Recent zbMATH articles in MSC 45https://zbmath.org/atom/cc/452021-01-08T12:24:00+00:00WerkzeugAn inventive numerical method for solving the most general form of integro-differential equations with functional delays and characteristic behavior of orthoexponential residual function.https://zbmath.org/1449.653642021-01-08T12:24:00+00:00"Kürkçü, Ömür Kıvanç"https://zbmath.org/authors/?q=ai:kurkcu.omur-kivanc"Aslan, Ersin"https://zbmath.org/authors/?q=ai:aslan.ersin"Sezer, Mehmet"https://zbmath.org/authors/?q=ai:sezer.mehmetSummary: In this study, we constitute the most general form of functional integro-differential equations with functional delays. An inventive method based on Dickson polynomials with the parameter-\( \alpha \) along with collocation points is employed to solve them. The stability of the solutions is simulated according to an interval of the parameter-\( \alpha \). A useful computer program is developed to obtain the precise values from the method. The residual error analysis is used to improve the obtained solutions. The characteristic behavior of the residual function is established with the aid of the orthoexponential polynomials. We compare the present numerical results of the method with those obtained by the existing methods in tables.Fractional differential equations and Volterra-Stieltjes integral equations of the second kind.https://zbmath.org/1449.340092021-01-08T12:24:00+00:00"Asanov, Avyt"https://zbmath.org/authors/?q=ai:asanov.avyt"Almeida, Ricardo"https://zbmath.org/authors/?q=ai:almeida.ricardo"Malinowska, Agnieszka B."https://zbmath.org/authors/?q=ai:malinowska.agnieszka-barbaraSummary: In this paper, we construct a method to find approximate solutions to fractional differential equations involving fractional derivatives with respect to another function. The method is based on an equivalence relation between the fractional differential equation and the Volterra-Stieltjes integral equation of the second kind. The generalized midpoint rule is applied to solve numerically the integral equation and an estimation for the error is given. Results of numerical experiments demonstrate that satisfactory and reliable results could be obtained by the proposed method.Bernoulli operational matrix method for the numerical solution of nonlinear two-dimensional Volterra-Fredholm integral equations of Hammerstein type.https://zbmath.org/1449.653562021-01-08T12:24:00+00:00"Bazm, Sohrab"https://zbmath.org/authors/?q=ai:bazm.sohrab"Hosseini, Alireza"https://zbmath.org/authors/?q=ai:hosseini.alirezaSummary: Two-dimensional Volterra-Fredholm integral equations of Hammerstein type are studied. Using the Banach Fixed Point Theorem, the existence and uniqueness of a solution to these equations in the space \(L^\infty ([0,1]\times [0,1])\) is proved. Then, the operational matrices of integration and product for two-variable Bernoulli polynomials are derived and utilized to reduce the solution of the considered problem to the solution of a system of nonlinear algebraic equations that can be solved by Newton's method. The error analysis is given and some examples are provided to illustrate the efficiency and accuracy of the method.The expected discounted penalty function of a risk model with linear dividend barrier.https://zbmath.org/1449.910962021-01-08T12:24:00+00:00"Chen, Jie"https://zbmath.org/authors/?q=ai:chen.jie.4|chen.jie.8|chen.jie|chen.jie.7|chen.jie.6|chen.jie.10|chen.jie.9|chen.jie.3|chen.jie.5|chen.jie.1|chen.jie.2"Yu, Yong"https://zbmath.org/authors/?q=ai:yu.yong"Shen, Ying"https://zbmath.org/authors/?q=ai:shen.ying.1|shen.ying"Liu, Jianmei"https://zbmath.org/authors/?q=ai:liu.jianmeiSummary: The operation of insurance companies will be affected by interest rates or other factors. In this study, a risk model with linear dividend barrier is established. The integro-differential equations for the expected discounted penalty function, ruin probability, survival probability and the expected discounted dividend function are obtained. If the claim sizes are exponentially distributed, the analytical expression of the integro-differential equation for the time of ruin is derived. Then the closed-form solutions of the deficit at ruin and the expected discounted dividend function are obtained.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.Convergence analysis of Wilson element for parabolic integro-differential equation.https://zbmath.org/1449.653172021-01-08T12:24:00+00:00"Liang, Conggang"https://zbmath.org/authors/?q=ai:liang.conggang"Yang, Xiaoxia"https://zbmath.org/authors/?q=ai:yang.xiaoxia"Shi, Dongyang"https://zbmath.org/authors/?q=ai:shi.dongyangSummary: In this paper, with the help of the Wilson element, new semi-discrete and fully-discrete schemes are proposed for parabolic integro-differential equation. Based on the properties of the element, through defining a new bilinear form, without using the technique of extrapolation and interpolated postprocessing, in the norm which is stronger than the usual \({H^1}\)-norm, the convergence results with order \(O (h^2)/O (h^2 + \tau)\) for the primitive solution are obtained for the corresponding schemes, respectively. The above results are just one order higher than the usual error estimates for the Wilson element. Here, \(h\) and \(\tau\) are parameters of the subdivision in space and time step, respectively. Finally, numerical results are provided to confirm the theoretical analysis.An algorithm for solving \(m\)th-order nonlinear Volterra-Fredholm integro-differential equations.https://zbmath.org/1449.651702021-01-08T12:24:00+00:00"Zhao, Xiaoxu"https://zbmath.org/authors/?q=ai:zhao.xiaoxu"Li, Meiyi"https://zbmath.org/authors/?q=ai:li.meiyi"Lv, Xueqin"https://zbmath.org/authors/?q=ai:lu.xueqinSummary: For the \(m\)th-order nonlinear Volterra-Fredholm integro-differential equations, the Legendre-Galerkin method is proposed to solve them. The Legendre polynomials are chosen as basis functions, the finite dimensional equations are obtained by orthogonal functions of the basis functions and the residuals, and the approximate solutions of the equations can be obtained by solving the finite dimensional equations. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.Mean square exponential stability of the split-step \(\theta\) method for a class of neutral stochastic delay integro-differential equations.https://zbmath.org/1449.650072021-01-08T12:24:00+00:00"Peng, Wei"https://zbmath.org/authors/?q=ai:peng.wei"Zhu, Mengjiao"https://zbmath.org/authors/?q=ai:zhu.mengjiao"Wang, Wenqiang"https://zbmath.org/authors/?q=ai:wang.wenqiangSummary: In this paper, we are concerned with the mean square exponential stability of a class of neutral stochastic delay integro-differential equations with a split-step theta (SST) method. It is shown that the mean square exponential stability of the exact solution with the drift coefficient satisfies linear growth conditions. It is also proved that the SST method with \(\theta \in [0, 1/2]\) can recover the exponential mean square stability of the exact solution with some restrictive conditions on step-size \(h < {h^*}\) and the drift coefficient, but for \(\theta \in (1/2, 1]\), the SST method can reproduce the exponential mean square stability with the step-size \(h = \tau /m\), where \(m\) is a positive integer. Finally, the numerical test verifies the correctness of the theoretical results.A kernel-based technique to solve three-dimensional linear Fredholm integral equations of the second kind over general domains.https://zbmath.org/1449.653582021-01-08T12:24:00+00:00"Esmaeili, Hamid"https://zbmath.org/authors/?q=ai:esmaeili.hamid"Moazami, Davoud"https://zbmath.org/authors/?q=ai:moazami.davoudSummary: In this article, we study a kernel-based method to solve three-dimensional