"Gao, Hang"https://zbmath.org/authors/?q=ai:gao.hangSummary: In the traditional inequality of arithmetic and geometric means, the number of items in the summation is fixed. The inequality is often used to solve extreme value problems. In this article, the number of summation terms is extended to be varying. In this situation, this inequality is revisited. Firstly, a volume function is defined. Next, two equivalent functions are given to verify the basic theoretical properties.Superconvergence analysis of Hermite-type rectangular element method for two-dimensional time fractional diffusion equations.https://zbmath.org/1449.652622021-01-08T12:24:00+00:00"Wang, Pingli"https://zbmath.org/authors/?q=ai:wang.pingli"Niu, Yuqi"https://zbmath.org/authors/?q=ai:niu.yuqi"Zhao, Yanmin"https://zbmath.org/authors/?q=ai:zhao.yanmin"Wang, Fenling"https://zbmath.org/authors/?q=ai:wang.fenling"Shi, Yanhua"https://zbmath.org/authors/?q=ai:shi.yanhuaSummary: Based on the classical \(L1\) approximation scheme, a Hermite-type rectangular element method is proposed for two-dimensional time fractional diffusion equations under the fully-discrete scheme. Firstly, unconditional stability analysis of the approximate scheme is provided. Secondly, by use of the integral identity result of the Hermite-type rectangular element, a superconvergence estimate in \(H^1\)-norm is established between the interpolation and Ritz projection. Moreover, combining with the relationship between the interpolation operator and Ritz projection, and by dealing with fractional derivatives skillfully, superclose and superconvergence results are obtained, which cannot be deduced by interpolation or Ritz projection alone. Finally, the global superconvergence property is derived by the technique of the postprocessing operator.An interval-valued fuzzy distance measure between two interval-valued fuzzy numbers.https://zbmath.org/1449.030262021-01-08T12:24:00+00:00"Hesamian, Gholamreza"https://zbmath.org/authors/?q=ai:hesamian.gholamreza"Akbari, Mohammad Ghasem"https://zbmath.org/authors/?q=ai:akbari.mohammad-ghasemSummary: Interval-valued fuzzy numbers have found remarkable importance in various real-world applications. Such cases often require an evaluation of the distance measure between two interval-valued fuzzy numbers to solve the problems in the intended application. In this regard, an interval-valued fuzzy distance measure is provided in this study. For this purpose, an exact distance measure was first suggested and some of its main properties were investigated via some theorems and lemmas and compared with the results obtained by other studies. Then the proposed exact distance measure was extended in space of interval-valued fuzzy numbers and its axiomatic properties, such as triangular and robustness properties, were also investigated. Two numerical evaluations were presented to illustrate the calculation of the distance between two interval-valued fuzzy numbers as an interval-valued fuzzy number, as well.Analysis and approximation of gradient flows associated with a fractional order Gross-Pitaevskii free energy.https://zbmath.org/1449.653022021-01-08T12:24:00+00:00"Ainsworth, Mark"https://zbmath.org/authors/?q=ai:ainsworth.mark"Mao, Zhiping"https://zbmath.org/authors/?q=ai:mao.zhipingSummary: We establish the well-posedness of the fractional PDE which arises by considering the gradient flow associated with a fractional Gross-Pitaevskii free energy functional and some basic properties of the solution. The equation reduces to the Allen-Cahn or Cahn-Hilliard equations in the case where the mass tends to zero and an integer order derivative is used in the energy. We study how the presence of a non-zero mass affects the nature of the solutions compared with the Cahn-Hilliard case. In particular, we show that, analogous to the Cahn-Hilliard case, the solutions consist of regions in which the solution is a piecewise constant (whose value depends on the mass and the fractional order) separated by an interface whose width is independent of the mass and the fractional derivative. However, if the average value of the initial data exceeds some threshold (which we determine explicitly), then the solution will tend to a single constant steady state.Uniform integrability of sequence of generalized functions described by \(K\)-quasi additive Sugeno integral.https://zbmath.org/1449.260502021-01-08T12:24:00+00:00"Li, Yanhong"https://zbmath.org/authors/?q=ai:li.yanhongSummary: \(K\)-quasi additive Sugeno integral is a new non-additive integral defined by the induced operator. It plays an important role in the generalized integral theory and some practical applications. In order to overcome the inborn deficiency of \(K\)-quasi additive measure: without additivity, a new non-additive integral model ``\(K\)-quasi additive Sugeno integral'' is introduced. This provides a new way to further study the theory of non-additive integral. On the one hand, on the \(K\)-quasi additive measure space, the \(K\)-quasi additive Sugeno integral with the generalized measurable function is defined by the induced operator, and the uniform integrability and uniform boundedness of sequence of generalized functions are discussed by using the analytic representation of the integral. On the other hand, on the \(K\)-quasi additive measure space, it is proved that the uniform boundedness of a sequence of nonnegative generalized functions contains uniform integrability. Then a sufficient and necessary condition for the uniform integrability of the sequence of nonnegative generalized functions is given in the sense of \(K\)-quasi additive Sugeno integral.Two-dimensional Müntz-Legendre hybrid functions: theory and applications for solving fractional-order partial differential equations.https://zbmath.org/1449.652782021-01-08T12:24:00+00:00"Sabermahani, Sedigheh"https://zbmath.org/authors/?q=ai:sabermahani.sedigheh"Ordokhani, Yadollah"https://zbmath.org/authors/?q=ai:ordokhani.yadollah"Yousefi, Sohrab-Ali"https://zbmath.org/authors/?q=ai:yousefi.sohrab-aliSummary: In this manuscript, we present a new numerical technique based on two-dimensional Müntz-Legendre hybrid functions to solve fractional-order partial differential equations (FPDEs) in the sense of Caputo derivative, arising in applied sciences. First, one-dimensional (1D) and two-dimensional (2D) Müntz-Legendre hybrid functions are constructed and their properties are provided, respectively. Next, the Riemann-Liouville operational matrix of 2D Müntz-Legendre hybrid functions is presented. Then, applying this operational matrix and collocation method, the considered equation transforms into a system of algebraic equations. Examples display the efficiency and superiority of the technique for solving these problems with a smooth or non-smooth solution over previous works.An alternating band parallel difference method for time fractional diffusion equation.https://zbmath.org/1449.652082021-01-08T12:24:00+00:00"Yang, Xiaozhong"https://zbmath.org/authors/?q=ai:yang.xiaozhong"Wu, Lifei"https://zbmath.org/authors/?q=ai:wu.lifeiSummary: The fractional anomalous diffusion equation has profound physical background and rich theoretical connotation, and its numerical methods are of important scientific significance and engineering application value. For the two-dimensional time fractional anomalous diffusion equation, an alternating band Crank-Nicolson difference parallel computing method (ABdC-N method) is studied in this paper. Based on the alternating segment technology, the ABdC-N scheme is constructed from the classic explicit scheme, implicit scheme and Crank-Nicolson difference scheme. It can be seen from both theoretical analyses and numerical experiments that the ABdC-N method is unconditionally stable and convergent. This method has good characteristics of parallel computing, and its computation efficiency is much higher than the classical serial differential method. Our results thus show that the ABdC-N difference method is effective for solving the two-dimensional time fractional anomalous diffusion equation.Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds.https://zbmath.org/1449.110452021-01-08T12:24:00+00:00"Qi, Feng"https://zbmath.org/authors/?q=ai:qi.feng"Niu, Da-Wei"https://zbmath.org/authors/?q=ai:niu.dawei"Lim, Dongkyu"https://zbmath.org/authors/?q=ai:lim.dongkyuSummary: In the paper, starting from the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds, by virtue of the Faà di Bruno formula, with the help of two identities for the Bell polynomials of the second kind, and making use of a new inversion theorem for combinatorial coefficients, the authors derive two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials of the first and second kinds.Horseshoe dynamics in Duffing oscillator with fractional damping and multi-frequency excitation.https://zbmath.org/1449.370272021-01-08T12:24:00+00:00"Priyatharsini, S. Valli"https://zbmath.org/authors/?q=ai:priyatharsini.s-valli"Meenakshi, M. V. Sethu"https://zbmath.org/authors/?q=ai:sethu-meenakshi.m-v"Chinnathambi, V."https://zbmath.org/authors/?q=ai:chinnathambi.v"Rajasekar, S."https://zbmath.org/authors/?q=ai:rajasekar.shanmuganathanSummary: The occurrence of horseshoe chaos in Duffing oscillator with fractional damping and multi-frequency excitation is analyzed by using analytical and numerical techniques. Applying Melnikov method, analytical threshold condition for the onset of horseshoe chaos is obtained. The effect of damping exponent and the number of periodic forces on the dynamics of the Duffing oscillator is also analyzed. Due to fractional damping and multi-frequency excitation, suppression of chaos and various nonlinear phenomena are predicted. Analytical predictions are demonstrated through numerical simulations.Numerical solutions of time-fractional coupled viscous Burgers' equations using meshfree spectral method.https://zbmath.org/1449.652732021-01-08T12:24:00+00:00"Hussain, Manzoor"https://zbmath.org/authors/?q=ai:hussain.manzoor"Haq, Sirajul"https://zbmath.org/authors/?q=ai:haq.sirajul"Ghafoor, Abdul"https://zbmath.org/authors/?q=ai:ghafoor.abdul"Ali, Ihteram"https://zbmath.org/authors/?q=ai:ali.ihteramSummary: In this article, we compute numerical solutions of time-fractional coupled viscous Burgers' equations using meshfree spectral method. Radial basis functions (RBFs) and spectral collocation approach are used for approximation of the spatial part. Temporal fractional part is approximated via finite differences and quadrature rule. Approximation quality and efficiency of the method are assessed using discrete \(E_2, E_{\infty }\) and \(E_{\text{rms}}\) error norms. Varying the number of nodal points \(M\) and time step-size \(\Delta t\), convergence in space and time is numerically studied. The stability of the current method is also discussed, which is an important part of this paper.On controllability of linear and nonlinear fractional integrodifferential systems.https://zbmath.org/1449.930052021-01-08T12:24:00+00:00"Matar, Mohammed M."https://zbmath.org/authors/?q=ai:matar.mohammed-mSummary: In this article, we investigate the controllability problem of linear and nonlinear fractional integrodifferential systems. We justify the controllability concepts on a fractional integrodifferential linear system, and use results, as well as Schauder's fixed point theorem, to obtain the controllability of the corresponding nonlinear system. Some applications are introduced to explain the theoretic parts.A class of inequalities of Hadamard with local fractional integrals for generalized \(s\)-convex functions on fractal sets and their applications.https://zbmath.org/1449.260382021-01-08T12:24:00+00:00"Qiu, Ke'e"https://zbmath.org/authors/?q=ai:qiu.kee"Chen, Songliang"https://zbmath.org/authors/?q=ai:chen.songliang"Deng, Xicai"https://zbmath.org/authors/?q=ai:deng.xicai"Tao, Lei"https://zbmath.org/authors/?q=ai:tao.lei"Liu, Zhuo"https://zbmath.org/authors/?q=ai:liu.zhuo.1Summary: By using the analytical method of local fractional integrals, we gave a Hadamard type identity of generalized \(s\)-convex functions on fractal sets, and then obtained a class of Hadamard inequalities. Some applications were given by combining numerical integration and several special means.A cardinal method to solve coupled nonlinear variable-order time fractional sine-Gordon equations.https://zbmath.org/1449.354372021-01-08T12:24:00+00:00"Heydari, Mohammad Hossein"https://zbmath.org/authors/?q=ai:heydari.mohammadhossein"Avazzadeh, Zakieh"https://zbmath.org/authors/?q=ai:avazzadeh.zakieh"Yang, Yin"https://zbmath.org/authors/?q=ai:yang.yin"Cattani, Carlo"https://zbmath.org/authors/?q=ai:cattani.carloSummary: In this study, a computational approach based on the shifted second-kind Chebyshev cardinal functions (CCFs) is proposed for obtaining an approximate solution of coupled variable-order time-fractional sine-Gordon equations where the variable-order fractional operators are defined in the Caputo sense. The main ideas of this approach are to expand the unknown functions in tems of the shifted second-kind CCFs and apply the collocation method such that it reduces the problem into a system of algebraic equations. To algorithmize the method, the operational matrix of variable-order fractional derivative for the shifted second-kind CCFs is derived. Meanwhile, an effective technique for simplification of nonlinear terms is offered which exploits the cardinal property of the shifted second-kind CCFs. Several numerical examples are examined to verify the practical efficiency of the proposed method. The method is privileged with the exponential rate of convergence and provides continuous solutions with respect to time and space. Moreover, it can be adapted for other types of variable-order fractional problems straightforwardly.Mean invariance identity.https://zbmath.org/1449.330032021-01-08T12:24:00+00:00"Matkowski, Janusz"https://zbmath.org/authors/?q=ai:matkowski.januszSummary: For a continuous and increasing function \(f\) in a real interval \(I\), and a bivariable mean \(P\) defined in \(I^2\), we prescribe a pair of bivariable means \(M\) and \(N\) such that the quasiarithmetic mean \(A_f\) generated by \(f\) is invariant with respect to the mean-type mapping \((M,N)\). This allows to find effectively the limit of the iterates of the mean-type mapping \((M,N)\). The means \(M\) and \(N\) are equal iff \(P\) is the arithmetic mean \(A\); they are symmetric iff so is \(P\). Treating \(f\) and \(P\) as the parameters, we obtain the family of all pairs of means \((M,N)\) such that the quasiarithmetic mean \(A_f\) is invariant with respect to \((M,N)\). In particular, we indicate the function \(f\) and the mean \(P\) such that the invariance identity \(A_f\circ (M,N) = A_f\) coincides with the equality \(G\circ (H,A)\), where \(G\) and \(H\) are the geometric and harmonic means, equivalent to the classical Pythagorean harmony proportion.
