Numerical Method to Modify the Fractional-Order Diffusion Equation

Time or space or time-space fractional-order diffusion equations (FODEs) are widely used to describe anomalous diffusion processes in many physical and biological systems. In recent years, many authors have proposed different numerical methods to solve the modified fractional-order diffusion equatio...

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Main Author: Yunkun Chen
Format: Article
Language:English
Published: Wiley 2022-01-01
Series:Advances in Mathematical Physics
Online Access:http://dx.doi.org/10.1155/2022/4846747
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author Yunkun Chen
author_facet Yunkun Chen
author_sort Yunkun Chen
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description Time or space or time-space fractional-order diffusion equations (FODEs) are widely used to describe anomalous diffusion processes in many physical and biological systems. In recent years, many authors have proposed different numerical methods to solve the modified fractional-order diffusion equations, and some achievements have been obtained. However, to our knowledge of the literature, up to date, all the proposed numerical methods to modify FODE have achieved at most a second-order time accuracy. In this study, we focus mainly on the numerical methods based on numerical integration in order to modify the fractional-order diffusion equation: 1+1/12δx2pjk−1+1/12δx2pjk−1=μα∑l=0kλαlδx2pjk−l+ μβ∑l=0kλβlδx2pjk−l+τ/21+1/12δx2fjk−1+fjk, k=1,2,…,K;j=1,2,…,j−1,pj0=ωj,j=0,1,…,J, p0k=φtk,pjk=ψtk,k=0,1,…,K,fjl=fxj,tl,ωj=ωxj. Accordingly, numerical methods can be built to modify FODE with second-order time accuracy and fourth-order spatial accuracy in ∂px,t/∂t=∂1−α/∂t1−α+B∂1−β/∂t1−β∂2px,t/∂x2+fx,t,0<t≤1,0<x<1,px,0=0,0≤x≤1. p0,t=t2,p1,t=et2,0≤t≤1. Our suggested method can improve the time precision with a certain value.
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spelling doaj-art-b21de2d98d0741e38d3d5c4668be8df72025-02-03T06:10:55ZengWileyAdvances in Mathematical Physics1687-91392022-01-01202210.1155/2022/4846747Numerical Method to Modify the Fractional-Order Diffusion EquationYunkun Chen0Anshun UniversityTime or space or time-space fractional-order diffusion equations (FODEs) are widely used to describe anomalous diffusion processes in many physical and biological systems. In recent years, many authors have proposed different numerical methods to solve the modified fractional-order diffusion equations, and some achievements have been obtained. However, to our knowledge of the literature, up to date, all the proposed numerical methods to modify FODE have achieved at most a second-order time accuracy. In this study, we focus mainly on the numerical methods based on numerical integration in order to modify the fractional-order diffusion equation: 1+1/12δx2pjk−1+1/12δx2pjk−1=μα∑l=0kλαlδx2pjk−l+ μβ∑l=0kλβlδx2pjk−l+τ/21+1/12δx2fjk−1+fjk, k=1,2,…,K;j=1,2,…,j−1,pj0=ωj,j=0,1,…,J, p0k=φtk,pjk=ψtk,k=0,1,…,K,fjl=fxj,tl,ωj=ωxj. Accordingly, numerical methods can be built to modify FODE with second-order time accuracy and fourth-order spatial accuracy in ∂px,t/∂t=∂1−α/∂t1−α+B∂1−β/∂t1−β∂2px,t/∂x2+fx,t,0<t≤1,0<x<1,px,0=0,0≤x≤1. p0,t=t2,p1,t=et2,0≤t≤1. Our suggested method can improve the time precision with a certain value.http://dx.doi.org/10.1155/2022/4846747
spellingShingle Yunkun Chen
Numerical Method to Modify the Fractional-Order Diffusion Equation
Advances in Mathematical Physics
title Numerical Method to Modify the Fractional-Order Diffusion Equation
title_full Numerical Method to Modify the Fractional-Order Diffusion Equation
title_fullStr Numerical Method to Modify the Fractional-Order Diffusion Equation
title_full_unstemmed Numerical Method to Modify the Fractional-Order Diffusion Equation
title_short Numerical Method to Modify the Fractional-Order Diffusion Equation
title_sort numerical method to modify the fractional order diffusion equation
url http://dx.doi.org/10.1155/2022/4846747
work_keys_str_mv AT yunkunchen numericalmethodtomodifythefractionalorderdiffusionequation