Two new approximations to Caputo–Fabrizio fractional equation on non-uniform meshes and its applications

Abstract

We present two numerical approximations with non-uniform meshes to the Caputo–Fabrizio derivative of order α (0 < α < 1). First, the L1 formula is obtained by using the linear interpolation approximation for constructing the second-order approximation. Next, the quadratic interpolation approximation is used for improving the accuracy in the temporal direction. Besides, we discretize the spatial derivative using the compact finite difference scheme. The accuracy of the suggested schemes is not dependent on the fractional α. The coefficients and the truncation errors are carefully investigated for two schemes, separately. Three examples are carried out to support the convergence orders and show the efficiency of the suggested scheme.

Keywords

• numerical approximations
• caputo–fabrizio fractional derivative
• diffusion equation
• advection equation
• non-uniform meshes