Numerical Solution for Solving Space-Fractional Diffusion Equations using Half-sweep Gauss-seidel Iterative Method

Abstract By exploiting Toeplitz-like structure and non-Hermitian dense property of the discrete coefficient matrix, a new double-layer iterative method called SHSS-PCG method is employed to solve the linear systems originating from the implicit finite difference discretization of fractional diffusion equations (FDEs). The method is a combination of the single-step Hermitian and skew-Hermitian splitting (SHSS) method with the preconditioned conjugate gradient (PCG) method. Further, the new circulant preconditioners are proposed to improve the efficiency of SHSS-PCG method, and the computation cost is further reduced via using the fast Fourier transform (FFT). Theoretical analysis shows that the SHSS-PCG iterative method with circulant preconditioners is convergent. Numerical experiments are given to show that our SHSS-PCG method with circulant preconditioners preforms very well, and the proposed circulant preconditioners are very efficient in accelerating the convergence rate.

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Stochastic model for multi-term time-fractional diffusion equations with noise

Thermal Science ◽

10.2298/tsci21s2287h ◽

2021 ◽

Vol 25 (Spec. issue 2) ◽

pp. 287-293

Author(s):

Vahid Hosseini ◽

Mohamad Remazani ◽

Wennan Zou ◽

Seddigheh Banihashemii

Keyword(s):

Numerical Solution ◽

Numerical Approach ◽

Fractional Diffusion ◽

Diffusion Equations ◽

Data Driven ◽

Algebraic Equations ◽

Fractional Diffusion Equations ◽

Collocation Points ◽

Linear Algebraic Equations ◽

Collocation Approach

This paper studies a spectral collocation approach for evaluating the numerical solution of the stochastic multi-term time-fractional diffusion equations associated with noisy data driven by Brownian motion. This model describes the symmetry breaking in molecular vibrations. The numerical solution of the stochastic multi-term time-fractional diffusion equations is proposed by means of collocation points method based on sixth-kind Chebyshev polynomial approach. For this purpose, the problem under consideration is reduced to a system of linear algebraic equations. Two examples highlight the robustness and accuracy of the proposed numerical approach.

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Tensor-Train Format Solution with Preconditioned Iterative Method for High Dimensional Time-Dependent Space-Fractional Diffusion Equations with Error Analysis

Journal of Scientific Computing ◽

10.1007/s10915-019-00994-3 ◽

2019 ◽

Vol 80 (3) ◽

pp. 1731-1763

Author(s):

Lot-Kei Chou ◽

Siu-Long Lei

Keyword(s):

Iterative Method ◽

Error Analysis ◽

Fractional Diffusion ◽

Time Dependent ◽

Diffusion Equations ◽

High Dimensional ◽

Fractional Diffusion Equations ◽

Preconditioned Iterative

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A numerical solution for a class of time fractional diffusion equations with delay

International Journal of Applied Mathematics and Computer Science ◽

10.1515/amcs-2017-0033 ◽

2017 ◽

Vol 27 (3) ◽

pp. 477-488 ◽

Cited By ~ 7

Author(s):

Vladimir G. Pimenov ◽

Ahmed S. Hendy

Keyword(s):

Numerical Solution ◽

Numerical Experiments ◽

Numerical Scheme ◽

Fixed Time ◽

Fractional Diffusion ◽

Diffusion Equations ◽

Discrete Energy ◽

Fractional Diffusion Equations ◽

Equations With Delay ◽

Theoretical Results

AbstractThis paper describes a numerical scheme for a class of fractional diffusion equations with fixed time delay. The study focuses on the uniqueness, convergence and stability of the resulting numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence orderO(τ2−α+h4) in L∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.

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