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Fast Second-order Evaluation for variable-order caputo fractional derivative with Applications to fractional Sub-Diffusion Equations

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Numerical Mathematics(Theory,Methods and Applications) 2022年第1期15卷 200-226页

作者： Jia-Li Zhang Zhi-Wei Fang Hai-Wei Sun Department of Mathematics University of MacaoMacaoSARChina

In this paper,we propose a fast second-order approximation to the variable-order(VO)caputo fractional derivative,which is developed based on L2-1σformula and the exponential-sum-approximation *** fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO caputo fractional *** fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme(F L2-1σscheme)for the multi-dimensional VO time-fractional sub-diffusion ***,F L2-1σscheme is proved to fulfill the similar properties of the coefficients as those of the well-studied L2-1σ***,F L2-1σscheme is strictly proved to be unconditionally stable and convergent.A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.

关键词： variable-order caputo fractional derivative exponential-sum-approximation method fast algorithm time-fractional sub-diffusion equation stability and convergence

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The Convergence of Euler-Maruyama Method of Nonlinear variable-order fractional Stochastic Differential Equations

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Advances in Applied Mathematics and Mechanics 2023年第4期15卷 852-879页

作者： Shanshan Xu Lin Wang Wenqiang Wang School of Mathematics and Computational Science&Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtanHunan 411105China

In this paper,we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations(VFSDEs).We futher constructe the Euler-Maruyama method to solve the equations and prove the convergence in mean and the strong convergence of the *** particular,when the fractional order is no longer varying,the conclusions obtained are consistent with the relevant conclusions in the existing ***,the numerical experiments at the end of the article verify the correctness of the theoretical results obtained.

关键词： variable-order caputo fractional derivative Stochastic differential equations Euler-Maruyama method convergence multiplicative noise

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A Hybrid ESA-CCD Method for variable-order Time-fractional Diffusion Equations

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Journal of Applied Mathematics and Physics 2024年第9期12卷 3053-3065页

作者： Xiaoxue Lu Chunhua Zhang Huiling Xue Bowen Zhong College of Mathematics and Information Science Nanchang Hangkong University Nanchang China School of Statistics and Data Science Jiangxi University of Finance and Economics Nanchang China School of Aircraft Engineering Nanchang Hangkong University Nanchang China

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.

关键词： variable-order caputo fractional derivative Combined Compact Difference Method Exponential-Sum-Approximation

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