An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order

2021 ◽  
pp. 107270
Author(s):  
Xian-Ming Gu ◽  
Hai-Wei Sun ◽  
Yong-Liang Zhao ◽  
Xiangcheng Zheng
Keyword(s):  
Difference Scheme ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Variable Order ◽  
Implicit Difference Scheme ◽  
Fractional Diffusion Equations ◽  
Type Variable ◽  
Time Invariant

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.

An Efficient Operational Matrix Technique for Multidimensional Variable-Order Time Fractional Diffusion Equations

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
M. A. Zaky ◽  
S. S. Ezz-Eldien ◽  
E. H. Doha ◽  
J. A. Tenreiro Machado ◽  
A. H. Bhrawy
Keyword(s):  
Present Method ◽  
Operational Matrix ◽  
Solution Process ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Variable Order ◽  
Algebraic Equations ◽  
Global Approximation ◽  
Fractional Diffusion Equations ◽  
Accurate Numerical Algorithm

This paper derives a new operational matrix of the variable-order (VO) time fractional partial derivative involved in anomalous diffusion for shifted Chebyshev polynomials. We then develop an accurate numerical algorithm to solve the 1 + 1 and 2 + 1 VO and constant-order fractional diffusion equation with Dirichlet conditions. The contraction of the present method is based on shifted Chebyshev collocation procedure in combination with the derived shifted Chebyshev operational matrix. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations, and it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, we analyze the convergence of the present method graphically. Finally, comparisons between the algorithm derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones.

scholarly journals On Time-Fractional Diffusion Equations with Space-Dependent Variable Order

2018 ◽  
Vol 19 (12) ◽  
pp. 3855-3881 ◽  
Author(s):  
Yavar Kian ◽  
Eric Soccorsi ◽  
Masahiro Yamamoto
Keyword(s):  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Variable Order ◽  
Fractional Diffusion Equations

scholarly journals The Maximum Principle for Variable-Order Fractional Diffusion Equations and the Estimates of Higher Variable-Order Fractional Derivatives

2020 ◽  
Vol 8 ◽  
Author(s):  
Guangming Xue ◽  
Funing Lin ◽  
Guangwang Su
Keyword(s):  
Maximum Principle ◽  
Fractional Derivatives ◽  
Arbitrary Point ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Variable Order ◽  
The Maximum Principle ◽  
Fractional Diffusion Equations

In this paper, the maximum principle of variable-order fractional diffusion equations and the estimates of fractional derivatives with higher variable order are investigated. Firstly, we deduce the fractional derivative of a function of higher variable order at an arbitrary point. We also give an estimate of the error. Some important inequalities for fractional derivatives of variable order at arbitrary points and extreme points are presented. Then, the maximum principles of Riesz-Caputo fractional differential equations in terms of the multi-term space-time variable order are proved. Finally, under the initial-boundary value conditions, it is verified via the proposed principle that the solutions are unique, and their continuous dependance holds.

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