Moving Finite Element Methods for a System of Semi-Linear Fractional Diffusion Equations

2016 ◽  
Vol 8 (6) ◽  
pp. 911-931 ◽  
Author(s):  
Jingtang Ma ◽  
Zhiqiang Zhou
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Convergence Rates ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Predator Prey ◽  
Fractional Diffusion Equations ◽  
Predator Prey Models ◽  
Derivatives Of ◽  
Moving Finite Element

AbstractThis paper studies a system of semi-linear fractional diffusion equations which arise in competitive predator-prey models by replacing the second-order derivatives in the spatial variables with fractional derivatives of order less than two. Moving finite element methods are proposed to solve the system of fractional diffusion equations and the convergence rates of the methods are proved. Numerical examples are carried out to confirm the theoretical findings. Some applications in anomalous diffusive Lotka-Volterra and Michaelis-Menten-Holling predator-prey models are studied.

Finite element methods for fractional diffusion equations

2020 ◽  
Vol 11 (04) ◽  
pp. 2030001
Author(s):  
Yue Zhao ◽  
Chen Shen ◽  
Min Qu ◽  
Weiping Bu ◽  
Yifa Tang
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Regularity Theory ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
The Past ◽  
Time Space ◽  
Engineering Physics ◽  
Substantial Interest

Due to the successful applications in engineering, physics, biology, finance, etc., there has been substantial interest in fractional diffusion equations over the past few decades, and literatures on developing and analyzing efficient and accurate numerical methods for reliably simulating such equations are vast and fast growing. This paper gives a concise overview on finite element methods for these equations, which are divided into time fractional, space fractional and time-space fractional diffusion equations. Besides, we also involve some relevant topics on the regularity theory, the well-posedness, and the fast algorithm.

scholarly journals Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Derivatives ◽  
Three Dimensional ◽  
Lattice Models ◽  
Fractional Diffusion ◽  
Lattice Diffusion ◽  
Diffusion Equations ◽  
Difference Operators ◽  
Fractional Diffusion Equations ◽  
Nonlocal Media ◽  
Derivatives Of

Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.

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