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

Ahmad Reza Haghighi; Hamideh Hoseini Ghejlo  and Nasim Asghari

doi:10.17485/ijst/2013/v6i7.10

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

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Indian Journal of Science and Technology

DOI: 10.17485/ijst/2013/v6i7.10

Year: 2013, Volume: 6, Issue: 7, Pages: 1-5

Original Article

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

Ahmad Reza Haghighi^1*, Hamideh Hoseini Ghejlo¹ and Nasim Asghari²

¹Department of Mathematics, [email protected]
[email protected]
²Department of Mathematics, [email protected]
*Author For Correspondence
Ahmad Reza Haghighi
Department of Mathematics,
Email:[email protected]

This work is licensed under a Creative Commons Attribution 4.0 International License.

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. 2010 Mathematics Subject Classifications: 34A08, 35AXX