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.

Index Terms—Fractional derivative, diffusion equation, grunwald approximation, crank-nicolson method.

H. M. Nasir is with the Sultan Qaboos University, Al-khoud 123, Muscat, Oman (e-mail: nasirh@ squ.edu.om, nasirhm11@yahoo.com).
B. L. K. Gunawardana and H. M. N. P. Abeyrathna are with University of Peradeniya, Sri lanka (e-mail: leksh.gg@gmail.com, npa512@yahoo.com).

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.