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Explicit and Implicit Methods for Fractional Diffusion Equations with the Riesz Fractional Derivative


Affiliations
1 Department of Mathematics, Urmia University of Technology, Urmia, Iran, Islamic Republic of
2 Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran, Islamic Republic of
 

In this paper, a fractional diffusion equation by using the explicit numerical method in a finite domain with second-order accuracy which includes the Riesz fractional derivative approximation is studied. For the Riesz fractional derivative approximation, ''fractional centered derivative'' approach is used. The error of the Riesz fractional derivative to the fractional centered difference is calculated. We used the implicit numerical method to solve the fractional diffusion equation and also investigated the stability of explicit and implicit methods. The maximum error of the implicit method for fractional diffusion equation with using fractional centered difference approach is shown by using the numerical results.

Keywords

Riesz Fractional Derivative Operator, Implicit Method, Explicit Method, Fractional Central Difference
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  • Explicit and Implicit Methods for Fractional Diffusion Equations with the Riesz Fractional Derivative

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Authors

Ahmad Reza Haghighi
Department of Mathematics, Urmia University of Technology, Urmia, Iran, Islamic Republic of
Hamideh Hoseini Ghejlo
Department of Mathematics, Urmia University of Technology, Urmia, Iran, Islamic Republic of
Nasim Asghari
Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran, Islamic Republic of

Abstract


In this paper, a fractional diffusion equation by using the explicit numerical method in a finite domain with second-order accuracy which includes the Riesz fractional derivative approximation is studied. For the Riesz fractional derivative approximation, ''fractional centered derivative'' approach is used. The error of the Riesz fractional derivative to the fractional centered difference is calculated. We used the implicit numerical method to solve the fractional diffusion equation and also investigated the stability of explicit and implicit methods. The maximum error of the implicit method for fractional diffusion equation with using fractional centered difference approach is shown by using the numerical results.

Keywords


Riesz Fractional Derivative Operator, Implicit Method, Explicit Method, Fractional Central Difference

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DOI: https://doi.org/10.17485/ijst%2F2013%2Fv6i7%2F34339