linear Fredholm integral equations of the second kind over general domains. The radial kernels are utilized as a basis in the discrete collocation method to reduce the solution of linear integral equations to that of a linear system of algebraic equations. Integrals appeared in the scheme are approximately computed by the Gauss-Legendre and Monte Carlo quadrature rules. The method does not require any background mesh or cell structures, so it is mesh free and accordingly independent of the domain geometry. Thus, for the three-dimensional linear Fredholm integral equation, an irregular domain can be considered. The convergence analysis is also given for the method. Finally, numerical examples are presented to show the efficiency and accuracy of the technique.Almost sure exponential stability for some neutral partial integro-differential equations.https://zbmath.org/1449.342672021-01-08T12:24:00+00:00"Ramkumar, K."https://zbmath.org/authors/?q=ai:ramkumar.kasinathan"Mohamed, M. S."https://zbmath.org/authors/?q=ai:mohamed.mohamed-salem"Diop, Mamadou Abdoul"https://zbmath.org/authors/?q=ai:diop.mamadou-abdoulSummary: This paper is concerned with the dynamics of a delay stochastic neutral integro-differential equation in Hilbert spaces by using the theory of resolvent operator. After establishing a result ensuring the existence and uniqueness of a mild solution of this class of equations, we investigate the exponential stability of the moments of a mild solution as well as its sample paths. An example is given to illustrate the results.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.A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation.https://zbmath.org/1449.354322021-01-08T12:24:00+00:00"Cheng, Xiao-liang"https://zbmath.org/authors/?q=ai:cheng.xiaoliang"Yuan, Le-le"https://zbmath.org/authors/?q=ai:yuan.lele"Liang, Ke-wei"https://zbmath.org/authors/?q=ai:liang.keweiSummary: In this paper, we consider a Cauchy problem of the time fractional diffusion equation (TFDE) in \(x \in [0, L]\). This problem is ubiquitous in science and engineering applications. The illposedness of the Cauchy problem is explained by its solution in frequency domain. Furthermore, the problem is formulated into a minimization problem with a modified Tikhonov regularization method. The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem. The error estimates for the regularized solutions are obtained under \(H^p\) norm priori bound assumptions. Finally, numerical examples illustrate the effectiveness of the proposed method.Some coincidence and common fixed point theorems concerning \(F\)-contraction and applications.https://zbmath.org/1449.541012021-01-08T12:24:00+00:00"Tomar, Anita"https://zbmath.org/authors/?q=ai:tomar.anita"Sharma, Ritu"https://zbmath.org/authors/?q=ai:sharma.rituSummary: The aim of this paper is to establish coincidence and common fixed point theorems for a discontinuous noncompatible pair of self-maps in noncomplete metric space without containment requirement of range space of involved maps acknowledging the notion of \(F\)-contraction introduced by \textit{D. Wardowski} [Fixed Point Theory Appl. 2012, Paper No. 94, 6 p. (2012; Zbl 1310.54074)]. Our results generalize, extend and improve analogous results existing in the literature and are supported with the help of illustrative examples associated with pictographic validations to demonstrate the authenticity of the postulates. Solutions of two-point boundary value problem of a second order differential equation arising in electric circuit and a Volterra type integral equation using Ćirić type as well as Hardy-Rogers-type \(F\)-crontactions are also given to exhibit the usability of obtained results.Nonexplicit evolution Volterra integral equation of the first kind with nonlinear integral delay.https://zbmath.org/1449.450042021-01-08T12:24:00+00:00"Yuldashev, T. K."https://zbmath.org/authors/?q=ai:yuldashev.tursun-kamaldinovichSummary: We prove a theorem of existence, uniqueness and stability of solution of the implicit evolution Volterra integral equation with respect to the initial value condition on the given finite segment. Here we use the method of successive approximation in combination with the method of compressing mapping.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.Option pricing under mixed exponential jump diffusion model based on the FST method.https://zbmath.org/1449.911692021-01-08T12:24:00+00:00"Zhang, Sumei"https://zbmath.org/authors/?q=ai:zhang.sumei"Zhao, Jieqiong"https://zbmath.org/authors/?q=ai:zhao.jieqiongSummary: The mixed exponential jump-diffusion model that can approximate any distribution is widely used to describe the actual trend of stock price. Based on the Fourier space time-stepping (FST) method, this paper considers European option pricing under the mixed exponential jump-diffusion model. By the Fourier transform and the characteristic exponent, the partial integro-differential equation for pricing European options is transformed into an ordinary differential equations and solved to obtain European option prices. Numerical results indicate that the FST method is accurate and fast. Moreover, by collecting real market data and the nonlinear least squares method, we apply the obtained option price to model calibration to obtain the model parameters which match the real market. By examining the impact of jump parameters on the implied volatility, we find that the mixed exponential jump-diffusion model can well reflect the volatility ``smile'' of asset returns.A high-accuracy analysis of unconventional Hermite-type rectangular element for nonlinear parabolic integro-differential equations.https://zbmath.org/1449.652512021-01-08T12:24:00+00:00"Li, Xianzhi"https://zbmath.org/authors/?q=ai:li.xianzhi"Fan, Zhongguang"https://zbmath.org/authors/?q=ai:fan.zhongguangSummary: An unconventional Hermite-type rectangular element approximation is discussed for a class of nonlinear parabolic integro-differential equations under a semi-discrete scheme. The superclose property with order \(O (h^3)\) in \({H^1}\) norm is obtained by means of the interpolation theory, a high-accuracy analysis and the derivative transfer techniques for the time \(t\). Furthermore, the global superconvergence result is derived with the interpolated post-processing technique. At the same time, the high-accuracy extrapolation solution with order \(O (h^4)\) is deduced through constructing a suitable extrapolation scheme.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.Existence of mild solutions of second order evolution integro-differential equations in the Fréchet spaces.https://zbmath.org/1449.342662021-01-08T12:24:00+00:00"Jawahdou, Adel"https://zbmath.org/authors/?q=ai:jawahdou.adelSummary: In this article, we shall establish sufficient conditions for the existence of mild solutions for second order semilinear integro-differential evolution equations in Fréchet spaces \(C(\mathbb{R}_+ , E)\), where \(E\) is a Banach space. Our approach is based on the concept of a measure of noncompactness and Tychonoff fixed point theorem. For illustration we give an example.Stability in nonlinear neutral Levin-Nohel integro-dynamic equations.https://zbmath.org/1449.450152021-01-08T12:24:00+00:00"Khelil, Kamil Ali"https://zbmath.org/authors/?q=ai:khelil.kamil-ali"Ardjouni, Abdelouaheb"https://zbmath.org/authors/?q=ai:ardjouni.abdelouaheb"Djoudi, Ahcene"https://zbmath.org/authors/?q=ai:djoudi.ahceneSummary: In this paper we use the Krasnoselskii-Burton's fixed point theorem to obtain asymptotic stability and stability results about the zero solution for the following nonlinear neutral Levin-Nohel integro-dynamic equation
\[
x^\Delta(t)+\int^t_{t-\tau(t)}a(t,s)g(x(s))\Delta s+c(t)x^{\widetilde\Delta}(t-\tau(t))=0.