Some examples and an application are also presented.Generalized preinvex functions and related Hermite-Hadamard type integral inequalities on fractal space.https://zbmath.org/1449.260422021-01-08T12:24:00+00:00"Sun, Wenbing"https://zbmath.org/authors/?q=ai:sun.wenbingSummary: This paper proposes the concept of generalized preinvex functions on fractal sets \({R^\alpha}\) (\(0 < \alpha \le 1\)) and establishes generalized Hermite-Hadamard's inequalities for generalized preinvex functions. Then, a local fractional integral identity for generalized preinvex functions is established. Using this identity, by a generalized Hölder inequality and a generalized power-mean inequality, some Hermite-Hadamard type inequalities for this type of function via local fractional integrals are obtained. These results extend some results of the existing researches.Integral representation of functions of \(K_b\) class.https://zbmath.org/1449.260152021-01-08T12:24:00+00:00"Lopotko, O. V."https://zbmath.org/authors/?q=ai:lopotko.o-vSummary: The necessary and sufficient conditions of multivalued continuation of even positive definite functions from intervals to all axis are obtained. All continuations of this type are described.Integral inequalities for generalized harmonically quasi-convex functions on fractal sets.https://zbmath.org/1449.260402021-01-08T12:24:00+00:00"Sun, Wenbing"https://zbmath.org/authors/?q=ai:sun.wenbingSummary: In this paper, the author introduces the concept of generalized harmonically quasi-convex functions on fractal sets \({\mathbb{R}}^{\alpha}\) (\(0<\alpha\leq 1\)) of real line numbers and establishes generalized Hermite-Hadamard and Simpson type inequalities for generalized harmonically quasi-convex functions. Some applications for \(\alpha\)-type special means of real line numbers are given.Some Hermite-Hadamard type inequalities for functions whose derivatives are quasi-convex.https://zbmath.org/1449.260352021-01-08T12:24:00+00:00"Meftah, B."https://zbmath.org/authors/?q=ai:meftah.badreddine"Merad, M."https://zbmath.org/authors/?q=ai:merad.meriem"Souahi, A."https://zbmath.org/authors/?q=ai:souahi.abdourazekSummary: In this paper, we establish new Hermite-Hadamard's inequalities using a new identity for parameter functions via quasi-convexity. Several known results are derived. Applications to special means are also given.Fuzzy logic embedding of fractional order sliding mode and state feedback controllers for synchronization of uncertain fractional chaotic systems.https://zbmath.org/1449.931642021-01-08T12:24:00+00:00"Pahnehkolaei, Seyed Mehdi Abedi"https://zbmath.org/authors/?q=ai:pahnehkolaei.seyed-mehdi-abedi"Alfi, Alireza"https://zbmath.org/authors/?q=ai:alfi.alireza"Machado, J. A. Tenreiro"https://zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiroSummary: This paper studies the synchronization of a class of uncertain fractional order (FO) chaotic systems that is applicable in secure communication. A novel hybrid FO controller, based on sliding mode and state feedback techniques combined with fuzzy logic, is developed. The algorithm, derived via the fractional Lyapunov theory, guarantees the stability of the overall system and the convergence of the synchronization errors toward a small residual set. Simulations demonstrate the capability of the proposed control algorithm in secure communications, not only in terms of speed of response, but also by reducing the chattering phenomenon.Some integral inequalities with Radon measure.https://zbmath.org/1449.260342021-01-08T12:24:00+00:00"Li, Huacan"https://zbmath.org/authors/?q=ai:li.huacan"Li, Qunfang"https://zbmath.org/authors/?q=ai:li.qunfangSummary: In this paper, we study the problem of Radon integrability of differential forms satisfying the Dirac-harmonic equation. By two kinds of Hölder inequalities, we first obtain the local Poincaré-type inequality applying to differential forms which satisfy Dirac-harmonic equation. Then, based on the local result, we also obtain the global Poincaré-type inequality on \(\delta \)-John domain by use of some proper integral skills and the property of Whitney cover, which generalized the integral theory in differential form.Arithmetic summable sequence space over non-Newtonian field.https://zbmath.org/1449.260022021-01-08T12:24:00+00:00"Yaying, Taja"https://zbmath.org/authors/?q=ai:yaying.taja"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipanSummary: In this article, we introduce the sequence spaces \(AS (G)\) and \(AC (G)\) of arithmetic summable and arithmetic convergent sequences, respectively, suggested by the geometric sum \(_G\sum_{k|m} f(k)\) as \(k\) ranges over the divisors of \(m\). We further obtain an analogous of Möbius inversion formula in the sense of geometric calculus and give interesting results in the geometric field.A fixed point theorem, intermediate value theorem, and nested interval property.https://zbmath.org/1449.550012021-01-08T12:24:00+00:00"Wu, Z."https://zbmath.org/authors/?q=ai:wu.zhiwei|wu.zhaofu|wu.zhiliang|wu.zhijan|wu.zhisong|wu.zhicheng|wu.zhenlong|wu.zhigen|wu.zhaocong|wu.zhoqun|wu.zhehui|wu.zhiqiang|wu.zhaohua|wu.zuyu|wu.zhilu|wu.zuping|wu.zhengjia|wu.zhao|wu.zhengwei|wu.zhishen|wu.zhonghuai|wu.zhengguang|wu.zhaojing|wu.zhenbin|wu.zhiqing|wu.zhuozhuo|wu.zikai|wu.zaixin|wu.zhonglin|wu.zongyu|wu.zhaoxia|wu.zhenguan|wu.zaide|wu.zongze|wu.zhworen|wu.zhangming|wu.ziheng|wu.zhengxiang|wu.zhide|wu.zhongbo|wu.zhanmin|wu.zhaojin|wu.ziwei|wu.zhengpeng|wu.zhongfu|wu.zhirong|wu.zhensen|wu.zhiang|wu.zigao|wu.zhidan|wu.zhonghua|wu.zezhong|wu.zhengwang|wu.ziqin|wu.zehui|wu.zhuanbao|wu.zhenghao|wu.zhixin|wu.ziqiang|wu.zhongcheng|wu.zhengxian|wu.zehao|wu.zongxin|wu.zedong|wu.zhibo|wu.zhengxiao|wu.zhen|wu.zhifeng|wu.zhenwei|wu.zhongtang|wu.zhenggang|wu.zuoren|wu.zhiqiao|wu.zan|wu.zhongtao|wu.zhangzhi|wu.zhemin|wu.zhuwu|wu.zhuangzhi|wu.zhouxiong|wu.zhilei|wu.zhuo|wu.zhengming|wu.zhongke|wu.zhibin|wu.zengbao|wu.zhenkui|wu.zhiquan.1|wu.zhengfei|wu.zizhao|wu.zhanggui|wu.zehu|wu.zongxian|wu.zhihui|wu.zhengrong|wu.zhengren|wu.zongqi|wu.zhengchang|wu.zhizhang|wu.zhenghua|wu.ziku|wu.zhenxiang|wu.zhixiang|wu.zemin|wu.zichen|wu.zhengmao|wu.zijun|wu.zhigang|wu.zhiming|wu.zhipeng|wu.zhimei|wu.zhaoqi|wu.zhixue|wu.zhongze|wu.zhijing|wu.zhenfeng|wu.ziongjian|wu.zhongxiang|wu.zhengzhong|wu.zongliang|wu.zhongchen|wu.zhi|wu.zhang|wu.zhongmin|wu.ziji|wu.zidian|wu.zhilin|wu.zhuang|wu.zhitao|wu.zhenxing|wu.zhuoqun|wu.zhanchun|wu.zhihua|wu.zuobing|wu.zhaoji|wu.zhihai|wu.zhenchao|wu.zhongxi|wu.zhizheng|wu.zhailian|wu.zongmin|wu.zhendong|wu.zili|wu.zengmao|wu.zikaio|wu.zhengtian|wu.zhan|wu.zhouhu|wu.zhenqiang|wu.zhengde|wu.zumin|wu.zheqian|wu.zhe|wu.zihua|wu.zhenming|wu.zhibiao|wu.zuohui|wu.zhenping|wu.zhenke|wu.zhong|wu.zhenglong|wu.zhonghai|wu.zhongming|wu.zizhen|wu.zhijian|wu.zonghua|wu.zhanji|wu.zhaoping|wu.zhansong|wu.zhuzhu|wu.zijing|wu.zhiqin|wu.zaigui|wu.zebin|wu.zheng|wu.zhongdao|wu.zhende|wu.zonglin|wu.zhangjun|wu.zhongchao|wu.zhiren|wu.zhengsheng|wu.zhengda|wu.zhengping|wu.zhifang|wu.zi|wu.zhimin|wu.zhaohui|wu.zhou|wu.zhaotong|wu.zhenhua|wu.zhansheng|wu.zijuan|wu.zhijin|wu.zhenhui|wu.zhaorong|wu.zhiyou|wu.zhaojun|wu.zefang|wu.zhengjiang|wu.zhixiong|wu.ziming|wu.zhixi|wu.zhuoren|wu.zhengang|wu.zhongle|wu.zongtao|wu.zhengguo|wu.zhiquan|wu.zhongshan|wu.zhangqing|wu.zhongqiang|wu.zhihong|wu.zhongru|wu.zhiping|wu.zhongan|wu.zixing|wu.zhongshiang|wu.zhijun.1|wu.ziniu|wu.zongsheng|wu.zhuanhao|wu.zhihuan|wu.zhidong|wu.zeping|wu.ze|wu.zhongwei|wu.zhouting|wu.zhenzhi|wu.zhihao|wu.zhengze|wu.zhaoming|wu.zijie|wu.zhengxing|wu.zhentao|wu.zhengyao|wu.zhiyu|wu.ziyan|wu.zhaoyang|wu.zhaoyan|wu.zhiyuan|wu.zeyan|wu.zunyou|wu.zhenning|wu.zhenying|wu.zhengyun|wu.zhinan|wu.zhengyu|wu.zhaoying|wu.zheyi|wu.zhenye|wu.zeyu|wu.zheyang|wu.zenan|wu.zhengyang|wu.zhenyang|wu.zhengyi|wu.zhenya|wu.ziyu|wu.zhenyong|wu.zhiyong|wu.zhenyu|wu.zhaoyu|wu.zhanyong|wu.zeyang|wu.zhenqun|wu.zhijuanSummary: For a continuous function \(f : [a,b] \to\mathbb{R}\), we prove that \(f\) has a fixed point if and only if the intervals \([a_0,b_0]= [a,b]\) and \([a_n,b_n] = [a_{n-1},b_{n-1}]\cap f([a_{n-1},b_{n-1}])\; (n = 1,2,\ldots)\) are all nonempty. More equivalent statements for the existence of fixed points of \(f\) have also been obtained and used to derive the intermediate value theorem and the nested interval property.Superconvergence analysis for time-fractional diffusion equations with nonconforming mixed finite element method.https://zbmath.org/1449.652662021-01-08T12:24:00+00:00"Zhang, Houchao"https://zbmath.org/authors/?q=ai:zhang.houchao"Shi, Dongyang"https://zbmath.org/authors/?q=ai:shi.dongyangSummary: In this paper, a fully discrete scheme based on the \(L1\) approximation in temporal direction for the fractional derivative of order \(\alpha\) in \( (0,1)\) and nonconforming mixed finite element method (MFEM) in spatial direction is established. First, we prove a novel result of the consistency error estimate with order \(O (h^2)\) of \(EQ_1^{rot}\) element (see Lemma 2.3). Then, by using the proved character of \(EQ_1^{rot}\) element, we present the superconvergent estimates for the original variable \(u\) in the broken \({H^1}\)-norm and the flux \(\vec p = \nabla u\) in the \( (L^2)^2\)-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.Computational solution of a fractional integro-differential equation.https://zbmath.org/1449.653652021-01-08T12:24:00+00:00"Kurulay, Muhammet"https://zbmath.org/authors/?q=ai:kurulay.muhammet"Akinlar, Mehmet Ali"https://zbmath.org/authors/?q=ai:akinlar.mehmet-ali"Ibragimov, Ranis"https://zbmath.org/authors/?q=ai:ibragimov.ranis-nSummary: Although differential transform method (DTM) is a highly efficient technique in the approximate analytical solutions of fractional differential equations, applicability of this method to the system of fractional integro-differential equations in higher dimensions has not been studied in detail in the literature. The major goal of this paper is to investigate the applicability of this method to the system of two-dimensional fractional integral equations, in particular to the two-dimensional fractional integro-Volterra equations. We deal with two different types of systems of fractional integral equations having some initial conditions. Computational results indicate that the results obtained by DTM are quite close to the exact solutions, which proves the power of DTM in the solutions of these sorts of systems of fractional integral equations.Fractional integral inequalities and global solutions of fractional differential equations.https://zbmath.org/1449.340412021-01-08T12:24:00+00:00"Zhu, Tao"https://zbmath.org/authors/?q=ai:zhu.taoSummary: New fractional integral inequalities are established, which generalize some famous inequalities. Then we apply these new fractional integral inequalities to study global existence results for fractional differential equations.Alternating direction implicit schemes for the two-dimensional time fractional nonlinear super-diffusion equations.https://zbmath.org/1449.651812021-01-08T12:24:00+00:00"Huang, Jianfei"https://zbmath.org/authors/?q=ai:huang.jianfei"Zhao, Yue"https://zbmath.org/authors/?q=ai:zhao.yue"Arshad, Sadia"https://zbmath.org/authors/?q=ai:arshad.sadia"Li, Kuangying"https://zbmath.org/authors/?q=ai:li.kuangying"Tang, Yifa"https://zbmath.org/authors/?q=ai:tang.yifaSummary: As is known, there exist numerous alternating direction implicit (ADI) schemes for the two-dimensional linear time fractional partial differential equations. However, if the ADI schemes for linear problems combined with local linearization techniques are applied to solve nonlinear problems, the stability and convergence of the methods are often not clear. In this paper, two ADI schemes are developed for solving the two-dimensional time fractional nonlinear super-diffusion equations based on their equivalent partial integro-differential equations. In these two schemes, the standard second-order central difference approximation is used for the spatial discretization, and the classical first-order approximation is applied to discretize the Riemann-Liouville fractional integral in time. The solvability, unconditional stability and \({L_2}\) norm convergence of the proposed ADI schemes are proved rigorously. The convergence order of the schemes is \(O\left ({\tau + h_x^2 + h_y^2} \right)\), where \(\tau\) is the temporal mesh size, \({h_x}\) and \({h_y}\) are spatial mesh sizes in the \(x\) and \(y\) directions, respectively. Finally, numerical experiments are carried out to support the theoretical results and demonstrate the performances of two ADI schemes.Parallel computation method of mixed difference schemes for time fractional reaction-diffusion equation.https://zbmath.org/1449.652972021-01-08T12:24:00+00:00"Dang, Xu"https://zbmath.org/authors/?q=ai:dang.xu"Yang, Xiaozhong"https://zbmath.org/authors/?q=ai:yang.xiaozhongSummary: The fractional reaction-diffusion equation has profound physical and engineering background, and its numerical methods are of great scientific significance and application value. A parallel computation method of mixed difference schemes for time fractional reaction-diffusion equation is proposed, and a class of alternative segment explicit-implicit scheme (ASE-I) and alternative segment implicit-explicit scheme (ASI-E) are constructed. This kind of parallel difference scheme is based on the effective combination of the Saul'yev asymmetric scheme, classical explicit difference scheme and classical implicit difference scheme. Theoretical analysis shows that the solution of ASE-I (ASI-E) scheme is uniquely solvable, unconditionally stable and convergent. Numerical experiments verify the theoretical analysis, which shows that the ASE-I scheme and the ASI-E scheme have ideal calculation accuracy and obvious parallel computing properties. It is proved that this kind of parallel difference