Fast Finite Difference Schemes for Time-Fractional Diffusion Equations with a Weak Singularity at Initial Time

2018 ◽  
Vol 8 (4) ◽  
pp. 834-858 ◽  
Author(s):  
Jin-ye Shen
Keyword(s):  
Finite Difference ◽  
Fractional Diffusion ◽  
Difference Schemes ◽  
Diffusion Equations ◽  
Initial Time ◽  
Finite Difference Schemes ◽  
Weak Singularity ◽  
Fractional Diffusion Equations

A Low-Dimensional Compact Finite Difference Method on Graded Meshes for Time-Fractional Diffusion Equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Rezvan Ghaffari ◽  
Farideh Ghoreishi
Keyword(s):  
Finite Difference ◽  
Dimensional Space ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Computational Time ◽  
Initial Time ◽  
Compact Finite Difference ◽  
Fractional Diffusion Equations ◽  
Graded Meshes ◽  
Cfd Method

Abstract In this paper, we propose an improvement of the classical compact finite difference (CFD) method by using a proper orthogonal decomposition (POD) technique for time-fractional diffusion equations in one- and two-dimensional space. A reduced CFD method is constructed with lower dimensions such that it maintains the accuracy and decreases the computational time in comparison with classical CFD method. Since the solution of time-fractional diffusion equation typically has a weak singularity near the initial time t = 0 {t=0} , the classical L1 scheme on uniform meshes may obtain a scheme with low accuracy. So, we use the L1 scheme on graded meshes for time discretization. Moreover, we provide the error estimation between the reduced CFD method based on POD and classical CFD solutions. Some numerical examples show the effectiveness and accuracy of the proposed method.

Finite difference scheme for the time-space fractional diffusion equations

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Jianxiong Cao ◽  
Changpin Li
Keyword(s):  
Finite Difference ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Order Accuracy ◽  
Fractional Diffusion Equations ◽  
Time Space ◽  
Liouville Fractional Derivative ◽  
The Stability

AbstractIn this paper, we derive two novel finite difference schemes for two types of time-space fractional diffusion equations by adopting weighted and shifted Grünwald operator, which is used to approximate the Riemann-Liouville fractional derivative to the second order accuracy. The stability and convergence of the schemes are analyzed via mathematical induction. Moreover, the illustrative numerical examples are carried out to verify the accuracy and effectiveness of the schemes.

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