This paper studies a type of multiterm fractional stochastic delay integro-differential equations (FSDIDEs). First, the Euler-Maruyama (EM) method is developed for solving the equations, and the strong convergence ord...
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This paper studies a type of multiterm fractional stochastic delay integro-differential equations (FSDIDEs). First, the Euler-Maruyama (EM) method is developed for solving the equations, and the strong convergence order of this method is obtained, which is min alpha l-12,alpha l-alpha l-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\min \left\{ \alpha _{l}-\frac{1}{2}, \alpha _{l}-\alpha _{l-1}\right\} }$$\end{document}. Then, a fast EM method is also presented based on the exponential-sum-approximation with trapezoid rule to cut down the computational cost of the EM method. In the end, some concrete numerical experiments are used to substantiate these theoretical results and show the effectiveness of the fast method.
In this paper, we consider the weakly singular stochastic Volterra integral equations with variable exponent. Firstly, the existence and uniqueness of the equations are studied by the Banach contraction mapping princi...
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In this paper, we consider the weakly singular stochastic Volterra integral equations with variable exponent. Firstly, the existence and uniqueness of the equations are studied by the Banach contraction mapping principle. Secondly, we develop an Euler-Maruyama (EM) method and obtain its strong convergence rate. Moreover, we propose a fast EM method via the exponential-sum-approximation technique to reduce the EM method's computational cost. More specifically, if one disregards the Monte Carlo sampling error, then the fast EM method reduces the computational cost from O(N-2) to O(N log(2) N) and the storage from O(N) to O(log(2) N), where N is the total number of time steps. Moreover, if the sampling error is taken into account, we employ the multilevel Monte Carlo method based on the fast EM method to reduce computational costs further. Significantly, the computational costs of the EM method and the fast EM method to achieve an accuracy of O(epsilon) (epsilon < 1) are reduced from O(epsilon(-2-2/)((alpha) over tilde)) and O(epsilon(-2-2/)((alpha) over tilde) log(2)(epsilon)), respectively, to O(epsilon(-1/)((alpha) over tilde)(log(epsilon(-1)))(3)), where (alpha) over tilde = min {1 - alpha*, 1/2 - beta*} is related to the exponents of the singular kernel in the equations. Finally, numerical examples are provided to illustrate our theoretical results and demonstrate the superiority of the fast EM method.
In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order a...
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In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments.
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