A new approximation to Caputo-type fractional diffusion and advection equations on non-uniform meshes

Abstract

In present work, a new high-order numerical approximation to the Caputo fractional derivative of order α(0<α<1) is constructed. The quadratic interpolation approximation of Caputo derivative on non-uniform meshes is considered. As a result, the new formula can be viewed as the modification of the existing jobs (Li et al., 2016 [8]; Zhang et al., 2014 [21]). The coefficients and truncation errors are analyzed, then we adopt the derived scheme to solve the fractional diffusion and advection equations with Dirichlet boundary conditions on non-uniform meshes. Moreover, the solvability, unconditional stability and convergence are proved. The effectiveness and accuracy of the scheme is demonstrated by numerical experiments to support the theoretical results.