A Second-Order Difference Scheme for Generalized Time-Fractional Diffusion Equation with Smooth Solutions

Abstract

In the current work, we build a difference analog of the Caputo fractional derivative with generalized memory kernel (μL2-1σ formula). The fundamental features of this difference operator are studied, and on its ground, some difference schemes generating approximations of the second order in time for the generalized time-fractional diffusion equation with variable coefficients are worked out. We have proved stability and convergence of the given schemes in the grid L2-norm with the rate equal to the order of the approximation error. The achieved results are supported by the numerical computations performed for some test problems.