\]
The results obtained here extend the work of \textit{K. A. Khelil} et al. [Korean J. Math. 25, No. 3, 303--321 (2017; Zbl 07148845)].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.Solvability of infinite system of nonlinear singular integral equations in the \(C(I \times I, c)\) space and modified semi-analytic method to find a closed-form of solution.https://zbmath.org/1449.450072021-01-08T12:24:00+00:00"Das, Anupam"https://zbmath.org/authors/?q=ai:das.anupam"Rabbani, Mohsen"https://zbmath.org/authors/?q=ai:rabbani.mohsen"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipan"Arab, Reza"https://zbmath.org/authors/?q=ai:arab.rezaSummary: In this article, we discuss about solvability of infinite systems of singular integral equations with two variables in the Banach sequence space \(C(I \times I, c)\) by applying measure of noncompactness and Meir-Keeler condensing operators. By presenting an example, we have illustrated our results. For validity of the results we introduce a modified semi-analytic method in the case of two variables to make an iteration algorithm to find a closed-form of solution for the above problem. The numerical results show that the produced sequence for approximating the solution of example is in the \(c\) space with a high accuracy.Convergence and error analysis of a spectral collocation method for solving system of nonlinear Fredholm integral equations of second kind.https://zbmath.org/1449.653632021-01-08T12:24:00+00:00"Khan, Sami Ullah"https://zbmath.org/authors/?q=ai:khan.sami-ullah"Ali, Ishtiaq"https://zbmath.org/authors/?q=ai:ali.ishtiaqSummary: This paper presents a new numerical approximation method to solve a system of nonlinear Fredholm integral equations of second kind. Spectral collocation method and their properties are applied to determine the general solution procedure for nonlinear Fredholm integral equations (FIEs). The convergence and error analysis of spectral collocation method are incorporated for the given nonlinear model. Legendre-Gauss-Lobatto (LGL) points are used as collocation points with various Legendre-Gauss quadrature with weight functions. The use of Legendre polynomials, together with the Gauss quadrature collocation points is well known for the accurate approximations that converge exponentially. Finally, we validate our theoretical results with a number of numerical examples, which further enhance the efficiency of our proposed scheme.Existence of mild solutions for a class of fractional semilinear integro-differential equation of mixed type.https://zbmath.org/1449.342772021-01-08T12:24:00+00:00"Zhu, Bo"https://zbmath.org/authors/?q=ai:zhu.bo"Han, Baoyan"https://zbmath.org/authors/?q=ai:han.baoyan"Liu, Lishan"https://zbmath.org/authors/?q=ai:liu.lishanSummary: In this paper, the authors studied the existence results of the mild solutions for a class of fractional semilinear integro-differential equation of mixed type by using the measure of noncompactness, \(k\)-set contraction and \(\beta\)-resolvent family. It is well known that the \(k\)-set contraction requires additional condition to ensure the contraction coefficient \(0 < k < 1\). We don't require additional condition to ensure the contraction coefficient \(0 < k < 1\). An example is introduced to illustrate the main results of this paper.The solving integro-differential equations of fractional order with the ultraspherical functions.https://zbmath.org/1449.653662021-01-08T12:24:00+00:00"Panahy, Saeid"https://zbmath.org/authors/?q=ai:panahy.saeid"Khani, Ali"https://zbmath.org/authors/?q=ai:khani.aliSummary: In this paper, an integration method is presented based on using ultraspherical polynomials for solving a class of linear fractional integro-differential equations of Volterra types. This method is based on a new investigation of ultraspherical integration to approximate the highest order derivative in the equations and generate approximations to the lower order derivatives through integration of the higher-order derivatives. Numerical example illustrate the efficiency and accuracy of the method.Application of Hilbert-Schmidt SVD approach to solve linear two-dimensional Fredholm integral equations of the second kind.https://zbmath.org/1449.653592021-01-08T12:24:00+00:00"Esmaeili, H."https://zbmath.org/authors/?q=ai:esmaeili.hamid"Moazami, D."https://zbmath.org/authors/?q=ai:moazami.davoudSummary: Meshfree techniques based on infinitely smooth radial kernels have the great potential to provide spectrally accurate function approximations with irregular domain in high dimensions. The maximum accuracy can mostly be found when the RBF shape parameter is small, i.e., when the radial kernel is relatively smooth. However, as the shape parameter goes to zero, the standard RBF interpolant matrix will be very ill-conditioned. The ill-conditioning can be alleviated using alternate bases. One of these alternative bases is the Hilbert-Schmidt SVD basis. The Hilbert-Schmidt SVD approach suggests a stable mechanism for replacing a set of near-flat kernels with scattered centres to a well-conditioned base for exactly the same space.
