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An ?-robust fast algorithm for distributed-order time-space fractional diffusion equation with weakly singular solution

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MATHEMATICS AND COMPUTERS IN SIMULATION 2023年 207卷 437-452页

作者： Sun, Lu-Yao Lei, Siu-Long Sun, Hai-Wei Zhang, Jia-Li Univ Macau Dept Math Macau Peoples R China

A fast algorithm is proposed for solving two-dimensional distributed-order time-space fractional diffusion equation where the solution has a weak singularity at initial time. The distributed-order fractional problem is firstly transformed into multi-term fractional problem by the Gauss-Legendre quadrature formula. Then the exponential-sum-approximation method on graded mesh is utilized to discretize time Caputo fractional derivatives in time direction, and a standard finite difference method is employed to approximate the spatial Riesz fractional derivatives. The scheme is proved to be alpha-robust convergent analytically. The discrete linear system possesses symmetric positive definite block-Toeplitz-Toeplitz-block structure and is efficiently solved by conjugate gradient method with the state-of-the-art sine-transformed based preconditioner. Numerical examples confirm the error analysis and the effectiveness of the preconditioner.(c) 2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Distributed-order time-space fractional equation Singularity Sine transform based preconditioner exponential-sum-approximation method ?-robust

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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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exponential-sum-approximation technique for variable-order time-fractional diffusion equations

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2022年第1期68卷 323-347页

作者： Zhang, Jia-Li Fang, Zhi-Wei Sun, Hai-Wei Univ Macau Dept Math Macau Peoples R China

In this paper, we study the variable-order (VO) time-fractional diffusion equations. For a VO function alpha(t) is an element of (0, 1), we develop an exponential-sum-approximation (ESA) technique to approach the VO Caputo fractional derivative. The ESA technique keeps both the quadrature exponents and the number of exponentials in the summation unchanged at different time level. Approximating parameters are properly selected to achieve the efficient accuracy. Compared with the general direct method, the proposed method reduces the storage requirement from O(n) to O(log(2) n) and the computational cost from O(n(2)) to O(n log(2) n), respectively, with n being the number of the time levels. When this fast algorithm is exploited to construct a fast ESA scheme for the VO time-fractional diffusion equations, the computational complexity of the proposed scheme is only of O(mn log(2) n) with O(m log(2) n) storage requirement, where m denotes the number of spatial grid points. Theoretically, the unconditional stability and error analysis of the fast ESA scheme are given. The effectiveness of the proposed algorithm is verified by numerical examples.

关键词： Variable-order Caputo fractional derivative exponential-sum-approximation method Fast algorithm Time-fractional diffusion equation Stability and convergence

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A fast algorithm for two-dimensional distributed-order time-space fractional diffusion equations

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APPLIED MATHEMATICS AND COMPUTATION 2022年 425卷

作者： Sun, Lu-Yao Fang, Zhi-Wei Lei, Siu-Long Sun, Hai-Wei Zhang, Jia-Li Univ Macau Dept Math Macau Peoples R China

In this paper, a fast algorithm is proposed for solving distributed-order time-space fractional diffusion equations. Integral terms in time and space directions are discretized by the Gauss-Legendre quadrature formula. The Caputo fractional derivatives are approximated by the exponential-sum-approximation method, and the finite difference method is applied for spatial approximation. The coefficient matrix of the discretized linear system is symmetric positive definite and possesses block-Toeplitz-Toeplitz-block structure. The preconditioned conjugate gradient method with a block-circulant-circulant-block pre conditioner is employed to solve the linear system. Theoretically, the stability and convergence of the proposed scheme are discussed. Numerical experiments are carried out to demonstrate the effectiveness of the scheme.(c) 2022 Elsevier Inc. All rights reserved.

关键词： Time-space fractional equation Distributed-order fractional derivative Fast algorithm Block-circulant-circulant-block preconditioner exponential-sum-approximation method Stability and convergence

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Robust fast method for variable-order time-fractional diffusion equations without regularity assumptions of the true solutions

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APPLIED MATHEMATICS AND COMPUTATION 2022年 430卷

作者： Zhang, Jiali Fang, Zhi-Wei Sun, Hai-Wei Univ Macau Dept Math Macau Peoples R China Foshan Univ Sch Math & Big Data Foshan Peoples R China

A robust fast method for mobile-immobile variable-order (VO) time-fractional diffusion equations (tFDEs) is developed, superiorly handling the cases of small or vanishing lower bound of the VO function. The fast finite difference scheme sharp decreases the storage requirement and computational complexity. Significantly, the error estimate for the proposed scheme is studied without any regularity assumptions of the true solution. Numerical experiments are tested to verify the effectiveness of the proposed method. (C) 2022 Elsevier Inc. All rights reserved.

关键词： Variable-order Caputo fractional derivative Fast algorithm exponential-sum-approximation method

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