scholarly journals A fast implicit difference scheme for solving the generalized time–space fractional diffusion equations with variable coefficients

2020 ◽  
Author(s):  
Xian‐Ming Gu ◽  
Ting‐Zhu Huang ◽  
Yong‐Liang Zhao ◽  
Pin Lyu ◽  
Bruno Carpentieri
Keyword(s):  
Difference Scheme ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Diffusion Equations ◽  
Implicit Difference Scheme ◽  
Fractional Diffusion Equations ◽  
Time Space ◽  
Generalized Time

Maximum principles for a class of generalized time-fractional diffusion equations

2020 ◽  
Vol 23 (3) ◽  
pp. 822-836
Author(s):  
Shengda Zeng ◽  
Stanisław Migórski ◽  
Van Thien Nguyen ◽  
Yunru Bai
Keyword(s):  
Fractional Derivatives ◽  
Extreme Points ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Variable Order ◽  
Fractional Diffusion Equations ◽  
Caputo Derivatives ◽  
Time Space ◽  
Generalized Time

AbstractTwo significant inequalities for generalized time fractional derivatives at extreme points are obtained. Then, we apply the inequalities to establish the maximum principles for multi-term time-space fractional variable-order operators. Finally, we employ the principles to investigate two kinds of diffusion equations involving generalized time-fractional Caputo derivatives and space-fractional Riesz-Caputo derivatives.

scholarly journals A Fast Preconditioned Semi-Implicit Difference Scheme for Strongly Nonlinear Space-Fractional Diffusion Equations

2021 ◽  
Vol 5 (4) ◽  
pp. 230
Author(s):  
Yu-Yun Huang ◽  
Xian-Ming Gu ◽  
Yi Gong ◽  
Hu Li ◽  
Yong-Liang Zhao ◽  
...  
Keyword(s):  
Difference Scheme ◽  
Fractional Diffusion ◽  
Euler Method ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Implicit Difference Scheme ◽  
Central Difference ◽  
Fractional Diffusion Equations ◽  
Backward Euler ◽  
Difference Formula

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from O(Ns2) to O(Ns) and the computational complexity from O(Ns3) to O(NslogNs) in each iterative step, where Ns is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method.

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