In this work, the Gaussian Hilbert-Schmidt SVD basis functions method is presented to numerically solve the linear two-dimensional Fredholm integral equations of the second kind. The method estimates the solution by the discrete collocation method based on Gaussian Hilbert-Schmidt SVD basis functions constructed on a set of scattered points. The emerged integrals in the scheme are approximately computed by the Gauss-Legendre quadrature rule. This approach reduces the problem under study to a linear system of algebraic equations which can be solved easily via applying an appropriate numerical technique. Also, the convergence of the proposed approach is established. Finally, numerical results are compared with standard RBF method to indicate the accuracy and efficiency of the suggested approach.A study on functional fractional integro-differential equations of Hammerstein type.https://zbmath.org/1449.653672021-01-08T12:24:00+00:00"Saeedi, Leila"https://zbmath.org/authors/?q=ai:saeedi.leila"Tari, Abolfazl"https://zbmath.org/authors/?q=ai:tari.abolfazl"Babolian, Esmail"https://zbmath.org/authors/?q=ai:babolian.esmailSummary: In this paper, functional Hammerstein integro-differential equations of fractional order is studied. Here, the existence and uniqueness of the solution is proved. A numerical method to approximate the solution of problem is also presented which is based on an improvement of the successive approximations method. Error estimation of the method is analyzed and error bound is obtained. The convergence and stability of the method are proved. At the end, application of the method is revealed by presenting some examples.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.B-spline wavelet collocation method for solution of nonlinear fractional Fredholm integro-differential equation.https://zbmath.org/1449.651672021-01-08T12:24:00+00:00"Lu, Wanshun"https://zbmath.org/authors/?q=ai:lu.wanshun"Yan, Jie"https://zbmath.org/authors/?q=ai:yan.jie"Ma, Xu"https://zbmath.org/authors/?q=ai:ma.xuSummary: A nonlinear fractional Fredholm integro-differential equation is studied. The equation is discretized into a set of algebraic equations with fractional integration operator matrix of B-spline wavelet, and the feasibility and effectiveness of this method are verified with numerical examples.On some conjugation problems of parabolic and hyperbolic equations with integro-differential conditions on the separating boundary.https://zbmath.org/1449.353162021-01-08T12:24:00+00:00"Eleev, Valeriĭ Abdurakhmanovich"https://zbmath.org/authors/?q=ai:eleev.valerii-abdurakhmanovich"Balkizova, Alëna Khamudbievna"https://zbmath.org/authors/?q=ai:balkizova.alena-khamudbievnaSummary: The unique solvability of the problems of conjugation of hyperbolic and parabolic equations in finite domains is proved by the method of equivalent reduction to the Volterra integral equation of the second kind.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.Neumann method for solving conformable fractional Volterra integral equations.https://zbmath.org/1449.450032021-01-08T12:24:00+00:00"Ilie, Mousa"https://zbmath.org/authors/?q=ai:ilie.mousa"Biazar, Jafar"https://zbmath.org/authors/?q=ai:biazar.jafar"Ayati, Zainab"https://zbmath.org/authors/?q=ai:ayati.zainabSummary: This paper deals with the solution of a class of Volterra integral equations in the sense of the conformable fractional derivative. For this goal, the well-organized Neumann method is developed and some theorems related to existence, uniqueness, and sufficient condition of convergence are presented. Some illustrative examples are provided to demonstrate the efficiency of the method in solving conformable fractional Volterra integral equations.Fractional evolution equations with nonlocal conditions in partially ordered Banach space.https://zbmath.org/1449.351742021-01-08T12:24:00+00:00"Nashine, Hemant Kumar"https://zbmath.org/authors/?q=ai:nashine.hemant-kumar"Yang, He"https://zbmath.org/authors/?q=ai:yang.he"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-pSummary: In the present work, we discuss the existence of mild solutions for the initial value problem of fractional evolution equation of the form \[\begin{cases} ^CD^\sigma_tx(t)+Ax(t)=f(t,x(t)),\quad t\in J:= [0,b],\\ x(0)=x_0\in X,\end{cases}\tag{A}\] where \({}^CD^\sigma_t\) denotes the Caputo fractional derivative of order \(\sigma\in(0,1),-A:D(A)\subset X\to X\) generates a positive \(C_0\)-semigroup \(T(t)(t\ge 0)\) of uniformly bounded linear operator in \(X,b >0\) is a constant, \(f\) is a given functions. For this, we use the concept of measure of noncompactness in partially ordered Banach spaces whose positive cone \(K\) is normal, and establish some basic fixed point results under the said concepts. In addition, we relaxed the conditions of boundedness, closedness and convexity of the set at the expense that the operator is monotone and bounded. We also supply some new coupled fixed point results via MNC. To justify the result, we prove an illustrative example that rational of the abstract results for fractional parabolic equations.On existence of solution of a class of quadratic-integral equations using contraction defined by simulation functions and measure of noncompactness.https://zbmath.org/1449.470912021-01-08T12:24:00+00:00"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad"Arab, Reza"https://zbmath.org/authors/?q=ai:arab.rezaSummary: In this paper we have introduced a new type of contraction condition using a class of simulation functions, in the sequel using the new contraction definition, involving measure of noncompactness; we establish few results on existence of fixed points of continuous functions defined on a subset of Banach space. This result also generalizes other related results obtained by \textit{R. Arab} [Miskolc Math. Notes 18, No. 2, 595--610 (2017; Zbl 1399.54082)] and by \textit{J. Banaś} and \textit{K. Goebel} [Measures of noncompactness in Banach spaces. New York, Basel: Marcel Dekker, Inc. (1980; Zbl 0441.47056)]. The obtained results are used in establishing existence theorems for a class of nonlinear quadratic equations (which generalizes several types of fractional-quadratic integral equations such as Abel's integral equation) defined on a closed and bounded subset of \(\mathbb{R}\). The existence of solutions is established with the aid of a measure of noncompactness defined on the function space \(C(I)\) introduced by \textit{J. Banaś} and \textit{L. Olszowy} [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 41, 13--23 (2001; Zbl 0999.47041)].Approaching simultaneous Fredholm integral equations using common fixed point theorems in complex valued metric spaces.https://zbmath.org/1449.540472021-01-08T12:24:00+00:00"Alfaqih, Waleed M."https://zbmath.org/authors/?q=ai:alfaqih.waleed-mohd"Imdad, Mohammad"https://zbmath.org/authors/?q=ai:imdad.mohammad"Gubran, Rqeeb"https://zbmath.org/authors/?q=ai:gubran.rqeebSummary: The aim of this paper is to discuss the existence and uniqueness of a common solution for the following system of linear Fredholm integral equations (of the second kind): \[u(t)=f_i(t)+\beta\int^b_aK_i(t,s)F_i(u(s)\,ds,\quad t,s\in[a,b],\] where \(\beta\in\mathbb{R}\), \(f_i,K_i\) and \(F_i\) are given continuous functions, \(i=1,2\), while \(u\) is an unknown function to be determined. To establish this, we prove a common fixed point theorem for two self-mappings defined on a complex metric space. Moreover, we prove coincidence and common fixed point theorems for two weakly compatible self-mappings defined on a complex metric space.Numerical computations of nonlocal Schrödinger equations on the real line.https://zbmath.org/1449.820042021-01-08T12:24:00+00:00"Yan, Yonggui"https://zbmath.org/authors/?q=ai:yan.yonggui"Zhang, Jiwei"https://zbmath.org/authors/?q=ai:zhang.jiwei"Zheng, Chunxiong"https://zbmath.org/authors/?q=ai:zheng.chunxiongSummary: The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artificial boundary method, we first derive the exact artificial nonreflecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in [\textit{X. Tian} and \textit{Q. Du}, SIAM J. Numer. Anal. 51, No. 6, 3458--3482 (2013; Zbl 1295.82021)] to discretize the nonlocal operator, and apply the \(z\)-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonreflecting boundary condition and leads us