method is effective for solving the time fractional reaction-diffusion equation.Fractional order glucose insulin system using fractional back-stepping sliding mode control.https://zbmath.org/1449.930222021-01-08T12:24:00+00:00"Vakili, S."https://zbmath.org/authors/?q=ai:vakili.soheyl"ToosianShandiz, H."https://zbmath.org/authors/?q=ai:toosianshandiz.hSummary: In this paper, based on a fractional order Bergman minimal model, a robust strategy for regulation of blood glucose in type 1 diabetic patients is presented. Glucose/insulin concentration in the patient body is controlled through the injection under the patients skin by the pump. Many various controllers for this system have been proposed in the literature. However, most of them have consider the system as an integer order system. Moreover, the majority of the presented methods suffer from an important disadvantage that is long settling time of the control system. Thus, the contribution of this paper in comparison with previous related works is presenting a fractional back-stepping sliding mode control that considerably reduces the required time for glucose to reach its desired level. Due to the sliding mode design, the proposed controller is robust against external disturbances. Due to the back-stepping design, convergence of each state variable of the system to its desired value can be guaranteed separately. Simulation results verify the satisfactory performance of the proposed controller.A lower bound of the power exponential function.https://zbmath.org/1449.260202021-01-08T12:24:00+00:00"Nishizawa, Yusuke"https://zbmath.org/authors/?q=ai:nishizawa.yusukeSummary: In this paper, we consider the lower bound of the power exponential function \(a^{2b}+b^{2a}\) for nonnegative real numbers \(a\) and \(b\). If \(a+b=1\), then it is known that the function has the maximum value 1, but it is not known that the minimum value. In this paper, we show that \(a^{2b}+b^{2a}>6083/6144 \simeq 0.990072\) for nonnegative real numbers \(a\) and \(b\) with \(a+b=1\).A high-order compact difference method for fractional sub-diffusion equations with variable coefficients and nonhomogeneous Neumann boundary conditions.https://zbmath.org/1449.652052021-01-08T12:24:00+00:00"Wang, Yuan-Ming"https://zbmath.org/authors/?q=ai:wang.yuanmingSummary: In a recent paper, \textit{L. Ren} and \textit{L. Liu} [Comput. Appl. Math. 37, No. 5, 6252--6269 (2018; Zbl 1413.65329)] proposed and analyzed a high-order compact finite difference method for a class of fractional sub-diffusion equations with variable coefficients and nonhomogeneous Neumann boundary conditions. In this paper, we point out some deficiencies and errors found in that paper and make the corresponding revisions.Schur convexity of Bonferroni means and its application.https://zbmath.org/1449.260142021-01-08T12:24:00+00:00"Wang, Dongsheng"https://zbmath.org/authors/?q=ai:wang.dongsheng"Fu, Chungru"https://zbmath.org/authors/?q=ai:fu.chungru"Shi, Huannan"https://zbmath.org/authors/?q=ai:shi.huannanSummary: Schur-convexity, Schur geometric convexity, and Schur harmonic convexity of Bonferroni mean with \(n\) variables are studied and several new inequalities of mean value with \(n\) variables are established.On some generalizations of Eneström-Kakeya theorem.https://zbmath.org/1449.260172021-01-08T12:24:00+00:00"Rather, Nisar A."https://zbmath.org/authors/?q=ai:rather.nisar-ahmed|rather.nisar-ahmad|rather.nisar-ahemadSummary: In this paper, we obtain some generalizations of a well-known result of Eneström-Kakeya concerning the bounds for the moduli of the zeros of polynomials with complex coefficients which improve some known results.On Hermite interpolation and divided differences.https://zbmath.org/1449.410012021-01-08T12:24:00+00:00"Dubeau, François"https://zbmath.org/authors/?q=ai:dubeau.francoisSummary: This paper is a survey of topics related to Hermite interpolation. In the first part we present the standard analysis of the Hermite interpolation problem. Existence, uniqueness and error formula are included. Then some computational aspects are studied including Leibnitz' formula and devided differences for monomials. Moreover continuity and differentiation properties of divided differences are analyzed. Finally we represent Hermite polynomial with respect to different basis and give links between them.Existence and continuation of solutions of Hilfer fractional differential equations.https://zbmath.org/1449.340132021-01-08T12:24:00+00:00"Bhairat, Sandeep P."https://zbmath.org/authors/?q=ai:bhairat.sandeep-pSummary: In the present paper, we consider initial value problems for Hilfer fractional differential equations and for system of Hilfer fractional differential equations. By using equivalent integral equations and some fixed point theorems, we study the local existence of solutions. We extend these local existence results globally with the help of continuation theorems and generalized Gronwall inequality.Jensen's inequality for GG-convex functions.https://zbmath.org/1449.260472021-01-08T12:24:00+00:00"Zabandan, Gholamreza"https://zbmath.org/authors/?q=ai:zabandan.gholamrezaSummary: In this paper, we obtain Jensen's inequality for GG-convex functions. Also, we get inequalities alike to Hermite-Hadamard inequality for GG-convex functions. Some examples are given.Automated proving for a class of transcendental-polynomial inequalities.https://zbmath.org/1449.681422021-01-08T12:24:00+00:00"Chen, Shiping"https://zbmath.org/authors/?q=ai:chen.shiping"Liu, Zhong"https://zbmath.org/authors/?q=ai:liu.zhongSummary: The problem of mechanical proving for transcendental-polynomial inequalities in accordance with the model of \(f (x, {\mathrm{trans}}_1 (x), \cdots, {\mathrm{trans}}_n (x)) > 0\) is discussed. Using Taylor expansion, the proving of the target inequality is reduced to the verification of a series of polynomial inequalities with only one variable, and then completed by algebraic inequality-proving package. The above algorithms are implemented on Maple. They are very efficient for the common transcendental-polynomial inequalities. Furthermore, the procedure is ``readable''.Insertion of a contra-Baire-1 (Baire-.5) function between two comparable real-valued functions.https://zbmath.org/1449.260052021-01-08T12:24:00+00:00"Mirmiran, Majid"https://zbmath.org/authors/?q=ai:mirmiran.majid"Naderi, Binesh"https://zbmath.org/authors/?q=ai:naderi.bineshFor a topological space \(X\), the function \(f : X \to \mathbb{R}\) is called \textit{Baire-.5} if the preimage of every open subset of \(\mathbb{R}\) is a \(G_{\delta}\)-set in \(X.\)
A property \(P\) defined relative to real-valued function on a topological space is a \textit{B-.5-property} if any constant function has property \(P\) and the sum of a function with property \(P\) and any Baire-.5 function also has property \(P.\) Let \(P_{1}\) and \(P_{2}\) be two B-.5-properties. Then, a space \(X\) has the \textit{weak B-.5-insertion property for \((P_{1},P_{2})\)} (\textit{B-.5-insertion property for \((P_{1},P_{2})\))} if for any functions \(g, f : X \to \mathbb{R}\) such that \(g \leq f,\) (\(g < f,\)) \(g\) has property \(P_{1}\) and \(f\) has property \(P_{2},\) then there exists a Baire-.5 function \(h\) such that \(g \leq h \leq f\) (\(g < h < f\)).
The main results are the following: for a topological space such that \(F_{\sigma}\)-kernel of sets are \(F_{\sigma}\)-sets, a sufficient condition is given for the weak B-.5-insertion property. Also for a space with the weak B-.5-insertion property, a necessary and sufficient condition is given for the space to have the B-.5-insertion property.
Reviewer: Zoltán Finta (Cluj-Napoca)Rheological model of viscoelastic body with memory and differential equations of fractional oscillator.https://zbmath.org/1449.740632021-01-08T12:24:00+00:00"Ogorodnikov, Evgeniĭ Nikolaevich"https://zbmath.org/authors/?q=ai:ogorodnikov.evgenii-nikolaevich"Radchenko, Vladimir Pavlovich"https://zbmath.org/authors/?q=ai:radchenko.vladimir-pavlovich"Yashagin, Nikolaĭ Sergeevich"https://zbmath.org/authors/?q=ai:yashagin.nikolai-sergeevichSummary: One-dimensional generalized rheologic model of viscoelastic body with Riemann-Liouville derivatives is considered. Instead of derivatives of order \(\alpha>1\) there are employed in defining relations derivatives of order \(0<\alpha<1\) from integer derivatives. It's shown, that the differential equation for the deformation with given dependence of the tension from the time with classical initial conditions of Cauchy are reduced to the Volterra integral equations. Some variants of the generalized fractional Voigt's model are considered. Explicit solutions for corresponding differential equation for the deformation are found out. It's indicated, that these solutions coincide with the classical ones when the fractional parameter vanishes.Noether's theorems based on El-Nabulsi extended exponentially quasi-fractional models in event space.https://zbmath.org/1449.370452021-01-08T12:24:00+00:00"Wang, Ze"https://zbmath.org/authors/?q=ai:wang.ze"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.1|zhang.yi.8|zhang.yi.11|zhang.yi.5|zhang.yi.7|zhang.yi.12|zhang.yi|zhang.yi.3|zhang.yi.10|zhang.yi.9|zhang.yi.2|zhang.yi.4Summary: In order to further study the dynamic behavior of non-conservative systems and reveal the relationship between the symmetries and conserved quantities of dynamic systems, we proposed and investigated the Noether theorems based on El-Nabulsi extended exponentially quasi fractional models in event space. Firstly, we put forward the El-Nabulsi quasi fractional variational problem based on the extended exponentially fractional integral, and established the differential equations of motion for holonomic systems and nonholonomic systems. Secondly, we provided the definition and criterion of Noether symmetric transformation and Noether quasi-symmetric transformation based on the invariance of the action functional under the infinitesimal transformations. Finally, we presented and proved the Noether theorems based on El-Nabulsi extended exponentially quasi fractional models in event space. Two examples were given to illustrate the application of the results.Solution in explicit form of non-local problem for differential equation with partial fractional derivative of Riemann-Liouville.https://zbmath.org/1449.353202021-01-08T12:24:00+00:00"Saĭganova, Svetlana Aleksandrovna"https://zbmath.org/authors/?q=ai:saiganova.svetlana-aleksandrovnaSummary: A non-local problem for a mixed type equation with partial fractional derivative of Riemann-Liouville is studied, boundary condition of which contains generalized operator of fractional integro-differentiation. Unique solution of the problem is then proved.On the nonlinear \(varPsi\)-Hilfer fractional differential equations.https://zbmath.org/1449.340232021-01-08T12:24:00+00:00"Kucche, Kishor D."https://zbmath.org/authors/?q=ai:kucche.kishor-d"Mali, Ashwini D."https://zbmath.org/authors/?q=ai:mali.ashwini-d"Sousa, J. Vanterler da C."https://zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.joseSummary: We consider the nonlinear Cauchy problem for \(\varPsi\)-Hilfer fractional differential equations and investigate the existence, interval of existence and uniqueness of solution in the weighted space of functions. The continuous dependence of solutions on initial conditions is proved via Weissinger fixed point theorem. Picard's successive approximation method has been developed to solve the nonlinear Cauchy problem for differential equations with \(\varPsi\)-Hilfer fractional derivative and an estimation has been obtained for the error bound. Further, by Picard's successive approximation, we derive the representation formulae for the solution of linear Cauchy problem for \(\varPsi\)-Hilfer fractional differential equation with constant coefficient and variable coefficient in terms of Mittag-Leffler function and generalized (Kilbas-Saigo) Mittag-Leffler function respectively.Existence and uniqueness of mild solutions for nonlinear fractional integro-differential evolution equations.https://zbmath.org/1449.370492021-01-08T12:24:00+00:00"Hou, Mimi"https://zbmath.org/authors/?q=ai:hou.mimi"Xi, Xuanxuan"https://zbmath.org/authors/?q=ai:xi.xuanxuan"Zhou, Xianfeng"https://zbmath.org/authors/?q=ai:zhou.xianfengSummary: In this paper, we study a class of nonlinear fractional integro-differential evolution equations in a Banach space \(X\). We use the fractional power of operators and the theory of analytic semigroups to prove the existence and uniqueness of the solution for the given problem. Furthermore, we give the Hölder continuity of the obtained mild solution.The solution of the full matrix analogue of the generalized Abel equation with constant coefficients.https://zbmath.org/1449.470862021-01-08T12:24:00+00:00"Ismagilova, Rina Rinatovna"https://zbmath.org/authors/?q=ai:ismagilova.rina-rinatovnaSummary: The system of generalized integral Abel equations in the matrix form with constant coefficients on the segment is considered in terms of the integral Riemann-Liouville operators of matrix order. Its reduction to the system of singular integral equations was found. Solution of this system was found for the case of the commutative matrices of the simple structure in the explicit form.On the functional equation \(G(x,G(y,x))=G(y,G(x,y))\) and means.https://zbmath.org/1449.330022021-01-08T12:24:00+00:00"Li, Lin"https://zbmath.org/authors/?q=ai:li.lin.2|li.lin|li.lin.1"Matkowski, Janusz"https://zbmath.org/authors/?q=ai:matkowski.januszA real valued function \(M(x,y)\) is called a mean, if \(\min(x,y)\le M(x,y)\le \max(x,y)\) for all \(x,y\) in some interval. A mean is called weighted quasi-arithmetic, if there exists a strictly monotone function \(h(x)\) and a number \(w\in (0,1)\) such that \(M(x,y) = h^{-1} (w h(x)+(1-w) h(y))\).
