scholarly journals A fast implicit difference scheme for a new class of time distributed-order and space fractional diffusion equations with variable coefficients

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Huan-Yan Jian ◽  
Ting-Zhu Huang ◽  
Xi-Le Zhao ◽  
Yong-Liang Zhao
Keyword(s):  
Difference Scheme ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Diffusion Equations ◽  
Implicit Difference Scheme ◽  
Fractional Diffusion Equations ◽  
New Class ◽  
Distributed Order

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.

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Eid Doha ◽  
Ali Bhrawy ◽  
Samer Ezz-Eldien
Keyword(s):  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Diffusion ◽  
High Accuracy ◽  
Variable Coefficients ◽  
Diffusion Equations ◽  
Algebraic Equations ◽  
Fractional Differentiation ◽  
Fractional Diffusion Equations ◽  
General Method

AbstractIn this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the time-dependent fractional diffusion equations.

An analytical solution of multi-dimensional space fractional diffusion equations with variable coefficients

2020 ◽  
pp. 2150006
Author(s):  
Pratibha Verma ◽  
Manoj Kumar
Keyword(s):  
Analytical Solution ◽  
Diffusion Coefficients ◽  
High Efficiency ◽  
Dimensional Space ◽  
Adomian Decomposition Method ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Adomian Decomposition

In this paper, we have considered the multi-dimensional space fractional diffusion equations with variable coefficients. The fractional operators (derivative/integral) are used based on the Caputo definition. This study provides an analytical approach to determine the analytical solution of the considered problems with the help of the two-step Adomian decomposition method (TSADM). Moreover, new results have been obtained for the existence and uniqueness of a solution by using the Banach contraction principle and a fixed point theorem. We have extended the dimension of the space fractional diffusion equations with variable coefficients into multi-dimensions. Finally, the generalized problems with two different types of the forcing term have been included demonstrating the applicability and high efficiency of the TSADM in comparison to other existing numerical methods. The diffusion coefficients do not require to satisfy any certain conditions/restrictions for using the TSADM. There are no restrictions imposed on the problems for diffusion coefficients, and a similar procedures of the TSADM has followed to the obtained analytical solution for the multi-dimensional space fractional diffusion equations with variable coefficients.

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