to reformulate the original infinite discrete system into an equivalent finite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are finally provided to demonstrate the effectiveness of our approach.Degenerate kernel method for the Fredholm integral equations of the second kind involving algebraic and logarithmic singularities.https://zbmath.org/1449.653602021-01-08T12:24:00+00:00"Guo, Jiawei"https://zbmath.org/authors/?q=ai:guo.jiawei"Lian, Huan"https://zbmath.org/authors/?q=ai:lian.huanSummary: Fredholm integral equation of the second kind involving algebraic and logarithmic endpoint singularities is considered. It is supposed that the kernel function possesses the Puiseux expansion at the endpoint of the interval. For this equation, the Puiseux interpolation is adopted in a small interval involving the singularity and piecewise linear interpolation is used in the remaining part of the interval, and then a hybrid degenerate kernel method is established. The deduced singular integrals are evaluated by the modified composite Gauss-Legendre algorithm. On the basis of numerical analysis, the convergence of the algorithm is proved. A numerical example shows that the method has good computational results for the kernel with algebraic and logarithmic singularity, and the precision is high.On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy.https://zbmath.org/1449.911072021-01-08T12:24:00+00:00"Peng, Xuanhua"https://zbmath.org/authors/?q=ai:peng.xuanhua"Su, Wen"https://zbmath.org/authors/?q=ai:su.wen"Zhang, Zhimin"https://zbmath.org/authors/?q=ai:zhang.zhimin.1Summary: In this paper, we model the insurance company's surplus flow by a perturbed compound Poisson model. Suppose that at a sequence of random time points, the insurance company observes the surplus to decide dividend payments. If the observed surplus level is larger than the maximum of a threshold \(b>0\) and the last observed level (after dividends payment if possible), then a fraction \(0<\theta<1\) of the excess amount is paid out as a lump sum dividend. We assume that the solvency is also discretely monitored at these observation times, so that the surplus process stops when the observed value becomes negative. Integro-differential equations for the expected discounted dividend payments before ruin and the Gerber-Shiu expected discounted penalty function are derived, and solutions are also analyzed by Laplace transform method. Numerical examples are given to illustrate the applicability of our results.An evolutionary perspective on cancer, with applications to anticancer drug resistance modelling and perspectives in therapeutic control.https://zbmath.org/1449.354202021-01-08T12:24:00+00:00"Clairambault, Jean"https://zbmath.org/authors/?q=ai:clairambault.jeanSummary: The question of a mathematical representation and theoretical overcoming by optimized therapeutic strategies of drug-induced drug resistance in cancer cell populations is tackled here from the point of view of adaptive dynamics and optimal population growth control, using integro-differential equations. Combined impacts of external continuous-time functions, standing for drug actions, on targets in a plastic (i.e., able to quickly change its phenotype in deadly environmental conditions) cell population model, represent a therapeutical control to be optimized. A justification for the introduction of the adaptive dynamics setting, retaining such plasticity for cancer cell populations, is firstly presented in light of the evolution of multicellular species and disruptions in multicellularity coherence that are characteristics of cancer and of its progression. Finally, open general questions on cancer and evolution in the Darwinian sense are listed, that may open innovative tracks in modelling and treating cancer by circumventing drug resistance. This study sums up results that were presented at the international NUMACH workshop, Mulhouse, France, in July 2018.The Liouville type theorem for a system of nonlinear integral equations on exterior domain.https://zbmath.org/1449.450102021-01-08T12:24:00+00:00"Yin, Rong"https://zbmath.org/authors/?q=ai:yin.rong"Zhang, Jihui"https://zbmath.org/authors/?q=ai:zhang.jihui"Shang, Xudong"https://zbmath.org/authors/?q=ai:shang.xudongSummary: In this paper we are concerned with a system of nonlinear integral equations on the exterior domain under the suitable boundary conditions. Through the method of moving planes in integral forms which has some innovative ideas we obtain that the exterior domain is radial symmetry and a pair of positive solutions of the system is radial symmetry and monotone non-decreasing. Consequently, we can obtain the corresponding Liouville type theorem about the solutions.Fixed point results for \(\phi\)-\((\gamma,\eta, n, m)\)-contractions with applications to nonlinear integral equations.https://zbmath.org/1449.540622021-01-08T12:24:00+00:00"Hammad, Hasanen A."https://zbmath.org/authors/?q=ai:hammad.hasanen-abuelmagd"La Sen, Manuel De"https://zbmath.org/authors/?q=ai:de-la-sen.manuelSummary: The aim of this paper is to introduce a new class of pair of contraction mappings, called \(\phi\)-\((\gamma,\eta, n, m)\)-contraction pairs, and obtain common fixed point theorems for a pair of mappings in this class, satisfying a weakly compatible condition. As an application, we use mappings of this class to find the existence of solutions for nonlinear integral equations on the space of continuous functions and in some of its subspaces. Moreover, some examples are given here to illustrate the applicability of these results.Existence of entropy solutions to a doubly nonlinear integro-differential equation.https://zbmath.org/1449.450212021-01-08T12:24:00+00:00"Scholtes, Martin"https://zbmath.org/authors/?q=ai:scholtes.martin"Wittbold, Petra"https://zbmath.org/authors/?q=ai:wittbold.petraThe authors consider a class of doubly nonlinear problems with memory. They consider kernels of the type \(k(t)=t^{-\alpha}/\Gamma(1-\alpha)\). Doing so, the time-derivatives side becomes the fractional derivative of order \(\alpha\in(0,1)\) in the sense of Riemann-Liouville. The uniqueness of entropy solutions has been shown in a previous work. In this paper, the authors prove the existence of entropy solutions for general \(L^1\)-data and Dirichlet boundary conditions. The main idea of the existence proof is a modification of the regularization method by \textit{R. Landes} [J. Reine Angew. Math. 393, 21--38 (1989; Zbl 0664.35027)].
Reviewer: Vincenzo Vespri (Firenze)Some fixed point theorems in Menger probabilistic partial metric spaces with application to Volterra type integral equation.https://zbmath.org/1449.540602021-01-08T12:24:00+00:00"Ghanenia, Amir"https://zbmath.org/authors/?q=ai:ghanenia.amir"Khanehgir, Mahnaz"https://zbmath.org/authors/?q=ai:khanehgir.mahnaz"Allahyari, Reza"https://zbmath.org/authors/?q=ai:allahyari.reza"Mehrabinezhad, Mohammad"https://zbmath.org/authors/?q=ai:mehrabinezhad.mohammadSummary: In this paper, we introduce the notion of Menger probabilistic partial metric space and prove some fixed point theorems in the framework of such spaces. Some examples and an application to Volterra type integral equations are given to support the obtained results. Finally, we apply successive approximations method to find a solution for a Volterra type integral equation with high accuracy.On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations.https://zbmath.org/1449.450192021-01-08T12:24:00+00:00"Andreev, A. S."https://zbmath.org/authors/?q=ai:andreev.aleksandr-s"Peregudova, O. A."https://zbmath.org/authors/?q=ai:peregudova.olga-aThe article presents the development of the Lyapunov functional method in the problem of the stability of integro-differential equations, including terms with finite and unlimited delay. The authors develop methods for deriving the limiting properties of solutions of differential and functional differential equations under the assumption of the existence of a Lyapunov function and functional with a constant derivative. The basis of the investigation is the construction of topological dynamics of equations. In addition, the authors obtain conditions for uniform asymptotic stability of systems of equations, which are a generalization of some models of physical and economic processes, as well as biological azimotion of populations.