In the present paper the authors show that every continuous and reducible solution of the functional equation \(G(x,G(y,x)) = G(y,G(x,y))\) generates a mean resembling a weighted quasi-arithmetic mean, but no weighted quasi-arithmetic mean is a solution of this equation.
Reviewer: Khristo N. Boyadzhiev (Ada)Fractional calculus and integral transforms of the \(M\)-Wright function.https://zbmath.org/1449.330092021-01-08T12:24:00+00:00"Khan, N. U."https://zbmath.org/authors/?q=ai:khan.nabiullah-u"Kashmin, T."https://zbmath.org/authors/?q=ai:kashmin.t"Khan, S. W."https://zbmath.org/authors/?q=ai:khan.shorab-waliSummary: This paper is concerned to investigate \(M\)-Wright function, which was earlier known as transcendental function of the Wright type. \(M\)-Wright function is a special case of the Wright function given by British mathematician (E. Maitland Wright) in 1933. We have explored the cosequences of Riemann-Liouville Integral and Differential operators on \(M\)-Wright function. We have also evaluated integral transforms of the \(M\)-Wright function.Duplex selections, equilibrium points, and viability tubes.https://zbmath.org/1449.340572021-01-08T12:24:00+00:00"Kánnai, Zoltán"https://zbmath.org/authors/?q=ai:kannai.zoltanSummary: Existence of viable trajectories to nonautonomous differential inclusions are proven for time-dependent viability tubes. In the convex case we prove a double-selection theorem and a new Scorza-Dragoni type lemma. Our result also provides a new and palpable proof for the equilibrium form of Kakutani's fixed point theorem.An indirect finite element method for variable-coefficient space-fractional diffusion equations and its optimal-order error estimates.https://zbmath.org/1449.653312021-01-08T12:24:00+00:00"Zheng, Xiangcheng"https://zbmath.org/authors/?q=ai:zheng.xiangcheng"Ervin, V. J."https://zbmath.org/authors/?q=ai:ervin.vincent-j"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.1Summary: We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension. By the representation formula of the solutions \(u(x)\) to the proposed variable coefficient models in terms of \(v(x)\), the solutions to the constant coefficient analogues, we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations \(v_h(x)\) to \(v(x)\) and then obtain the approximations \(u_h(x)\) of \(u(x)\) by plugging \(v_h(x)\) into the representation of \(u(x)\). Optimal-order convergence estimates of \(u(x)-u_h(x)\) are proved in both \(L^2\) and \(H^{\alpha /2}\) norms. Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.Numerical algorithm for the time-Caputo and space-Riesz fractional diffusion equation.https://zbmath.org/1449.652122021-01-08T12:24:00+00:00"Zhang, Yuxin"https://zbmath.org/authors/?q=ai:zhang.yuxin"Ding, Hengfei"https://zbmath.org/authors/?q=ai:ding.hengfeiSummary: In this paper, we develop a novel finite-difference scheme for the time-Caputo and space-Riesz fractional diffusion equation with convergence order \(\mathcal{O}(\tau^{2-\alpha}+h^2)\). The stability and convergence of the scheme are analyzed by mathematical induction. Moreover, some numerical results are provided to verify the effectiveness of the developed difference scheme.A high order formula to approximate the Caputo fractional derivative.https://zbmath.org/1449.651952021-01-08T12:24:00+00:00"Mokhtari, R."https://zbmath.org/authors/?q=ai:mokhtari.reza"Mostajeran, F."https://zbmath.org/authors/?q=ai:mostajeran.fSummary: We present here a high-order numerical formula for approximating the Caputo fractional derivative of order \(\alpha\) for \(0<\alpha<1\). This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial differential equations. In comparison with the previous formulae, the main superiority of the new formula is its order of accuracy which is \(4-\alpha\), while the order of accuracy of the previous ones is less than 3. It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost. The effectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples. Moreover, an application of the new formula in solving some fractional partial differential equations is presented by constructing a finite difference scheme. A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.Partial best approximations and magnitude of double Vilenkin-Fourier coefficients.https://zbmath.org/1449.420502021-01-08T12:24:00+00:00"Kuznetsova, Maria A."https://zbmath.org/authors/?q=ai:kuznetsova.mariya-andreevna"Volosivets, Sergey S."https://zbmath.org/authors/?q=ai:volosivets.sergey-sergeevichSummary: We give estimates for the magnitude of double Vilenkin-Fourier coefficients of functions from generalized Hölder spaces, some \(p\)-fluctuational spaces and bounded \(\Lambda\text{-}\Gamma\text{-}\varphi\)-fluctuation spaces. For Hölder and \(p\)-fluctuational spaces we establish the sharpness of these estimates. Also we establish relation between full and partial best approximations and Watari-Efimov type inequality concerning partial best approximation and partial modulus of continuity.Numerical method for MHD flows of fractional viscous equation.https://zbmath.org/1449.652112021-01-08T12:24:00+00:00"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.1|zhang.jun|zhang.jun.7|zhang.jun.5|zhang.jun.3|zhang.jun.9|zhang.jun.2|zhang.jun.6|zhang.jun.10Summary: In this paper, the numerical approximation of fractional viscosity MHD equation is discussed. We present an efficient numerical scheme for solving this equation and analyze its stability and error estimates. We prove that the scheme is unconditionally stable and the convergence order of the scheme is \(2 - \beta\) in time and spectral accuracy in space. Finally, numerical examples are given to verify the theoretical results.Nonlocal problem for a equation of mixed type of third order with generalized operators of fractional integro-differentiation of arbitrary order.https://zbmath.org/1449.353252021-01-08T12:24:00+00:00"Repin, Oleg Aleksandrovich"https://zbmath.org/authors/?q=ai:repin.oleg-aleksandrovich"Kumykova, Svetlana Kanshubievna"https://zbmath.org/authors/?q=ai:kumykova.svetlana-kanshubievnaSummary: The unique solvability of internally boundary value problem for equation of mixed type of third order with multiple characteristics is investigated. The uniqueness theorem is proved with the restrictions on certain features and different orders of fractional integro-differentiation. The existence of solution is equivalent reduced to a Fredholm integral equation of the second kind.\({A^I}\)-statistical convergence and strong \({A^I}\)-convergence of sequences of fuzzy numbers with respect to the Orlicz function.https://zbmath.org/1449.400022021-01-08T12:24:00+00:00"Feng, Xue"https://zbmath.org/authors/?q=ai:feng.xue"Gong, Zengtai"https://zbmath.org/authors/?q=ai:gong.zengtaiSummary: As an extension of the ideal statistical convergent sequence space of fuzzy number, based on Orlicz functions and a non-negative regular matrix \(A = \{a_{nk}\}\), we defined and discussed \({A^I}\)-statistical convergence and strong \({A^I}\)-convergence of sequences of fuzzy numbers. In addition, the relationship of the two different convergences was investigated. If a sequence of fuzzy number is strongly \({A^I}\)-convergent then it is \({A^I}\)-statistically convergent.High accuracy analysis of linear triangular element for time fractional diffusion equations.https://zbmath.org/1449.653232021-01-08T12:24:00+00:00"Shi, Yanhua"https://zbmath.org/authors/?q=ai:shi.yanhua"Zhang, Yadong"https://zbmath.org/authors/?q=ai:zhang.yadong"Wang, Fenling"https://zbmath.org/authors/?q=ai:wang.fenling"Zhao, Yanmin"https://zbmath.org/authors/?q=ai:zhao.yanmin"Wang, Pingli"https://zbmath.org/authors/?q=ai:wang.pingliSummary: In this paper, based on linear triangular element and improved \(L1\) approximation, a fully-discrete scheme is proposed for time fractional diffusion equations with \(\alpha\) order Caputo fractional derivative. Firstly, the unconditional stability is proved. Secondly, by employing the properties of the element and Ritz projection operator, superclose analysis for the projection operator is deduced with order \(O ({h^2} + {\tau^{2 - \alpha}})\). Furthermore, combining with relationship between the interpolation operator and Ritz projection, superclose analysis for the interpolation operator is also investigated with order \(O ({h^2} + {\tau^{2 - \alpha}})\). And then, the superconvergence result is obtained through the interpolated postprocessing technique. Finally, numerical results are provided to show the validity of our theoretical analysis.Subdifferential representation of the convexification of non-convex functions on \(X^*\).https://zbmath.org/1449.260102021-01-08T12:24:00+00:00"Dai, Duanxu"https://zbmath.org/authors/?q=ai:dai.duanxu"Cheng, Qingjin"https://zbmath.org/authors/?q=ai:cheng.qingjin"Wang, Jianjian"https://zbmath.org/authors/?q=ai:wang.jianjianSummary: In this paper, we first obtain a subdifferential representation of a proper \(w^*\)-lower semi-continuous convex function on \(X^*\) (a dual version of Rockafellar's integral formula). Then an application to the convexification of \(w^*\) lower semi-continuous non-convex functions is given.Numerical method of value boundary problem decision for 2D equation of heat conductivity with fractional derivatives.https://zbmath.org/1449.651722021-01-08T12:24:00+00:00"Beĭbalaev, Vetlugin Dzhabrilovich"https://zbmath.org/authors/?q=ai:beibalaev.vetlugin-dzhabrilovich"Shabanova, Mumina Ruslanovna"https://zbmath.org/authors/?q=ai:shabanova.mumina-ruslanovnaSummary: In this work a solution is obtained for the boundary problem for two-dimensional thermal conductivity equation with derivatives of fractional order on time and space variables by grid method. Explicit and implicit difference schemes are developed. Stability criteria of these difference schemes are proven. It is shown that the approximation order by time is equal one but by space variables it is equal two. A solution method is suggested using fractional steps. It is proved that the transition module, corresponding to two half-steps, approximates the transition module for the given equation.Properties of inversion operator of the Abel matrix equation.https://zbmath.org/1449.470852021-01-08T12:24:00+00:00"Ismagilova, Rina Rinatovna"https://zbmath.org/authors/?q=ai:ismagilova.rina-rinatovnaSummary: Generalization of integral-differential Riemann-Liouville operator on the matrix order is reviewed and its properties are studied. Theorem of the composition of operators of the matrix of integration and differentiation can be proved. The necessary and sufficient conditions for the unique solvability of the matrix Abel equation in a special class of functions are obtained.Equality and homogeneity of generalized integral means.https://zbmath.org/1449.260522021-01-08T12:24:00+00:00"Páles, Zs."https://zbmath.org/authors/?q=ai:pales.zsolt"Zakaria, A."https://zbmath.org/authors/?q=ai:zakaria.amrThe authors mainly generalize the results of the papers by \textit{L. Losonczi} and \textit{Z. Páles} [Aequationes Math. 81, No. 1--2, 31--53 (2011; Zbl 1234.39003)] and \textit{L. Losonczi} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 41, 103--117 (2013; Zbl 1289.39045)] and also many former results obtained in various particular cases of this problem.
As direct applications of the results obtained on the equality of generalized Bajraktarevic means, they consider and solve the homogeneity problem of these means under general conditions.
A section is entirely dedicated to characterize the equality of generalized quasi-arithmetic means in various settings and the next to the homogeneity of generalized Bajraktarevic means under 3 times differentiability assumptions.
Finally, in the last section, the authors consider particular cases of the characterization of the equality of generalized Bajraktarevic means under 3 times differentiability assumptions, precisely when all the means are generalized quasi-arithmetic.
