Kronecker product-based structure preserving preconditioner for three-dimensional space-fractional diffusion equations

2019 ◽  
Vol 97 (3) ◽  
pp. 585-601 ◽  
Author(s):  
Hao Chen ◽  
Wen Lv
Keyword(s):  
Kronecker Product ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Structure Preserving ◽  
Fractional Diffusion Equations ◽  
Three Dimensional Space

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.

An analytical solution of multi-dimensional space fractional diffusion equations with variable coefficients

2020 ◽  
pp. 2150006
Author(s):  
Pratibha Verma ◽  
Manoj Kumar
Keyword(s):  
Analytical Solution ◽  
Diffusion Coefficients ◽  
High Efficiency ◽  
Dimensional Space ◽  
Adomian Decomposition Method ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Adomian Decomposition

In this paper, we have considered the multi-dimensional space fractional diffusion equations with variable coefficients. The fractional operators (derivative/integral) are used based on the Caputo definition. This study provides an analytical approach to determine the analytical solution of the considered problems with the help of the two-step Adomian decomposition method (TSADM). Moreover, new results have been obtained for the existence and uniqueness of a solution by using the Banach contraction principle and a fixed point theorem. We have extended the dimension of the space fractional diffusion equations with variable coefficients into multi-dimensions. Finally, the generalized problems with two different types of the forcing term have been included demonstrating the applicability and high efficiency of the TSADM in comparison to other existing numerical methods. The diffusion coefficients do not require to satisfy any certain conditions/restrictions for using the TSADM. There are no restrictions imposed on the problems for diffusion coefficients, and a similar procedures of the TSADM has followed to the obtained analytical solution for the multi-dimensional space fractional diffusion equations with variable coefficients.

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