doi: 10.7763/IJAPM.2013.V3.212
A Second Order Finite Difference Approximation for the Fractional Diffusion Equation
H. M. Nasir
1
,
B. L. K. Gunawardana
2
,
H. M. N. P. Abeyrathna
2
- 1 the Sultan Qaboos University, Al-khoud 123, Muscat, Oman
- 2 University of Peradeniya, Sri lanka
Abstract
We consider an approximation of one-dimensional fractional diffusion equation. We claim and show that the finite difference approximation obtained from the Grünwald-Letnikov formulation, often claimed to be of first order accuracy, is in fact a second order approximation of the fractional derivative at a point away from the grid points. We use this fact to device a second order accurate finite difference approximation for the fractional diffusion equation. The proposed method is also shown to be unconditionally stable. By this approach, we treat three cases of difference approximations in a unified setting. The results obtained are justified by numerical examples.
Keywords
- Fractional derivative
- diffusion equation
- grunwald approximation
- crank-nicolson method
How to Cite
H. M. Nasir, B. L. K. Gunawardana, and H. M. N. P. Abeyrathna, "A Second Order Finite Difference Approximation for the Fractional Diffusion Equation," International Journal of Applied Physics and Mathematics, vol. 3, no. 4, pp. 237-243, 2013. https://doi.org/ 10.7763/IJAPM.2013.V3.212
Copyright & License
Copyright © 2013 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).