Reviewer: Maria Alessandra Ragusa (Catania)Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means.https://zbmath.org/1449.260532021-01-08T12:24:00+00:00"Qian, Wei-Mao"https://zbmath.org/authors/?q=ai:qian.weimao"Zhang, Wen"https://zbmath.org/authors/?q=ai:zhang.wen.3"Chu, Yu-Ming"https://zbmath.org/authors/?q=ai:chu.yumingSummary: In the article, we find the best possible parameters \(\lambda_{1}\), \(\mu_{1}\), \(\lambda_{2}\) and \(\mu_{2}\) on the interval \([0,1/2]\) such that the double inequalities \[H(a, b; \lambda_{1})<\alpha A(a,b)+(1-\alpha)T(a,b)<H(a, b; \mu_{1}),\] \[G(a, b; \lambda_{2})<\alpha A(a,b)+(1-\alpha)T(a,b)<G(a, b; \mu_{2})\] hold for all \(\alpha\in [0,1]\) and \(a, b>0\) with \(a\neq b\), where \(A(a,b)=(a+b)/2\), \(T(a,b)=2\int_{0}^{\pi/2}a^{\cos^{2}\theta}b^{\sin^{2}\theta}d\theta/\pi\), \(H(a, b; \lambda)=2[\lambda a+(1-\lambda)b][\lambda b+(1-\lambda)a]/(a+b)\), \(G(a, b; \mu)=\sqrt{[\mu a +(1-\mu)b][\mu b+(1-\mu)a]}\) are the arithmetic, integral, one-parameter harmonic and one-parameter geometric means of \(a\) and \(b\), respectively.Sharp bounds for Toader-type mean in terms of harmonic, geometric, centroidal and contra-harmonic means.https://zbmath.org/1449.260512021-01-08T12:24:00+00:00"He, Xiaohong"https://zbmath.org/authors/?q=ai:he.xiaohong"Xu, Huizuo"https://zbmath.org/authors/?q=ai:xu.huizuo"Qian, Weimao"https://zbmath.org/authors/?q=ai:qian.weimaoSummary: In this paper, we present the best possible parameters \({\alpha_1}, {\alpha_2}, {\alpha_3}, {\alpha_4}, {\beta_1}, {\beta_2}, {\beta_3}, {\beta_4} \in (0,1)\) such that the double inequalities \[\begin{array}{l}{\alpha_1}E (a, b) + (1 - {\alpha_1})G (a, b) < T[ A (a, b), G (a, b)] < {\beta_1}E (a, b) + (1 - {\beta_1})G (a, b), \\ {\alpha_2}E (a, b) + (1 - {\alpha_2})H (a, b) < T[A (a, b), G (a, b)] < {\beta_2}E (a, b) + (1 - {\beta_2})H (a, b), \\ {\alpha_3}C (a, b) + (1 - {\alpha_3})G (a, b) < T[A (a, b), G (a, b)] < {\beta_3}C (a, b) + (1 - {\beta_3})G (a, b), \\ {\alpha_4}C (a, b) + (1 - {\alpha_4})H (a, b) < T[A (a, b), G (a, b)] < {\beta_4}C (a, b) + (1 - {\beta_4})H (a, b)\end{array}\] hold for all \(a, b > 0\) with \(a \ne b\). As an application, we establish a new bound for the complete elliptic integral of second kind, where \[\begin{array}{l}H (a, b) = \frac{2ab}{a + b}, \;\;G (a, b) = \sqrt{ab}, \;\;E (a, b) = \frac{2 ({a^2} + ab + {b^2})}{3 (a + b)},\\ C (a, b) = \frac{{a^2} + {b^2}}{a + b}, \;\;T (a, b) = \frac{2}{\pi}\int_0^{\pi/2}\sqrt {{a^2}{\mathrm{cos}}^2t + {b^2}{\mathrm{sin}}^2t}{\mathrm{d}}t\end{array}\] are the harmonic, geometric, centroidal, contra-harmonic and Toader means of two numbers \(a\) and \(b\), respectively.A functional bound for Young's cosine polynomial.https://zbmath.org/1449.260192021-01-08T12:24:00+00:00"Fong, J. Z. Y."https://zbmath.org/authors/?q=ai:fong.jolie-zhi-yi"Lee, T. Y."https://zbmath.org/authors/?q=ai:lee.tuo-yeong"Wong, P. X."https://zbmath.org/authors/?q=ai:wong.pei-xianYoung's inequality asserts that \(1+\sum_{k=1}^{n}\frac{\cos k\theta}{k}>0\) for all \(n\in\mathbb{N}\) and \(\theta\in(0,\pi).\) The paper under review is devoted to the following stronger result: \(\frac{5}{6}+\sum_{k=1}^{n}\frac{\cos k\theta}{k}\geq\frac{1}{4}(1+\cos\theta)^{2}\) for all \(n\geq2\) and \(\theta\in(0,\pi).\) Equality occurs if and only if \(n=2\) and \(\theta=\pi-\arccos\frac{1}{3}.\)
Reviewer: Constantin Niculescu (Craiova)A weighted companion of Ostrowski's inequality using three step weighted kernel.https://zbmath.org/1449.260362021-01-08T12:24:00+00:00"Obeidat, S."https://zbmath.org/authors/?q=ai:obeidat.sofian"Latif, M. A."https://zbmath.org/authors/?q=ai:latif.muhammad-amer"Qayyum, A."https://zbmath.org/authors/?q=ai:qayyum.atherSummary: A weighted version of Ostrowski type integral inequalities is established. We use a newly developed special type of three steps kernel. Our findings give some new error bounds for various quadrature rules. We apply our results to cumulative distributive functions.New logarithmic information divergence measure.https://zbmath.org/1449.940522021-01-08T12:24:00+00:00"Singh, Devendra"https://zbmath.org/authors/?q=ai:singh.devendra-pratap"Jain, K. C."https://zbmath.org/authors/?q=ai:jain.kamal-cSummary: A new information divergence measure is proposed. This measure belongs to the class of Csiszar's f-divergence family. Its properties are studied and bounds in terms of well-known divergence measures are obtained.Some new integral inequalities for \(N\)-times differentiable \(R\)-convex and \(R\)-concave functions.https://zbmath.org/1449.260212021-01-08T12:24:00+00:00"Kadakal, Huriye"https://zbmath.org/authors/?q=ai:kadakal.huriye"Kadakal, Mahir"https://zbmath.org/authors/?q=ai:kadakal.mahir"Işcan, Imdat"https://zbmath.org/authors/?q=ai:iscan.imdatSummary: In this work, by using an integral identity together with both the Hölder and the Power-Mean integral inequality we establish several new inequalities for \(n\)-time differentiable \(r\)-convex and concave functions.Some aspects of initial value problems theory for differential equations with Riemann-Liouville derivatives.https://zbmath.org/1449.340302021-01-08T12:24:00+00:00"Ogorodnikov, Evgeniĭ Nikolaevich"https://zbmath.org/authors/?q=ai:ogorodnikov.evgenii-nikolaevichSummary: Some subjects of the well-formed initial value problem for ordinary differential equations with Riemann-Liouville derivatives are discussed. As an example the simplest linear homogeneous differential equation with two fractional derivatives is considered.On integral inequalities of Hermite-Hadamard type for coordinated \(r\)-mean convex functions.https://zbmath.org/1449.260292021-01-08T12:24:00+00:00"Gao, Dan-Dan"https://zbmath.org/authors/?q=ai:gao.dan-dan"Xi, Bo-Yan"https://zbmath.org/authors/?q=ai:xi.boyan"Wu, Ying"https://zbmath.org/authors/?q=ai:wu.ying"Guo, Bai-Ni"https://zbmath.org/authors/?q=ai:guo.bai-niSummary: In the paper, the authors first introduce a concept ``\(r\)-mean convex function on coordinates'' and then establish several integral inequalities of the Hermite-Hadamard type for \(r\)-convex functions and \(r\)-mean convex functions on coordinates.New bounds for Hermite-Hadamard's trapezoid and mid-point type inequalities via fractional integrals.https://zbmath.org/1449.260262021-01-08T12:24:00+00:00"Delavar, M. Rostamian"https://zbmath.org/authors/?q=ai:delavar.mohsen-rostamian|rostamian-delavar.mSummary: Some trapezoid and mid-point type inequalities with new bounds for Hermite-Hadamard inequality related to Riemann-Liouville integrals of order \(\alpha >0 \) are obtained. Also a refinement of Hermite-Hadamard inequality for nonnegative monotone convex functions is presented. Furthermore some applications in connection with special means are given.The second order semi-implicit asymmetric iteration scheme for solving one-dimensional fractional convection diffusion equations.https://zbmath.org/1449.652172021-01-08T12:24:00+00:00"Zhu, Lin"https://zbmath.org/authors/?q=ai:zhu.linSummary: In this paper, we construct a second order semi-implicit finite difference scheme for solving one-dimensional fractional convection diffusion equations combining with the asymmetric iteration technique. The second order weighted and shifted Grünwald-Letnikov operator is used to discretize the Riemann-Liouville fractional derivative and the central-difference operator is used to discretize the convection term. The presented scheme is formally implicit, but it can be computed explicitly by choosing unknowns in different nodal-point sequences at the odd time level and the even time level, respectively. The stability is proved by Fourier analysis method and the error estimate between the discrete and analytical solution in discrete \({l^2}\) norm is presented. At last, a numerical example is given for confirming the theoretical conclusions.New generalized midpoint type inequalities for fractional integral.https://zbmath.org/1449.260232021-01-08T12:24:00+00:00"Budak, H."https://zbmath.org/authors/?q=ai:budak.huseyi|budak.huseyin|budak.huseyin-budak"Agarwal, H."https://zbmath.org/authors/?q=ai:agarwal.harish|agarwal.hukum-chand|agarwal.himanshu|agarwal.hansSummary: Here, our first aim to establish a new identity for differentiable function involving Riemann-Liouville fractional integrals. Then, we obtain same generalized midpoint type inequalities utilizing convex and concave function.Cauchy problem for the nonlocal equation diffusion-advection radon in fractal media.https://zbmath.org/1449.653482021-01-08T12:24:00+00:00"Parovik, Roman Ivanovich"https://zbmath.org/authors/?q=ai:parovik.roman-ivanovichSummary: In this article, using the Green's function method solved the Cauchy problem for the equation of anomalous diffusion-advection of radon in a fractal medium, which is represented by a fractional derivative of the Caputo time fractional derivative and Riesz-Weil on the spatial coordinate.Characterizations of certain Hankel transform involving Riemann-Liouville fractional derivatives.https://zbmath.org/1449.420092021-01-08T12:24:00+00:00"Upadhyay, S. K."https://zbmath.org/authors/?q=ai:upadhyay.santosh-kumar"Khatterwani, Komal"https://zbmath.org/authors/?q=ai:khatterwani.komalSummary: In this paper, the relation between the two dimensional fractional Fourier transform and the fractional Hankel transform is discussed in terms of radial functions. Various operational properties of the Hankel transform and the fractional Hankel transform are studied involving Riemann-Liouville fractional derivatives. The application of the fractional Hankel transform in networks with time varying parameters is given.Right-continuous pseudo-inverse of monotone functions on closed intervals and its applications in the construction of triangular conorms.https://zbmath.org/1449.260062021-01-08T12:24:00+00:00"Zhou, Hongjun"https://zbmath.org/authors/?q=ai:zhou.hongjun"Lan, Shumin"https://zbmath.org/authors/?q=ai:lan.shuminSummary: t-norms and t-conorms have many important applications in a wide scope of communities such as functional equation, ordered semigroup, many-valued logic, probabilistic metric space, fuzzy set theory and information aggregations. Constructions and characterizations of t-norms and t-conorms have been the research topics in related fields. Due to the well-known duality between t-norms and t-conorms, most researches focus on t-norms and then are transformed to t-conorms. However, not all properties or construction methods of t-norms can be translated by duality to t-conorms, and the construction of t-norms by means of pseudo-inverses of monotone functions is such a good counterexample. In this paper, construction methods of t-conorms in terms of right-continuous pseudo-inverses and quasi-inverses of monotone functions are proposed, which can be viewed as counterpart of the construction of t-norms based on left-continuous pseudo-inverses of monotone functions. Firstly, some basic properties of right-continuous pseudo-inverse of the monotone function on a closed interval are discussed, and then several methods for constructions of t-conorms are given by using the right-continuous pseudo-inverse and quasi-inverse of the non-decreasing function on the closed unit interval. Finally, several examples are given to verify the results.Approximate solution of time-fractional fuzzy partial differential equations.https://zbmath.org/1449.652942021-01-08T12:24:00+00:00"Senol, Mehmet"https://zbmath.org/authors/?q=ai:senol.mehmet"Atpinar, Sevda"https://zbmath.org/authors/?q=ai:atpinar.sevda"Zararsiz, Zarife"https://zbmath.org/authors/?q=ai:zararsiz.zarife"Salahshour, Soheil"https://zbmath.org/authors/?q=ai:salahshour.soheil"Ahmadian, Ali"https://zbmath.org/authors/?q=ai:ahmadian.aliSummary: In this study, we develop perturbation-iteration algorithm (PIA) for numerical solutions of some types of fuzzy fractional partial differential equations (FFPDEs) with generalized Hukuhara derivative. We also present the convergence analysis of the method. The proposed approach reveals fast convergence rate and accuracy of the present method when compared with exact solutions of crisp problems. The main efficiency of this method is that while scaling support zone of uncertainty for the fractional partial differential equations, it eliminates over calculation and produces highly approximate and accurate results. Error analysis of the PIA for the FFPDEs is also illustrated within examples.Setting and solving of the Cauchy type problems for the second order differential equations with Riemann-Liouville fractional derivatives.https://zbmath.org/1449.450162021-01-08T12:24:00+00:00"Ogorodnikov, Evgeniĭ Nikolaevich"https://zbmath.org/authors/?q=ai:ogorodnikov.evgenii-nikolaevich"Yashagin, Nikolaĭ Sergeevich"https://zbmath.org/authors/?q=ai:yashagin.nikolai-sergeevichSummary: The correctness of the Cauchy problems in local (classical) and nonlocal staging for two linear ordinary second order differential equations with Riemann-Liouville fractional derivatives is substantiated. The explicit solutions in terms of some special functions related Mittag-Leffler type function are found out. The continuos dependence from the fractional parameter \(\beta\) for these solutions is indicated. For the second equation the changing statement of the Cauchy type problem coinciding with classical when \(\beta=0\) is considered. These equations are proposed such as some model fractional oscillating equation.Theory of hybrid fractional differential equations with complex order.https://zbmath.org/1449.340332021-01-08T12:24:00+00:00"Vivek, Devaraj"https://zbmath.org/authors/?q=ai:vivek.devaraj"Baghani, Omid"https://zbmath.org/authors/?q=ai:baghani.omid"Kanagarajan, Kuppusamy"https://zbmath.org/authors/?q=ai:kanagarajan.kuppusamySummary: We develop the theory of hybrid fractional differential equations with the complex order \(\theta\in \mathbb{C}\), \(\theta=m+i\alpha\), \(0<m\leq 1\), \(\alpha\in \mathbb{R}\), in Caputo sense. Using Dhage's type fixed point theorem for the product of abstract nonlinear operators in Banach algebra; one of the operators is \(\mathfrak{D}\)-Lipschitzian and the other one is completely continuous, we prove the existence of mild solutions of initial value problems for hybrid fractional differential equations. Finally, an application to solve one-variable linear fractional Schrödinger equation with complex order is given.Initial time difference quasilinearization method for fractional differential equations involving generalized Hilfer fractional derivative.https://zbmath.org/1449.340222021-01-08T12:24:00+00:00"Kucche, Kishor D."https://zbmath.org/authors/?q=ai:kucche.kishor-d"Mali, Ashwini D."https://zbmath.org/authors/?q=ai:mali.ashwini-dSummary: We present the quasilinearization method with initial time difference for nonlinear fractional differential equations (FDEs) involving generalized Hilfer fractional derivative under various conditions on the nonlinear function involved in the right hand side of the equation. An essential comparison result concerning lower and upper solutions is obtained for this generalized FDEs without demanding the Hölder continuity assumption.Time-stepping error bound for a stochastic parabolic Volterra equation disturbed by fractional Brownian motions.https://zbmath.org/1449.652562021-01-08T12:24:00+00:00"Qi, Ruisheng"https://zbmath.org/authors/?q=ai:qi.ruisheng"Lin, Qiu"https://zbmath.org/authors/?q=ai:lin.qiuSummary: In this paper, we consider a stochastic parabolic Volterra equation driven by the infinite dimensional fractional Brownian motion with Hurst parameter \(H \in \left[ {\frac{1}{2}, 1} \right)\). We apply the piecewise constant, discontinuous Galerkin method to discretize this equation in the temporal direction. Based on the explicit form of the scalar resolvent function and the refined estimates for the Mittag-Leffler function, we derive sharp mean-square regularity results for the mild solution. The sharp regularity results enable us to obtain the optimal error bound of the time discretization. These theoretical findings are finally accompanied by several numerical examples.Analytical study of fractional-order multiple chaotic FitzHugh-Nagumo neurons model using multistep generalized differential transform method.https://zbmath.org/1449.920072021-01-08T12:24:00+00:00"Momani, Shaher"https://zbmath.org/authors/?q=ai:momani.shaher-m"Freihat, Asad"https://zbmath.org/authors/?q=ai:freihat.asad-a"AL-Smadi, Mohammed"https://zbmath.org/authors/?q=ai:al-smadi.mohammed-hSummary: The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation. The fractional derivatives are described in the Caputo sense. Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method. The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient.On new generalized Ostrowski type integral inequalities.https://zbmath.org/1449.260372021-01-08T12:24:00+00:00"Qayyum, A."https://zbmath.org/authors/?q=ai:qayyum.atif|qayyum.ather|qayyum.afshan"Shoaib, M."https://zbmath.org/authors/?q=ai:shoaib.muhammad|shoaib.m-u|shoaib.mobien"Matouk, A. E."https://zbmath.org/authors/?q=ai:matouk.a-e"Latif, M. A."https://zbmath.org/authors/?q=ai:latif.md-abdul|latif.muhammad-amer|latif.muhammed-amerSummary: The Ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The aim of this paper is to establish some new inequalities similar to the Ostrowski's inequality. The current paper obtains bounds for the deviation of a function from a combination of integral means over the end intervals covering the entire interval in terms of the norms of the second derivative of the function. Some new perturbed results are obtained. Application for cumulative distribution function is also discussed.Existence and stability results for nonlocal initial value problems for differential equations with Hilfer fractional derivative.https://zbmath.org/1449.340122021-01-08T12:24:00+00:00"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffak"Bouriah, Soufyane"https://zbmath.org/authors/?q=ai:bouriah.soufyane"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseSummary: In this paper, we establish sufficient conditions for the existence and stability of solutions for a class of nonlocal initial value problems for differential equations with Hilfer's fractional derivative. The arguments are based upon the Banach contraction principle. Two examples are included to show the applicability of our results.Finite difference methods for the time fractional advection-diffusion equation.https://zbmath.org/1449.651942021-01-08T12:24:00+00:00"Ma, Yan"https://zbmath.org/authors/?q=ai:ma.yan"Musbah, F. S."https://zbmath.org/authors/?q=ai:musbah.f-sSummary: In this paper, three implicit finite difference methods are developed to solve a one dimensional time fractional advection-diffusion equation. The fractional derivative is treated by applying the right shifted Grunwald-Letnikov formula of order \(\alpha \in (0, 1)\). We investigate the stability analysis by using the von Neumann method with mathematical induction and prove that these three proposed methods are unconditionally stable. Numerical results are presented to demonstrate the effectiveness of the schemes mentioned in this paper.Schur power convexity of binary geometry Bonferroni mean.https://zbmath.org/1449.260132021-01-08T12:24:00+00:00"Wang, Chunyong"https://zbmath.org/authors/?q=ai:wang.chunyong"Tao, Shengda"https://zbmath.org/authors/?q=ai:tao.shengda"Chen, Disan"https://zbmath.org/authors/?q=ai:chen.disan"Lu, Liang"https://zbmath.org/authors/?q=ai:lu.liangSummary: In this paper, we study the Schur power convexity of binary geometric Bonferroni mean, give the necessary and sufficient conditions for the determination, and reveal some properties.Existence and stability of Langevin equations with two Hilfer-Katugampola fractional derivatives.https://zbmath.org/1449.340202021-01-08T12:24:00+00:00"Ibrahim, Rabha W."https://zbmath.org/authors/?q=ai:ibrahim.rabha-waell"Harikrishnan, Sugumaran"https://zbmath.org/authors/?q=ai:harikrishnan.sugumaran"Kanagarajan, Kuppusamy"https://zbmath.org/authors/?q=ai:kanagarajan.kuppusamySummary: In this note, we discuss the existence, uniqueness and stability results for a general class of Langevin equations. We suggest the generalization via the Hilfer-Katugampola fractional derivative. We introduce some conditions for existence and uniqueness of solutions. We utilize the concept of fixed point theorems (Krasnoselskii fixed point theorem (KFPT), Banach contraction principle (BCP)). Moreover, we illustrate definitions of the Ulam type stability. These definitions generalize the fractional Ulam stability.Hermite-Hadamard type inequality for \( ({\eta_1}, {\eta_2})\)-convex functions.https://zbmath.org/1449.260392021-01-08T12:24:00+00:00"Shi, Tongye"https://zbmath.org/authors/?q=ai:shi.tongye"Zeng, Zhihong"https://zbmath.org/authors/?q=ai:zeng.zhihong"Cao, Junfei"https://zbmath.org/authors/?q=ai:cao.junfeiSummary: \( ({\eta_1}, {\eta_2})\)-convex functions with \({\eta_1}\) satisfying condition \(C\) are considered. Starting from the definition of \( ({\eta_1}, {\eta_2})\)-convex functions, the right Hermite-Hadamard type inequality of \( ({\eta_1}, {\eta_2})\)-convex functions and its refinement are established in the case where \({\eta_2}\) is bounded from above, the left Hermite-Hadamard type inequality involving the integral of function \({\eta_2}\) for \( ({\eta_1}, {\eta_2})\)-convex functions and its generalization are established under the condition that \({\eta_2}\) is bounded from above and measurable. Three integral inequalities of differentiable \( ({\eta_1}, {\eta_2})\)-convex functions are also given.Accurate splitting approach to characterize the solution set of boundary layer problems.https://zbmath.org/1449.340312021-01-08T12:24:00+00:00"Sayevand, Khosro"https://zbmath.org/authors/?q=ai:sayevand.khosro"Machado, Jose Antonio Tenreiro"https://zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiroSummary: The boundary layer (BL) is an important concept and refers to the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. This paper studies singularly perturbed fractional differential equations where the fractional derivatives are defined in the Caputo sense. The solution of such equations, with appropriate boundary conditions, displays BL behavior. The solution out of the BL is estimated by the solution of the reduced problem and the layer solution is approximated by means of a modified truncated Chebyshev series. The coefficients of the truncated series are evaluated using a novel operational matrix technique. Moreover, the stability and the error analysis of the proposed method are analyzed. Several examples illustrate the validity and applicability of the method.Conformable fractional approximation by max-product operators.https://zbmath.org/1449.410202021-01-08T12:24:00+00:00"Anastassiou, George A."https://zbmath.org/authors/?q=ai:anastassiou.george-aSummary: Here we study the approximation of functions by a big variety of Max-product operators under conformable fractional differentiability. These are positive sublinear operators. Our study is based on our general results about positive sublinear operators. We produce Jackson type inequalities under conformable fractional initial conditions. So our approach is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of a high order conformable fractional derivative of the function under approximation.Several Hardy-type inequalities on nonconvex domains.https://zbmath.org/1449.260492021-01-08T12:24:00+00:00"Zheng, Qianqian"https://zbmath.org/authors/?q=ai:zheng.qianqian"Ma, Yali"https://zbmath.org/authors/?q=ai:ma.yali"Shen, Xiaomin"https://zbmath.org/authors/?q=ai:shen.xiaomin"Jin, Yongyang"https://zbmath.org/authors/?q=ai:jin.yongyangSummary: A class of Hardy inequalities for nonconvex domain on anisotropic Heisenberg group was obtained by constructing special vector-valued auxiliary functions and careful calculation. Moreover, several Hardy-type inequalities associated with Greiner type vector field on nonconvex domains are also obtained in this paper.q-homotopy analysis method for solving the seventh-order time-fractional Lax's Korteweg-de Vries and Sawada-Kotera equations.https://zbmath.org/1449.354272021-01-08T12:24:00+00:00"Akinyemi, Lanre"https://zbmath.org/authors/?q=ai:akinyemi.lanreSummary: This article presents exact and approximate solutions of the seventh order time-fractional Lax's Korteweg-de Vries (7TfLKdV) and Sawada-Kotera (7TfSK) equations using the modification of the homotopy analysis method called the q-homotopy analysis method. Using this method, we construct the solutions to these problems in the form of recurrence relations and present the graphical representation to verify all obtained results in each case for different values of fractional order. Error analysis is also illustrated in the present investigation.A variant of the Fejér-Jackson inequality.https://zbmath.org/1449.260182021-01-08T12:24:00+00:00"Alzer, Horst"https://zbmath.org/authors/?q=ai:alzer.horst"Kwong, Man Kam"https://zbmath.org/authors/?q=ai:kwong.man-kamThe following nice variant of the Fejér-Jackson inequality is proved: For all natural numbers \(n\) and real numbers \(x\in [0,\pi]\) we have \[-0.05781\ldots=\] \[-(5/48)\sqrt{130-58\sqrt{5}}\leq F_{n}(x),\] where \(F_{n}(x)=\sum_{k=1}^{n}\left( -1\right) ^{k+1}\left( \frac {\sin\left( (2k-1)x\right)}{2k-1}+\frac{\sin\left( 2kx\right)}{2k}\right) ;\) the sign of equality holds if and only if \(n=2\) and \(x=4\pi/5\).
Reviewer: Constantin Niculescu (Craiova)A class of quasi-fractional Noether's theorems for nonconservative systems in event space.https://zbmath.org/1449.370462021-01-08T12:24:00+00:00"Wang, Ze"https://zbmath.org/authors/?q=ai:wang.ze"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.10|zhang.yi.3|zhang.yi.11|zhang.yi|zhang.yi.8|zhang.yi.2|zhang.yi.7|zhang.yi.12|zhang.yi.5|zhang.yi.1|zhang.yi.9|zhang.yi.4Summary: To study the symmetry and conserved quantity of fractional non-conservative dynamic systems, the Noether theorem based on El-Nabulsi periodic law quasi-fractional model in event space is proposed and studied. Firstly, the fractional order variational problem based on the El-Nabulsi periodic law quasi-fractional model is established in the event space, and the differential equations of the holonomic nonconservative system and the nonholonomic nonconservative system are derived. Secondly, based on the invariance of the action functional under the infinitesimal transformation, the definition and criterion of the Noether symmetric transform and the Noether quasi-symmetric transformation are given. Finally, the Noether theorem based on the El-Nabulsi periodic law quasi-fractional model in the event space is proposed and proved. Two examples are given to illustrate the application of the results.Sobolev-type fractional stochastic differential equations driven by fractional Brownian motion with non-Lipschitz coefficients.https://zbmath.org/1449.601122021-01-08T12:24:00+00:00"Zhan, Wentao"https://zbmath.org/authors/?q=ai:zhan.wentao"Li, Zhi"https://zbmath.org/authors/?q=ai:li.zhi|li.zhi.1Summary: In this paper, we are concerned with the existence and uniqueness of mild solution for a class of nonlinear fractional Sobolev-type stochastic differential equations driven by fractional Brownian motion with Hurst parameter \(H \in (1/2, 1)\) in Hilbert spaces. We obtain the required result by using semigroup theory, stochastic analysis principle, fractional calculus and Picard iteration techniques with some non-Lipschitz conditions.Condition for half-discrete Hilbert-type inequality with homogeneous kernel to take the best constant factor and its applications.https://zbmath.org/1449.260322021-01-08T12:24:00+00:00"Hong, Yong"https://zbmath.org/authors/?q=ai:hong.yong"Zeng, Zhihong"https://zbmath.org/authors/?q=ai:zeng.zhihongSummary: By using the method of weight function and the technique of real analysis, a half-discrete Hilbert-type inequality with homogeneous kernel \(\int_0^{+\infty}\sum\limits_{n=1}^\infty K (n,x){a_n}f (x){\mathrm{d}}x \le M{||\tilde a||_{p,\alpha}} {||f||_{q,\beta}}\) and conditions to achieve the beat constant factor are discussed, and their applications in operator theory are considered.The sign of infinity based on the theory of limit of real-valued functions.https://zbmath.org/1449.260012021-01-08T12:24:00+00:00"Zhi, Yuanhong"https://zbmath.org/authors/?q=ai:zhi.yuanhong"He, Qinghai"https://zbmath.org/authors/?q=ai:he.qinghaiSummary: By comparing two methods of extension of real numbers, that is, the affine extension, and the projective extension, together with studying and summarizing the advantages and drawbacks when introducing, in real analysis, especially in the limit theory of real-value functions, the signed infinities \(-\infty, +\infty\) and the unsigned infinity \(\infty\). It is obtained that, without introduction of \(\lim\limits_{x\to {x_0}}f (x) = \infty\), there are, in real analysis, no confusions of notation for limit expression any longer, and the limit theory of real-valued functions is more precise, by just using only \(+\infty\) and \(-\infty\), and hence, the depiction of local properties of functions near the limit point is easier and more precise. Moreover, without unsigned infinity, the whole limit theory of real-valued functions is more clear and unified, which makes it possible to generalize some important theorems in real analysis, with neat proofs and multitude of applications.Problem with shift for the third-order equation with discontinuous coefficients.https://zbmath.org/1449.353262021-01-08T12:24:00+00:00"Repin, Oleg Aleksandrovich"https://zbmath.org/authors/?q=ai:repin.oleg-aleksandrovich"Kumykova, Svetlana Kanshubievna"https://zbmath.org/authors/?q=ai:kumykova.svetlana-kanshubievnaSummary: The unique solvability of boundary value problem with Saigo operators for the third-order equation with multiple characteristics was investigated. The uniqueness theorem with constraints of inequality type on the known functions and different orders of generalized fractional integro-differentiation was proved. The existence of solution is equivalently reduced to the