A second order difference scheme for time fractional diffusion equation with generalized memory kernel

TL;DR

This paper introduces a second order difference scheme for solving time fractional diffusion equations with a generalized memory kernel, demonstrating stability, convergence, and second-order accuracy through theoretical analysis and numerical tests.

Contribution

It develops a novel second order difference scheme for generalized time-fractional diffusion equations with variable coefficients, including stability and convergence proofs.

Findings

Scheme achieves second order accuracy in time.

Proven stability and convergence in the grid L2 norm.

Numerical tests confirm theoretical results.

Abstract

In the current work we build a difference analog of the Caputo fractional derivative with generalized memory kernel ($_\lambda$L2-1$_\sigma$ 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 $L_2$ - 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.