Compact Crank–Nicolson and Du Fort–Frankel Method for the Solution of the Time Fractional Diffusion Equation

Abstract

In this paper, two compact implicit finite difference methods are developed and analyzed for solving the one-dimensional time fractional diffusion equation. The temporal derivative is approximated by using Grünwald–Letnikov formula. Compact finite difference approximation is used for the second-order derivative in space. The local truncation errors are discussed. The stability analysis and the convergence of the proposed methods are investigated by means of Fourier series method. A comparison between the results of these methods and the exact solution is made. Numerical tests are given to verify the feasibility and accuracy of the methods.