solvability of Fredholm integral equation of the second kind.New Brunn-Minkowski type inequalities for general width-integral of index \(i\).https://zbmath.org/1449.260482021-01-08T12:24:00+00:00"Zhang, Xuefu"https://zbmath.org/authors/?q=ai:zhang.xiufu"Wu, Shanhe"https://zbmath.org/authors/?q=ai:wu.shanheSummary: Recently, the general width-integral of index \(i\) was introduced and some of its isoperimetric inequalities were established. In this paper, we establish some new Brunn-Minkowski type inequalities for general width-integral of index \(i\).Asymptotic behavior of ``intermediate point'' in the high order Cauchy mean value theorem for two variable functions.https://zbmath.org/1449.260112021-01-08T12:24:00+00:00"Zhang, Shuyi"https://zbmath.org/authors/?q=ai:zhang.shuyi"Nie, Hui"https://zbmath.org/authors/?q=ai:nie.huiSummary: The purpose of this paper is to study asymptotic behavior of the ``intermediate point'' \( ({x_0}+\theta \Delta x, {y_0}+\theta \Delta y)\) in the high order Cauchy mean value theorem for two variable functions when point \(B ({x_0}+\Delta x, {y_0}+\Delta y)\) approaches point \(A ({x_0}, {y_0})\) along line segment \(BA\). By using the concept for comparison function, several asymptotic estimation formulas of the ``intermediate point'' \( ({x_0}+\theta \Delta x, {y_0}+\theta \Delta y)\) in the high order Cauchy mean value theorem for two variable functions are established under certain conditions.A full-planar Hilbert integral inequality related to middle-variable of exponential function.https://zbmath.org/1449.260432021-01-08T12:24:00+00:00"Wang, Aizhen"https://zbmath.org/authors/?q=ai:wang.aizhen"Yang, Bicheng"https://zbmath.org/authors/?q=ai:yang.bicheng(no abstract)Estimation of unknown function of a class of double integral inequalities.https://zbmath.org/1449.260332021-01-08T12:24:00+00:00"Huang, Xingshou"https://zbmath.org/authors/?q=ai:huang.xingshou"Wang, Wusheng"https://zbmath.org/authors/?q=ai:wang.wusheng"Luo, Ricai"https://zbmath.org/authors/?q=ai:luo.ricaiSummary: A class of nonlinear double integral inequalities is studied, which includes an unknown function and its derivative function in integrand function, and a nonconstant factor outside integral sign. The upper bound of the unknown function in the integro-differential inequality is estimated explicitly using the techniques of change of variable, the method of amplification, and inverse function technique. The derived results can be applied in the study of the explicit upper bounds of solutions of a class of integro-differential equations.An effective method of controller design for uncertain fractional T-S fuzzy systems.https://zbmath.org/1449.930592021-01-08T12:24:00+00:00"Zhang, Xuefeng"https://zbmath.org/authors/?q=ai:zhang.xuefeng"Liu, Yangyang"https://zbmath.org/authors/?q=ai:liu.yangyangSummary: Considering the stability and stabilization of a class of nonlinear fractional order systems, based on the linear matrix inequality (LMI) approach, fractional order T-S fuzzy systems are studied. Using the method of parallel distributed compensation, controllers of fractional order T-S fuzzy systems are designed. Considering the fractional order T-S fuzzy systems with the order \(\alpha\) satisfying \(0 < \alpha < 1\), stabilization criterion is given in terms of LMI, which can be solved by Matlab. This criterion can handle the problems of the stability and stabilization of fractional order T-S fuzzy systems which have positive real eigenvalues, while maintaining the consistency with the stability criterion of fractional order systems from Matignon. The limitation and conservatism of the eigenvalues in negative real parts in the other methods are solved. Numerical simulation results verify the effectiveness of the proposed controller design method.Stability and convergence of difference schemes for the multi-term time-fractional diffusion equation with generalized memory kernels.https://zbmath.org/1449.651882021-01-08T12:24:00+00:00"Khibiev, Aslanbek Khizirovich"https://zbmath.org/authors/?q=ai:khibiev.aslanbek-khizirovichSummary: In this paper, a priori estimate for the corresponding differential problem is obtained by using the method of the energy inequalities. We construct a difference analog of the multi-term Caputo fractional derivative with generalized memory kernels (analog of L1 formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the \((2-\alpha_0)\)-th order in time for the generalized multi-term time-fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid \(L_2 \)-norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by numerical calculations carried out for some test problems.On two special functions, generalizing the Mittag-Leffler type function, their properties and applications.https://zbmath.org/1449.330192021-01-08T12:24:00+00:00"Ogorodnikov, Evgeniĭ Nikolaevich"https://zbmath.org/authors/?q=ai:ogorodnikov.evgenii-nikolaevichSummary: Two special functions, concerning Mittag-Leffler type functions, are studied. The first is the modification of generalized Mittag-Leffler function, which was introduced by A. A. Kilbas and M. Saigo; the second is the special case of the first one. The relation of these functions with some elementary and special functions and their role in solving of Abel-Volterra integral equations is indicated. The formulas of the fractional integration and differentiation in sense of Riemann-Liouville and Kober are presented. The applications to Cauchy type problems for some linear fractional differential equations with Riemann-Liouville and Kober derivatives are noticed.Invariant subspaces and exact solutions for a system of fractional PDEs in higher dimensions.https://zbmath.org/1449.354352021-01-08T12:24:00+00:00"Choudhary, Sangita"https://zbmath.org/authors/?q=ai:choudhary.sangita"Prakash, P."https://zbmath.org/authors/?q=ai:prakash.pankaj|prakash.pradyot|prakash.p-v|prakash.prem|prakash.periasamy"Daftardar-Gejji, Varsha"https://zbmath.org/authors/?q=ai:daftardar-gejji.varshaSummary: In this article, we develop an invariant subspace method for a system of time-fractional nonlinear partial differential equations in \((1+2)\) dimensions. Efficacy of the method is demonstrated by solving coupled system of nonlinear time-fractional diffusion equations and coupled system of time-fractional Burger's equations in higher dimensions. Furthermore, the algorithmic approach to find more than one invariant subspace is proposed and corresponding exact solutions are constructed.Some new properties of Morgan-Voyce polynomials.https://zbmath.org/1449.260162021-01-08T12:24:00+00:00"Pei, Yanni"https://zbmath.org/authors/?q=ai:pei.yanni"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.6|wang.yi.1|wang.yi.10|wang.yi.2|wang.yi.3|wang.yi.4|wang.yi.9|wang.yi.7|wang.yi.5|wang.yi.8Summary: We show that zeros of Morgan-Voyce polynomials are dense in the closed interval \([-4, 0]\). We show also that coefficients of Morgan-Voyce polynomials are approximately normally distributed and that the coefficient arrays are totally positive matrices.Some integral inequalities for interval-valued functions on time scales.https://zbmath.org/1449.260302021-01-08T12:24:00+00:00"Guo, Yuanyuan"https://zbmath.org/authors/?q=ai:guo.yuanyuan"Ye, Guoju"https://zbmath.org/authors/?q=ai:ye.guoju"Zhao, Dafang"https://zbmath.org/authors/?q=ai:zhao.dafang"Liu, Wei"https://zbmath.org/authors/?q=ai:liu.wei.7Summary: In this paper, we introduce the concept of the Riemann \({\diamondsuit_\alpha}\)-integral for interval-valued functions on time scales and discuss the basic properties of this integral. By using the interval analysis and the integral theory on time scales, we obtain Jensen's, Hölder's and Minkowski's inequalities of the Riemann \({\diamondsuit_\alpha}\)-integral for interval-valued functions, which generalize the results of existing literatures.On Riemann integration of vector-valued functions.https://zbmath.org/1449.260082021-01-08T12:24:00+00:00"Yang, Zhichun"https://zbmath.org/authors/?q=ai:yang.zhichun"Wei, Zhou"https://zbmath.org/authors/?q=ai:wei.zhouSummary: We study the relationship between the Riemann integration and the continuity almost everywhere of vector-valued functions and the property of Lebesgue (that is, every Riemann integrable vector-valued function is continuous almost everywhere). We prove that \({l^p}\) (\(1 < p < \infty\)) and \({l^\infty}\) do not possess the property of Lebesgue by constructing two counterexamples. Further, based on the existing literature, we show that \({l^1}\) has the property of Lebesgue.\(k\)-fractional integral inequalities of Hadamard type for \((h-m)\)-convex functions.https://zbmath.org/1449.260282021-01-08T12:24:00+00:00"Farid, Ghulam"https://zbmath.org/authors/?q=ai:farid.ghulam"Rehman, Atiq Ur"https://zbmath.org/authors/?q=ai:rehman.atiq-ur"Ul Ain, Qurat"https://zbmath.org/authors/?q=ai:ul-ain.quratSummary: In this paper, we establish Hadamard type fractional integral inequalities for a more general class of functions that is the class of \((h-m)\)-convex functions. These results are due to Riemann-Liouville (RL) \(k\)-fractional integrals: a generalization of RL fractional integrals. Several known results are special cases of proved results.On some applicable approximations of Gaussian type integrals.https://zbmath.org/1449.330232021-01-08T12:24:00+00:00"Chesneau, Christophe"https://zbmath.org/authors/?q=ai:chesneau.christophe"Navarro, Fabien"https://zbmath.org/authors/?q=ai:navarro.fabienSummary: In this paper, we introduce new applicable approximations for Gaussian type integrals. A key ingredient is the approximation of the function \(e^{-x^2}\) by the sum of three simple polynomial-exponential functions. Five special Gaussian type integrals are then considered as applications. Approximation of the so-called Voigt error function is investigated.Direct estimates for Stancu variant of Lupaş-Durrmeyer operators based on Polya distribution.https://zbmath.org/1449.410222021-01-08T12:24:00+00:00"Mishra, Lakshmi Narayan"https://zbmath.org/authors/?q=ai:mishra.lakshmi-narayan"Kumar, Alok"https://zbmath.org/authors/?q=ai:kumar.alokSummary: In this paper, we study approximation properties of a family of linear positive operators and establish the Voronovskaja type asymptotic formula, local approximation and pointwise estimates using the Lipschitz type maximal function. In the last section, we consider the King type modification of these operators to obtain better estimates.The semi-continuity and convexity of interval-value mapping.https://zbmath.org/1449.260042021-01-08T12:24:00+00:00"Li, Na"https://zbmath.org/authors/?q=ai:li.na"Bao, Yu'e"https://zbmath.org/authors/?q=ai:bao.yue(no abstract)Non-Archimedean stability of nonhomogeneous second order linear differential equations.https://zbmath.org/1449.340442021-01-08T12:24:00+00:00"Majani, Hamid"https://zbmath.org/authors/?q=ai:majani.hamidSummary: Let \((\mathbb{R},|\,|)\) be non-Archimedean normed space of real numbers. In this paper, we prove the Hyers-Ulam stability of nonhomogeneous second order linear differential equations with non-constant coefficients,
\[
y'' +f(x) y' +g(x)y=h(x)
\]
in the non-Archimedean normed space \((\mathbb{R},|\,|)\), where \(f\), \(g\), \(h:(a,b)\subseteq \mathbb{R}\to\mathbb{R}\) are given continuous functions.Jacobi collocation methods for solving generalized space-fractional Burgers' equations.https://zbmath.org/1449.652802021-01-08T12:24:00+00:00"Wu, Qingqing"https://zbmath.org/authors/?q=ai:wu.qingqing"Zeng, Xiaoyan"https://zbmath.org/authors/?q=ai:zeng.xiaoyanSummary: The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers' equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi-Gauss-Lobatto collocation points. By means of the simplified Jacobi operational matrix, we produce the differentiation matrix and transfer the space-fractional Burgers' equation into a system of ordinary differential equations that can be solved by the fourth-order Runge-Kutta method. The numerical simulations indicate that the Jacobi spectral collocation method is highly accurate and fast convergent for the generalized space-fractional Burgers' equation.Galerkin finite element method for 2D Riesz fractional differential equation based on unstructured meshes.https://zbmath.org/1449.653112021-01-08T12:24:00+00:00"Bu, Weiping"https://zbmath.org/authors/?q=ai:bu.weipingSummary: Galerkin finite element method is developed for the two-dimensional Riesz fractional diffusion equations based on Dirichlet boundary conditions. The Lagrange linear piecewise polynomial is employed as the basic function. Based on triangle unstructured meshes, the implementation of the finite element method for fractional differential equations is described in detail. Compared with the existing methods, the developed method efficiently reduces the computational cost and increases the accuracy of the stiffness matrix. Finally, some numerical tests are given to verify the effectiveness of the devised method.Numerical study on the modulational instability of space fractional Schrödinger equation.https://zbmath.org/1449.652752021-01-08T12:24:00+00:00"Li, Wenbin"https://zbmath.org/authors/?q=ai:li.wenbin"Wang, Dongling"https://zbmath.org/authors/?q=ai:wang.donglingSummary: Modulational instability is widely used in mathematics and physics. In this work, we mainly used splitting Fourier spectral method to numerically calculate the space fractional Schrödinger equation and deduced the modulational instability condition of space fractional Schrödinger equation by the Benjamin-Feir-Lighthill criterion. Then we studied the different modulational instability behavior of space fractional Schrödinger equation in different initial conditions, and we also compared it with the integer order Schrödinger equation. The comparison results show that the modulational instability behavior of the integer order Schrödinger equation can be applied to fractional Schrödinger equations as well.Multigrid methods for time-fractional evolution equations: a numerical study.https://zbmath.org/1449.652482021-01-08T12:24:00+00:00"Jin, Bangti"https://zbmath.org/authors/?q=ai:jin.bangti"Zhou, Zhi"https://zbmath.org/authors/?q=ai:zhou.zhiSummary: In this work, we develop an efficient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of order \(\alpha (0,1)\) in time. The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear finite elements in space and corrected high-order BDF convolution quadrature in time. At each time step, instead of solving the linear algebraic system exactly, we employ a multigrid iteration with a Gauss-Seidel smoother to approximate the solution efficiently. Illustrative numerical results for nonsmooth problem data are presented to demonstrate the approach.An implicit difference approximation for the variable coefficients space fractional diffusion equation.https://zbmath.org/1449.651922021-01-08T12:24:00+00:00"Liu, Dongbing"https://zbmath.org/authors/?q=ai:liu.dongbing"Tan, Qianrong"https://zbmath.org/authors/?q=ai:tan.qianrong"Liu, Tao"https://zbmath.org/authors/?q=ai:liu.tao.1Summary: The