Volume 6, Issue 4
The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation

Hongmei Zhang & Shujun Shen

Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 571-585.

Published online: 2013-06

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  • Abstract

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. Numerical methods and analysis of stability and convergence of numerical scheme for the variable fractional order partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time variable fractional order diffusion equation on a finite domain. It is worth mentioning that here we use the Coimbra-definition variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation is proposed and then the stability and convergence of the numerical scheme are investigated.  Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

  • Keywords

Variable fractional derivative, diffusion equation, implicit Euler scheme, stability, convergence.

  • AMS Subject Headings

26A33, 34K28, 65M12, 60J70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-6-571, author = {}, title = {The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2013}, volume = {6}, number = {4}, pages = {571--585}, abstract = {

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. Numerical methods and analysis of stability and convergence of numerical scheme for the variable fractional order partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time variable fractional order diffusion equation on a finite domain. It is worth mentioning that here we use the Coimbra-definition variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation is proposed and then the stability and convergence of the numerical scheme are investigated.  Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2013.y12107}, url = {http://global-sci.org/intro/article_detail/nmtma/5919.html} }
TY - JOUR T1 - The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 571 EP - 585 PY - 2013 DA - 2013/06 SN - 6 DO - http://doi.org/10.4208/nmtma.2013.y12107 UR - https://global-sci.org/intro/article_detail/nmtma/5919.html KW - Variable fractional derivative, diffusion equation, implicit Euler scheme, stability, convergence. AB -

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. Numerical methods and analysis of stability and convergence of numerical scheme for the variable fractional order partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time variable fractional order diffusion equation on a finite domain. It is worth mentioning that here we use the Coimbra-definition variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation is proposed and then the stability and convergence of the numerical scheme are investigated.  Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Hongmei Zhang & Shujun Shen. (2020). The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation. Numerical Mathematics: Theory, Methods and Applications. 6 (4). 571-585. doi:10.4208/nmtma.2013.y12107
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