Reviewer: Tatuana Badokina (Saransk)New numerical process solving nonlinear infinite-dimensional equations.https://zbmath.org/1449.651122021-01-08T12:24:00+00:00"Khellaf, Ammar"https://zbmath.org/authors/?q=ai:khellaf.ammar"Merchela, Wassim"https://zbmath.org/authors/?q=ai:merchela.wassim"Benarab, Sarra"https://zbmath.org/authors/?q=ai:benarab.sarraSummary: Solving a nonlinear equation in a functional space requires two processes: discretization and linearization. In recent paper [\textit{L. Grammont} et al., J. Integral Equations Appl. 26, No. 3, 413--436 (2014; Zbl 1307.65077)], the authors study the difference between applying them in one and in the other order. Linearizing first the nonlinear problem and discretizing the linear problem will be called option (B). Discretizing first the nonlinear problem and linearizing the discrete nonlinear problem will be called option (C). In this paper, we propose a new numerical process equivalent to the option (B): we linearize first the original nonlinear problem with an alternative linearization scheme than that used in the option (B), then we discretize the resulting iterative linear equations using a projection method to implement the corresponding finite-dimensional problem. The aims of this new process are intended to get weaker the theoretical assumptions and to give a powerful numerical performance. We give sufficient conditions to deal with the convergence results. Finally, as a numerical application, we solve a system of Fredholm equations of the second kind. The accuracy and efficiency of this process are illustrated in some numerical examples.Linear barycentric rational collocation method for solving second-order Volterra integro-differential equation.https://zbmath.org/1449.450222021-01-08T12:24:00+00:00"Li, Jin"https://zbmath.org/authors/?q=ai:li.jin.1"Cheng, Yongling"https://zbmath.org/authors/?q=ai:cheng.yonglingSummary: Second-order Volterra integro-differential equation is solved by the linear barycentric rational collocation method. Following the barycentric interpolation method of Lagrange polynomial and Chebyshev polynomial, the matrix form of the collocation method is obtained from the discrete Volterra integro-differential equation. With the help of the convergence rate of the linear barycentric rational interpolation, the convergence rate of linear barycentric rational collocation method for solving Volterra integro-differential equation is proved. At last, several numerical examples are provided to validate the theoretical analysis.An \(hp\)-version Chebyshev spectral collocation method for nonlinear Volterra integro-differential equations with weakly singular kernels.https://zbmath.org/1449.652742021-01-08T12:24:00+00:00"Jia, Hongli"https://zbmath.org/authors/?q=ai:jia.hongli"Yang, Yang"https://zbmath.org/authors/?q=ai:yang.yang.5|yang.yang.2|yang.yang.1|yang.yang.3|yang.yang.4"Wang, Zhongqing"https://zbmath.org/authors/?q=ai:wang.zhongqingSummary: This paper presents an \(hp\)-version Chebyshev spectral collocation method for nonlinear Volterra integro-differential equations with weakly singular kernels. The \(hp\)-version error bound of the collocation method under the \({H^1}\)-norm is established on an arbitrary mesh. Numerical experiments demonstrate the effectiveness of the proposed method.Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations in two-dimensional spaces based on block pulse functions.https://zbmath.org/1449.653692021-01-08T12:24:00+00:00"Wang, Junxia"https://zbmath.org/authors/?q=ai:wang.junxiaSummary: In order to obtain the numerical solutions of two-dimensional nonlinear Volterra-Fredholm-Hammerstein integral equations, this study constructs the corresponding operational matrix based on block pulse functions. The Volterra-Fredholm-Hammerstein integral equations are transformed into a nonlinear system of algebra equations, then the numerical solutions of original equations are obtained after discretizing unknown variables. The numerical examples are given to verify the feasibility and effectiveness of the proposed algorithm.General functional-integral Volterra equation.https://zbmath.org/1449.450052021-01-08T12:24:00+00:00"Yuldashev, T. K."https://zbmath.org/authors/?q=ai:yuldashev.tursun-kamaldinovich"Artykova, Zh. A."https://zbmath.org/authors/?q=ai:artykova.zh-aSummary: We research the one-valued solvability of the integral Volterra equation of general type with given initial condition. The theorem on the existence and uniqueness of continuous solution, which satisfies the Lipschitz condition on the considered interval is proved. The method of successive approximations and the method of contracting mappings are used.Boundary triples for integral systems on the half-line.https://zbmath.org/1449.450172021-01-08T12:24:00+00:00"Strelnikov, D."https://zbmath.org/authors/?q=ai:strelnikov.dmytro|strelnikov.d-iThe author studies the following integro-differential system of Sturm-Liouville type on the half-line \([0,\infty)\): \[ J\vec{f}(x)-J\vec{a}=\int\limits_0^x\begin{pmatrix} \lambda dW-dQ & 0\\ 0 & dP\end{pmatrix} \vec{f}(t),\quad J=\begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}. \] Here \(P,Q,W\) are real functions of locally bounded variation on \([0,\infty )\), and \(W\) is non-decreasing. The author defines minimal and maximal linear relations associated with the system in the limit point and limit circle cases, and finds boundary triplets and corresponding Weyl functions. The case of a compact interval was studied by the author earlier [\textit{D. Strelnikov}, J. Math. Sci., New York 231, No. 1, 83--100 (2018; Zbl 1401.45002)].
Reviewer: Anatoly N. Kochubei (Kyïv)Calculus and nonlinear integral equations for functions with values in \(L\)-spaces.https://zbmath.org/1449.450092021-01-08T12:24:00+00:00"Babenko, V."https://zbmath.org/authors/?q=ai:babenko.v-t|babenko.vira|babenko.v-n|babenko.vladislav-f|babenko.vladimir-i|babenko.v-e|babenko.viktor-v|babenko.v-aSummary: In this paper, the calculus of functions with values in \(L\)-spaces is developed. We then consider nonlinear integral equations of Fredholm and Volterra types for functions with values in \(L\)-spaces. Such class of equations includes set-valued integral equations, fuzzy integral equations, and many others. We prove theorems of existence and uniqueness of the solutions of such equations and investigate data dependence of their solutions.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.Modified numerical approaches for a class of Volterra integral equations with proportional delays.https://zbmath.org/1449.653682021-01-08T12:24:00+00:00"Taghizadeh, Elham"https://zbmath.org/authors/?q=ai:taghizadeh.elham"Matinfar, Mashallah"https://zbmath.org/authors/?q=ai:matinfar.mashallahSummary: This paper addresses modified-meshless numerical schemes for dynamical systems with proportional delays. The proposed mesh reduction techniques are based on a redial-point interpolation and moving least-squares methods. An optimal influence domain radius is constructed utilizing nodal connectivity and node-depending integration background mesh. Optimal shape parameters are obtained by the use of properties of the delta Kronecker and the compactly supported weight function. Numerical results are provided to justify the accuracy and efficiency of the proposed schemes.Liouville type theorem for an integral system with the poly-harmonic extension operator.https://zbmath.org/1449.351322021-01-08T12:24:00+00:00"Tang, Sufang"https://zbmath.org/authors/?q=ai:tang.sufangSummary: Liouville type theorem is considered for a system of integral equations with the poly-harmonic extension operator on the upper half Euclidean space in this paper. Under the natural structure conditions, we classify the positive solutions to the system based on the method of moving sphere in integral form and some integral inequality.