variable coefficient space fractional diffusion equation is considered. An implicit difference scheme is constructed, which is unconditionally stable and is super-linearly convergent about space step, and the convergence order of the method is \(O ({h^\alpha} + \tau)\) \((1 < \alpha \le 2)\). Finally, an example is presented to show that the numerical analysis is right and the method is feasible and efficient.On the \((\alpha,\beta)\)-Scott-Blair anti-Zener arrangement.https://zbmath.org/1449.340192021-01-08T12:24:00+00:00"Hassouna, M."https://zbmath.org/authors/?q=ai:hassouna.m"Ouhadan, A."https://zbmath.org/authors/?q=ai:ouhadan.abdelaziz"El Kinani, E. H."https://zbmath.org/authors/?q=ai:el-kinani.el-hassanSummary: In this paper, we study some fractional rheological anti-Zener equations that are derived from the ordinary anti-Zener model. We analyze three situations : the theoretical anti-Zener, the \(\alpha\)-Scott-Blair anti-Zener and the \((\alpha,\beta)\)-Scott-Blair anti-Zener models. The Mellin transform technique and Fox-H functions are investigated to derive relaxation modulus and the creep compliance function corresponding to each model. The limit cases are discussed.Finite difference schemes for the variable coefficients single and multi-term time-fractional diffusion equations with non-smooth solutions on graded and uniform meshes.https://zbmath.org/1449.651752021-01-08T12:24:00+00:00"Cui, Mingrong"https://zbmath.org/authors/?q=ai:cui.mingrongSummary: A finite difference scheme for the variable coefficients subdiffusion equations with non-smooth solutions is constructed and analyzed. The spatial derivative is discretized on a uniform mesh, and an \(L1\) approximation is used for the discretization of the fractional time derivative on a possibly graded mesh. The stability of the proposed scheme is given using the discrete energy method. The numerical scheme is \(\mathcal{O} (N^{-\min \{2-\alpha, r\alpha\}})\) accurate in time, where \(\alpha\) \((0 < \alpha < 1)\) is the order of the fractional time derivative, \(r\) is an index of the mesh partition, and it is second order accurate in space. The extension to multi-term time-fractional problems with nonhomogeneous boundary conditions is also discussed, with the stability and error estimate proved both in the discrete \({l^2}\)-norm and the \({l^\infty}\)-norm on the nonuniform temporal mesh. Numerical results are given for both the two-dimensional single and multi-term time-fractional equations.Bounds for the difference between two Čebyšev functionals.https://zbmath.org/1449.260222021-01-08T12:24:00+00:00"Alomari, Mohammad W."https://zbmath.org/authors/?q=ai:alomari.mohammad-wajeehSummary: In this work, a generalization of pre-Grüss inequality is established. Several bounds for the difference between two Čebyšev functional are proved.Identifying inverse source for fractional diffusion equation with Riemann-Liouville derivative.https://zbmath.org/1449.354502021-01-08T12:24:00+00:00"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Zhou, Yong"https://zbmath.org/authors/?q=ai:zhou.yong.1"Long, Le Dinh"https://zbmath.org/authors/?q=ai:long.le-dinh"Can, Nguyen Huu"https://zbmath.org/authors/?q=ai:can.nguyen-huuSummary: In this work, we study an inverse problem to determine an unknown source term for fractional diffusion equation with Riemann-Liouville derivative. In general, the problem is severely ill posed in the sense of Hadamard. To regularize the unstable solution of the problem, we have applied the quasi-boundary value method. In the theoretical result, we show the error estimate between the exact solution and regularized solution with a priori parameter choice rules and analyze it. Eventually, a numerical example has been carried out, the result shows that our regularization method is converged.A new modification of Durrmeyer type mixed hybrid operators.https://zbmath.org/1449.410212021-01-08T12:24:00+00:00"Kajla, Arun"https://zbmath.org/authors/?q=ai:kajla.arun"Acar, Tuncer"https://zbmath.org/authors/?q=ai:acar.tuncerSummary: In 2008, \textit{V. Miheşan} [Creat. Math. Inform. 17, No. 3, 466--472 (2008; Zbl 1265.41055)] constructed a general class of linear positive operators generalizing the Szász operators. In this article, a Durrmeyer variant of these operators is introduced which is a method to approximate the Lebesgue integrable functions. First, we derive some indispensable auxiliary results in the second section. We present a quantitative Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. The rate of convergence for differential functions whose derivatives are of bounded variation is also obtained.Numerical method of solution of the problem on transposition of two-sided derivative of the fractional order.https://zbmath.org/1449.653032021-01-08T12:24:00+00:00"Beĭbalaev, V. D."https://zbmath.org/authors/?q=ai:beibalaev.vetlugin-dzhabrilovich|beibalaev.vertlugin-dzhabrailovichSummary: Numerical method of the solution of the problem of heat transposition with two-sided derivative of the fractional order along the space variable and with the fractional order derivative in time is studied. Finite-different scheme was constructed and the stability of this different scheme was proven.On some parameters in the space of regulated functions and their applications.https://zbmath.org/1449.260032021-01-08T12:24:00+00:00"Cichoń, Kinga"https://zbmath.org/authors/?q=ai:cichon.kinga"Cichoń, Mieczysław"https://zbmath.org/authors/?q=ai:cichon.mieczyslaw"Metwali, Mohamed M. A."https://zbmath.org/authors/?q=ai:metwali.mohamed-m-aSummary: In this paper, we study a class of discontinuous functions being a space of solutions for some differential and integral equations. We investigate functions having finite one-sided limits, i.e. regulated functions. In the space of such functions, we introduce some new concepts like a modulus of equi-regularity or a measure of noncompactness, allowing us to unify the proofs for the results about existence for both continuous and discontinuous solutions. An example of applications for quadratic integral equations, essentially improving earlier ones, completes the paper.A new picture fuzzy information measure based on Tsallis-Havrda-Charvat concept with applications in presaging poll outcome.https://zbmath.org/1449.940482021-01-08T12:24:00+00:00"Joshi, Rajesh"https://zbmath.org/authors/?q=ai:joshi.rajeshSummary: The picture fuzzy set (PFS) proposed by Cuong and Kreinovich are well suitable to capture the uncertain information in vague circumstances. The main objective of this communication is to propose a new framework as a criteria of fuzzy entropy for PFSs. Further, a new picture fuzzy information measure based on Tsallis-Havrda-Charvat entropy is proposed and validated in accordance with newly proposed framework. Besides this, some major properties of proposed information measure are also discussed. Apart from this, a new multi-criteria decision-making method using the concept of VIKOR (Vlsekriterijumska Optimizacija i Kompromisno Resenje) based on relative projection is proposed. To show the practical utility of proposed decision-making method, two numerical examples based on election forecast through opinion polls have been discussed.Design of a soft variable structure controller for synchronization of fractional-order chaotic systems with different structures.https://zbmath.org/1449.930212021-01-08T12:24:00+00:00"Shao, Keyong"https://zbmath.org/authors/?q=ai:shao.keyong"Guo, Haoxuan"https://zbmath.org/authors/?q=ai:guo.haoxuan"Han, Feng"https://zbmath.org/authors/?q=ai:han.feng"Wang, Tingting"https://zbmath.org/authors/?q=ai:wang.tingtingSummary: Based on the fractional calculus and the Mittag-Leffler stability theory, the synchronization of fractional order chaotic systems with different structures is investigated. A soft variable structure controller is proposed for synchronization of different fractional order chaotic systems. The different structures synchronization of Chen chaotic system and Liu chaotic system are realized based on the designed soft variable structure controller. The Matlab simulation results verify the effectiveness of the designed controller.Highly accurate technique for solving distributed-order time-fractional-sub-diffusion equations of fourth order.https://zbmath.org/1449.652702021-01-08T12:24:00+00:00"Abdelkawy, M. A."https://zbmath.org/authors/?q=ai:abdelkawy.mohamed-a"Babatin, Mohammed M."https://zbmath.org/authors/?q=ai:babatin.mohammed-m"Lopes, António M."https://zbmath.org/authors/?q=ai:lopes.antonio-mSummary: This paper presents a new method for calculating the numerical solution of distributed-order time-fractional-sub-diffusion equations (DO-TFSDE) of fourth order. The method extends the shifted fractional Jacobi (SFJ) collocation scheme for discretizing both the time and space variables. The approximate solution is expressed as a finite expansion of SFJ polynomials whose derivatives are evaluated at the SFJ quadrature points. The process yields a system of algebraic equations that are solved analytically. The new method is compared with alternative numerical algorithms when solving different types of DO-TFSDE. The results show that the proposed method exhibits superior accuracy with an exponential convergence rate.Finite difference schemes for the tempered fractional Laplacian.https://zbmath.org/1449.652132021-01-08T12:24:00+00:00"Zhang, Zhijiang"https://zbmath.org/authors/?q=ai:zhang.zhijiang"Deng, Weihua"https://zbmath.org/authors/?q=ai:deng.weihua"Fan, Hongtao"https://zbmath.org/authors/?q=ai:fan.hongtaoSummary: The second and all higher order moments of the \(\beta \)-stable Lévy process diverge, the feature of which is sometimes referred to as shortcoming of the model when applied to physical processes. So, a parameter \(\lambda \) is introduced to exponentially temper the Lévy process. The generator of the new process is the tempered fractional Laplacian \({\left ({\Delta + \lambda} \right)^{\beta /2}}\) In this paper, we first design the finite difference schemes for the tempered fractional Laplacian equation with the generalized Dirichlet type boundary condition, their accuracy depends on the regularity of the exact solution on \({\bar \Omega}\). Then the techniques of effectively solving the resulting algebraic equation are presented, and the performances of the schemes are demonstrated by several numerical examples.Inequalities for functions of selfadjoint operators on Hilbert spaces: a survey of recent results.https://zbmath.org/1449.470362021-01-08T12:24:00+00:00"Dragomir, Silvestru Sever"https://zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: The main aim of this survey is to present recent results concerning inequalities for continuous functions of selfadjoint operators on complex Hilbert spaces. It is intended for use by both researchers in various fields of Linear Operator Theory and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas.Inversion of initial-value problem by means of quasi-reversibility regularization method combined with discrete random noise.https://zbmath.org/1449.652282021-01-08T12:24:00+00:00"Yang, Fan"https://zbmath.org/authors/?q=ai:yang.fan.1"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.4|zhang.yan.3|zhang.yan.2"Li, Xiaoxiao"https://zbmath.org/authors/?q=ai:li.xiaoxiaoSummary: The inversion of initial value problem of fractional diffusion equation is explored with discrete random noise. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the measured data. The quasi-reversibility regularization method is used to obtain a regularized approximate solution and the convergence estimate is given under a priori parameter choice rule. Numerical results show that this method will be effective and stable.Several companions of quasi-Grüss type inequalities for complex functions defined on unit circle.https://zbmath.org/1449.300012021-01-08T12:24:00+00:00"Zhu, Jian"https://zbmath.org/authors/?q=ai:zhu.jian"Chen, Lijuan"https://zbmath.org/authors/?q=ai:chen.lijuan"Xue, Qiaoling"https://zbmath.org/authors/?q=ai:xue.qiaolingSummary: Several companions of quasi-Grüss type inequalities for the Riemann-Stieltjes integral of continuous complex valued integrands defined on the complex unit circle \(C\left[ {0, 2\pi} \right]\) are given, while the integrator \(u\) is the case of bounded variation Lipschitzian. Some inequalities are generalized for second derivative.Implicit Runge-Kutta and spectral Galerkin methods for Riesz space fractional/distributed-order diffusion equation.https://zbmath.org/1449.652832021-01-08T12:24:00+00:00"Zhao, Jingjun"https://zbmath.org/authors/?q=ai:zhao.jingjun"Zhang, Yanming"https://zbmath.org/authors/?q=ai:zhang.yanming"Xu, Yang"https://zbmath.org/authors/?q=ai:xu.yang.1Summary: A numerical method with high accuracy both in time and in space is constructed for the Riesz space fractional diffusion equation, in which the temporal component is discretized by an \(s\)-stage implicit Runge-Kutta method and the spatial component is approximated by a spectral Galerkin method. For an algebraically stable Runge-Kutta method of order \(p\) \((p\ge s+1)\), the unconditional stability of the full discretization is proven and the convergence order of \(s+1\) in time is obtained. The optimal error estimate in space, with convergence order only depending on the regularity of initial value and \(f\), is also derived. Meanwhile, this kind of method is applied to the Riesz space distributed-order diffusion equation, and similar stability and convergence results are obtained. Finally, numerical experiments are provided to illustrate the theoretical results.Optimal bounds for a quasi-arithmetic mean in terms of other bivariate means.https://zbmath.org/1449.260452021-01-08T12:24:00+00:00"Xu, Renxu"https://zbmath.org/authors/?q=ai:xu.renxu"Xu, Huizuo"https://zbmath.org/authors/?q=ai:xu.huizuo"Qian, Weimao"https://zbmath.org/authors/?q=ai:qian.weimaoSummary: This paper deals with the inequalities involving the quasi-arithmetic means using methods of real analysis. The convex combinations of the arithmetic mean \(A (a, b)\) and the square-root \(N (a, b)\) (or Heron mean \(He (a, b)\)) for the quasi-arithmetic means \(E (a, b)\) are discussed. we find the best possible parameters \({\alpha_1}\), \({\alpha_2}\), \({\alpha_3}\), \({\alpha_4}\) and \({\beta_1}\), \({\beta_2}\), \({\beta_3}\), \({\beta_4}\) such that the double inequalities \[\begin{array}{l}{\alpha_1}A (a, b) + (1-{\alpha_1})N (a, b) < E (a, b) < {\beta_1}A (a, b) + (1-\beta_1)N (a, b)\\ \\ \frac{\alpha_2}{N (a, b)} + \frac{1-\alpha_2}{A (a, b)} < \frac{1}{E (a, b)} < \frac{\beta_2}{N (a, b)} + \frac{1-\beta_2}{A (a, b)}\\ \\ {\alpha_3}A (a, b) + (1-{\alpha_3})He (a, b) < E (a, b) < {\beta_3}A (a, b) + (1-{\beta_3})He (a, b)\\ \\ \frac{\alpha_4}{He (a, b)} + \frac{1-{\alpha_4}}{A (a, b)} < \frac{1}{E (a, b)} < \frac{\beta_4}{He (a, b)} + \frac{1-{\beta_4}}{A (a, b)}\end{array}\] hold for all \(a, b > 0\) with \(a \ne b\). As the application, four optimal inequalities for the complete elliptic integrals of the second kind are found.