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

<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>

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

AIMS Mathematics ◽

10.3934/math.2022134 ◽

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

Filomat ◽

10.2298/fil2105543v ◽

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.

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

Walailak Journal of Science and Technology (WJST) ◽

10.48048/wjst.2021.9246 ◽

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.

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