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.Some features of second kind Fredholm equations kernels.https://zbmath.org/1449.450012021-01-08T12:24:00+00:00"Buzova, Mariya Aleksandrovna"https://zbmath.org/authors/?q=ai:buzova.mariya-aleksandrovnaSummary: Kernels of Fredholm integral equations of the second kind with exceptions are analysed in this article. The equations under consideration have a meaning of magnetic field boundary condition and are used in problems of scattering on scatterers with finite thickness. It is shown that these kernels could be stated in a form of Dirac delta-functions. This mathematical formalization results in interesting physical effect that induced current calculated via physical optics equals the difference of face and back currents of the scatterer, calculated using method of integral equations.Triangular functions with convergence for solving linear system of two-dimensional fuzzy Fredholm integral equation.https://zbmath.org/1449.450022021-01-08T12:24:00+00:00"Hengamian Asl, E."https://zbmath.org/authors/?q=ai:hengamian-asl.e"Saberi-Nadjafi, J."https://zbmath.org/authors/?q=ai:saberi-nadjafi.jafarSummary: In this paper, we present a review on triangular functions (TFs) to
solve linear two-dimensional fuzzy Fredholm integral equations system of the second
kind (2D-FFIES-2). The properties of triangular functions are utilized to reduce the 2D-FFIES-2 to a linear system of algebraic equations. Moreover, we state the convergence analysis of the method. Finally, some examples show the simplicity and the validity of the present numerical method.Research on the buckling behavior of thermoelectric films on infinite elastic substrates.https://zbmath.org/1449.740992021-01-08T12:24:00+00:00"Zhang, Chenxi"https://zbmath.org/authors/?q=ai:zhang.chenxi"Ding, Shenghu"https://zbmath.org/authors/?q=ai:ding.shenghuSummary: In this paper, the buckling behavior of thermoelectric thin film bonded to elastic substrates is studied. Combining the interface shear stress with the axial stress of the thin film, the calculation model of the thermoelectric thin film is established and the problem is transformed into a singular integral equation by using boundary conditions. The singular integral equation is separated by using Chebyshev polynomials, and the normalized stress intensity factors are obtained. The effects of film stress and interfacial stress intensity factors are determined by the film thickness and the stiffness ratio of substrate to film. The influence of the film length and the thickness ratio on film stress and interface stress intensity factors is discussed. The result shows that the stiffness ratio between the film and the substrate has a significant effect on the stress level of the film.Fractional linear interpolation method for Fredholm integral equations of the second kind with two-endpoint singularities.https://zbmath.org/1449.653612021-01-08T12:24:00+00:00"Guo, Jiawei"https://zbmath.org/authors/?q=ai:guo.jiawei"Wang, Tongke"https://zbmath.org/authors/?q=ai:wang.tongkeSummary: The Fredholm integral equations of the second kind with two-endpoint singularities are considered. It is supposed that the kernel function possesses fractional Taylor's expansions at both endpoints of the interval. For this type of kernel, the approximation is done by fractional order interpolation in a small interval involving the singularity and piecewise linear interpolation in the remaining part of the interval, which leads to a kind of fractional degenerate kernel method based on linear interpolation. We discuss the condition under which the method can converge and give the convergence order estimation. Numerical examples demonstrate that the fractional order hybrid linear interpolation algorithm has good computational results for the kernel functions with two-endpoint singularities.On a Bagley-Torvik fractional integro-differential inclusion.https://zbmath.org/1449.450112021-01-08T12:24:00+00:00"Cernea, A."https://zbmath.org/authors/?q=ai:cernea.aurelianSummary: Existence of solutions for a Bagley-Torvik fractional integro-differential inclusion is investigated in the case when the values of the set-valued map are not convex.Using integro-differential operators for synthesis of models of large information systems with multiple-valued characteristics of nodes.https://zbmath.org/1449.450182021-01-08T12:24:00+00:00"Yudashkin, A. A."https://zbmath.org/authors/?q=ai:yudashkin.aleksandr-anatolevichSummary: We suggest the method for synthesis the models of information systems storing and reconstructing their configurations for the case of infinite number of nodes when the state of each node can be not uniquely defined. The models use the given structure of non-linear integro-differential equations, constructed for complex functions describing the state of distributed network as basis. The universal form of models is designed, its basic properties are obtained, the algorithm for storing the configurations is constructed.Weak Galerkin finite-element method for time-fractional nonlinear integro-differential equations.https://zbmath.org/1449.653252021-01-08T12:24:00+00:00"Wang, Haifeng"https://zbmath.org/authors/?q=ai:wang.haifeng"Xu, Da"https://zbmath.org/authors/?q=ai:xu.da"Guo, Jing"https://zbmath.org/authors/?q=ai:guo.jingSummary: In this article, a fully discrete scheme for one-dimensional (1D) time-fractional nonlinear integro-differential equation is established based on the weak Galerkin finite-element method. The stability and convergence of this scheme are proved. Several numerical experiments are presented to illustrate the theoretical analysis and to show the strong potential of this method.Spectral solution of fractional fourth order partial integro-differential equations.https://zbmath.org/1449.450202021-01-08T12:24:00+00:00"Bazgir, Hamed"https://zbmath.org/authors/?q=ai:bazgir.hamed"Ghazanfari, Bahman"https://zbmath.org/authors/?q=ai:ghazanfari.bahmanSummary: In this paper, mixed spectral method is applied to solve the fractional fourth order partial integro-differential equations together with weak singularity. Eigenfunctions of the fourth order self-adjoint positive-definite differential operator are used for the discretization of spatial variable and its derivatives. Also, shifted Legendre polynomials are applied to the discretization of time variable. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. The method is easy to apply and produces very accurate numerical results.On some fractional integro-differential inclusions with nonlocal multi-point boundary conditions.https://zbmath.org/1449.450122021-01-08T12:24:00+00:00"Cernea, Aurelian"https://zbmath.org/authors/?q=ai:cernea.aurelianSummary: Existence of solutions for two classes of fractional integro-differential inclusions with nonlocal multi-point boundary conditions is investigated in the case when the values of the set-valued map are not convex.Nonlinear equations with weighted potential type operators in Lebesgue spaces.https://zbmath.org/1449.450082021-01-08T12:24:00+00:00"Askhabov, Sultan Nazhmudinovich"https://zbmath.org/authors/?q=ai:askhabov.sultan-nazhmudinovichSummary: By method of monotone operators, existence and uniqueness theorems are proved for some classes of nonlinear equations with weighted potential type operators in Lebesgue spaces.A discontinuous Galerkin method with penalty for one-dimensional nonlocal diffusion problems.https://zbmath.org/1449.652422021-01-08T12:24:00+00:00"Du, Qiang"https://zbmath.org/authors/?q=ai:du.qiang"Ju, Lili"https://zbmath.org/authors/?q=ai:ju.lili"Lu, Jianfang"https://zbmath.org/authors/?q=ai:lu.jianfang"Tian, Xiaochuan"https://zbmath.org/authors/?q=ai:tian.xiaochuanSummary: There have been many theoretical studies and numerical investigations of nonlocal diffusion (ND) problems in recent years. In this paper, we propose and analyze a new discontinuous Galerkin method for solving one-dimensional steady-state and time-dependent ND problems, based on a formulation that directly penalizes the jumps across the element interfaces in the nonlocal sense. We show that the proposed discontinuous Galerkin scheme is stable and convergent. Moreover, the local limit of such DG scheme recovers classical DG scheme for the corresponding local diffusion problem, which is a distinct feature of the new formulation and assures the asymptotic compatibility of the discretization. Numerical tests are also presented to demonstrate the effectiveness and the robustness of the proposed method.Discounted penalty function of generalized compound Poisson model of two type insurance when funds fall to initial surplus.https://zbmath.org/1449.911042021-01-08T12:24:00+00:00"Li, Jingbin"https://zbmath.org/authors/?q=ai:li.jingbin"Wang, Xiulian"https://zbmath.org/authors/?q=ai:wang.xiulian"Zou, Hua"https://zbmath.org/authors/?q=ai:zou.huaSummary: The generalized compound Poisson model of two type insurance is considered. The discounted penalty function about the stopping time when funds fall to initial surplus is studied. The integral differential equation and renewal equation for the discounted penalty function are deduced by using probability theory and the Laplace transform, and then the concrete expression of the discounted penalty function and the moment of stopping time are obtained. The explicit expression of the discounted penalty function is given when the claim amount distribution is exponential.Galerkin and multi-Galerkin methods for weakly singular Volterra-Hammerstein integral equations and their convergence analysis.https://zbmath.org/1449.653622021-01-08T12:24:00+00:00"Kant, Kapil"https://zbmath.org/authors/?q=ai:kant.kapil"Nelakanti, Gnaneshwar"https://zbmath.org/authors/?q=ai:nelakanti.gnaneshwarSummary: In this paper, we consider the Galerkin and iterated Galerkin methods to approximate the solution of the second kind nonlinear Volterra integral equations of Hammerstein type with the weakly singular kernel of algebraic type, using piecewise polynomial basis functions based on graded mesh. We prove that the Galerkin method converges with the order of convergence \(\mathcal{O}(n^{-m})\), whereas iterated Galerkin method converges with the order \(\mathcal{O}(n^{-2m})\) in uniform norm. We also prove that in iterated multi-Galerkin method the order of convergence is \(\mathcal{O}(n^{-3m})\). Numerical results are provided to justify the theoretical results.On the numerical treatment and analysis of two-dimensional Fredholm integral equations using quasi-interpolant.https://zbmath.org/1449.653572021-01-08T12:24:00+00:00"Derakhshan, M."https://zbmath.org/authors/?q=ai:derakhshan.maryam"Zarebnia, M."https://zbmath.org/authors/?q=ai:zarebnia.mohammadSummary: In this paper, we study the quadratic rule for the numerical solution of linear and nonlinear two-dimensional Fredholm integral equations based on spline quasi-interpolant. Also the convergence analysis of the method is given. We show that the order of the method is \(O(h_x^{m+1}) + O(h_y^{m'+1})\). The theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical part.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.Using Bernstein multi-scaling polynomials to obtain numerical solution of Volterra integral equations system.https://zbmath.org/1449.653712021-01-08T12:24:00+00:00"Yaghoobnia, A. R."https://zbmath.org/authors/?q=ai:yaghoobnia.a-r"Ezzati, R."https://zbmath.org/authors/?q=ai:ezzati.rezaSummary: In this paper, first, Bernstein multi-scaling polynomials (BMSPs), which are generalization of Bernstein polynomials (BPs), are introduced and some of their properties are explained. Then, a new method based on BMSPs to achieved numerical solution for system of nonlinear integral equations is proposed. The proposed method converted the system of integral equations to a nonlinear system. To evaluate the efficiency of the proposed method, some systems of nonlinear integral equations are solved, and their numerical solutions are compared with other similar methods.Boundary value problem solution of a second type confluent \(B\)-elliptic equation.https://zbmath.org/1449.352332021-01-08T12:24:00+00:00"Chebatoreva, E. V."https://zbmath.org/authors/?q=ai:chebatoreva.e-vSummary: Fundamental solution and potential of simple fiber and double layer types for confluent \(B\)-elliptic equation of the second kind are built in the given research work. with the help of above mentioned potentials boundary problems are converted into Fredholm integral equations of the second kind.Tau approximate solution of linear pantograph Volterra delay-integro-differential equation.https://zbmath.org/1449.651472021-01-08T12:24:00+00:00"Zhao, Jingjun"https://zbmath.org/authors/?q=ai:zhao.jingjun"Cao, Yang"https://zbmath.org/authors/?q=ai:cao.yang"Xu, Yang"https://zbmath.org/authors/?q=ai:xu.yang.1Summary: The operational Tau method is used to find numerical solution of linear pantograph Volterra delay-integro-differential equation. Meanwhile, error estimate and convergence analysis are given for the operational Tau method. Numerical results reveal that the method is effective.Inversion and characterization of some potentials with the densities in \(L^p\) in the non-elliptic case.https://zbmath.org/1449.450062021-01-08T12:24:00+00:00"Gil', Alekseĭ Viktorovich"https://zbmath.org/authors/?q=ai:gil.aleksei-viktorovich"Zadorozhnyĭ, Anatoliĭ Ivanovich"https://zbmath.org/authors/?q=ai:zadorozhnii.anatolii-ivanovich"Nogin, Vladimir Aleksandrovich"https://zbmath.org/authors/?q=ai:nogin.vladimir-aleksandrovichSummary: We construct the inversion of generalized Strichartz potentials with singularities of the kernels on a finite union of spheres in \(\mathbb{R}^n\) with densities from space \(L^p\), \(1\leq p\leq 2\) and Hardy space \(H^1\) in the non-elliptic case, when its symbols degenerate on a set of zero measure in \(\mathbb{R}^n\). We also give the description of these potentials in terms of the inverting constructions.Local and nonlocal value boundary problems for a third-order mixed-type equation equipped with Tricomi operator in its hyperbolic part.https://zbmath.org/1449.353232021-01-08T12:24:00+00:00"Balkizov, Zh. A."https://zbmath.org/authors/?q=ai:balkizov.zhuraslan-anatolevich|balkizov.zhiraslan-anatolevichSummary: The existence and uniqueness of local and nonlocal value boundary problems for third-order mixed-type equations with multiple characteristics is proved. Uniqueness of the problem solution is proved with energy-integral method. The existence of the solution is proved with equivalent reduction method to Fredholm integral equations of the second kind with the help of Green's function.Solution of the non-local problem for the hyperbolic equation in the closed form.https://zbmath.org/1449.353142021-01-08T12:24:00+00:00"Salikhov, R. N."https://zbmath.org/authors/?q=ai:salikhov.r-nSummary: The non-local boundary value problem for the degenerate hyperbolic equation in the area \(D\), which is the union of two areas in the upper and lower half-planes, is examined. The proof of the existence and uniqueness of this problem solution is reduced to the question of the solvability of a singular integral equation.