scholarly journals Numerical Solution for Riesz Fractional Diffusion Equation via Fractional Centered Difference Scheme

2021 ◽  
Vol 18 (7) ◽  
Author(s):  
Sohrab VALIZADEH ◽  
Abdollah BORHANIFAR
Keyword(s):  
Diffusion Equation ◽  
Difference Scheme ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Matrix Analysis ◽  
Riesz Fractional Derivative ◽  
Transform Method ◽  
Theoretical Predictions ◽  
The Matrix ◽  
Centered Difference

In this paper, a mixed matrix transform method with fractional centered difference scheme for solving fractional diffusion equation with Riesz fractional derivative was examined. It was obtained that the numerical scheme was unconditionally stable and feasible using the matrix analysis method. Numerical experiments were, then, carried out to support the theoretical predictions.

scholarly journals Mittag-Leffler-Pade approximations for the numerical solution of space and time fractional diffusion equations

2015 ◽  
Vol 4 (4) ◽  
pp. 466 ◽  
Author(s):  
Abdollah Borhanifar ◽  
Sohrab Valizadeh
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Point Of View ◽  
Matrix Analysis ◽  
Space And Time ◽  
The Matrix ◽  
Exponential Relaxation ◽  
Time Fractional Diffusion Equation ◽  
Approximate Scheme

<p>Anomalous diffusion and non-exponential relaxation patterns can be described by a space - time fractional diffusion equation. This paper aims to present a Pade approximation for Mittag-Leffler function mixed finite difference method to develop a numerical method to obtain an approximate solution for the space and time fractional diffusion equation. The truncation error of the method is theoretically analyzed. It is proved that the numerical proposed method is unconditionally stable from the matrix analysis point of view. Finally, some numerical results are given, which demonstrate the efficiency of the approximate scheme.</p>

scholarly journals Compact ADI method for two-dimensional Riesz space fractional diffusion equation

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1543-1554
Author(s):  
Sohrab Valizadeh ◽  
Alaeddin Malek ◽  
Abdollah Borhanifar
Keyword(s):  
Diffusion Equation ◽  
Maximum Error ◽  
Fractional Diffusion ◽  
Riesz Space ◽  
Fractional Diffusion Equation ◽  
Matrix Analysis ◽  
Two Dimensional ◽  
Adi Method ◽  
Space Fractional Diffusion Equation ◽  
The Matrix

In this paper, a compact alternating direction implicit (ADI) method has been developed for solving two-dimensional Riesz space fractional diffusion equation. The precision of the discretization method used in spatial directions is twice the order of the corresponding fractional derivatives. It is proved that the proposed method is unconditionally stable via the matrix analysis method and the maximum error in achieving convergence is discussed. Numerical example is considered aiming to demonstrate the validity and applicability of the proposed technique.

A New Numerical Method for the Riesz Space Fractional Diffusion Equation

2011 ◽  
Vol 213 ◽  
pp. 393-396 ◽  
Author(s):  
Heng Fei Ding ◽  
Yu Xin Zhang ◽  
Wan Sheng He ◽  
Xiao Ya Yang
Keyword(s):  
Diffusion Equation ◽  
Difference Scheme ◽  
Analytic Solution ◽  
Fractional Diffusion ◽  
Riesz Space ◽  
Fractional Diffusion Equation ◽  
Transform Method ◽  
Unconditionally Stable ◽  
Space Fractional Diffusion Equation ◽  
The Difference

Firstly, using matrix transform method, we transform the Riesz space fractional diffusion equation into an ordinary differential equation, and get its analytic solution. Secondly, we use (2,1) Pade approxiation to the exponentinal matrix of the analytic solution and obtain a new difference scheme for solving Riesz space fractional diffusion equation. Finally, we prove that the difference scheme is unconditionally stable.

scholarly journals An Analytical Solution of a Time-fractional Diffusion Equation Withexternal Force and Absorbent Term by Generalized Two-dimensional differential Transform Method

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Diffusion Equation ◽  
Initial Conditions ◽  
Fractional Diffusion ◽  
Differential Transform Method ◽  
Fractional Diffusion Equation ◽  
Transform Method ◽  
Differential Transform ◽  
Time Fractional Diffusion Equation ◽  
Generalized Differential ◽  
External Forces

In this paper, we have used generalized differential transform method in obtaining a general recurrencerelation for determining the solutions of time fractional diffusion equation with external force and absorbent term.Diffusion equations play an improtant part in energy transfer problems. Inclusion of fractional derivatives bring thenon-locality aspect into the physical system containing this equation. The obtained relation will help us to solvesuch equations with various external forces and initial conditions. Three illustrative examples have been discussed.

scholarly journals An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions

2021 ◽  
Vol 7 (2) ◽  
pp. 2370-2392
Author(s):  
Fouad Mohammad Salama ◽  
◽  
Nur Nadiah Abd Hamid ◽  
Norhashidah Hj. Mohd Ali ◽  
Umair Ali ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Iterative Method ◽  
Method Comparison ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Matrix Analysis ◽  
Stability And Convergence ◽  
Two Space Dimensions ◽  
Time Fractional Diffusion Equation

<abstract><p>In this paper, a new modified hybrid explicit group (MHEG) iterative method is presented for the efficient and accurate numerical solution of a time-fractional diffusion equation in two space dimensions. The time fractional derivative is defined in the Caputo sense. In the proposed method, a Laplace transformation is used in the temporal domain, and, for the spatial discretization, a new finite difference scheme based on grouping strategy is considered. The unique solvability, unconditional stability and convergence are thoroughly proved by the matrix analysis method. Comparison of numerical results with analytical and other approximate solutions indicates the viability and efficiency of the proposed algorithm.</p></abstract>

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