In this paper, we consider a type of fractional diffusion equation (FDE) with variable coefficient on a finite domain. Firstly, we utilize a second-order scheme to approximate the Riemann-Liouville fractional derivative and present the finite difference scheme. Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the scheme and conclude that the scheme is unconditionally stable and convergent with the accuracy of O(t 2+h2). Finally, comparing to the general first order scheme, two numerical examples are given to show the effectiveness and accuracy of our numerical method, and the results are in excellent agreement with theoretical analysis.
Feng, Libo; Zhuang, Pinghui; Liu, Fawang; Turner, Ian; and Yang, Qianqian
"Second-Order Approximation for the Space Fractional Diffusion Equation with Variable Coefficient,"
Progress in Fractional Differentiation & Applications: Vol. 1
, Article 3.
Available at: https://dc.naturalspublishing.com/pfda/vol1/iss1/3