The Crank-Nicholson type compact difference scheme for a loaded time-fractional Hallaire's equation

TL;DR

This paper develops a high-order compact finite difference scheme of Crank-Nicholson type for a loaded time-fractional Hallaire's equation, providing theoretical analysis and numerical validation of stability, convergence, and accuracy.

Contribution

It introduces a novel high-order compact difference scheme for the fractional Hallaire's equation with rigorous theoretical analysis and numerical experiments.

Findings

The scheme achieves an accuracy of order h^4 + au^{2-\u03b1}

Proves stability and convergence of the numerical method

Numerical experiments confirm theoretical results

Abstract

In this paper, we study loaded modified diffusion equation (the Hallaire equation with the fractional derivative with respect to time). The compact finite difference scheme of Crank-Nicholson type of higher order is developed for approximating the stated problem on uniform grids. A priori estimates are obtained in difference and differential interpretations, from which there follow uniqueness, stability, and convergence of the solution of the difference problem to solution of the differential problem with the rate $\mathcal{O}(h^4+\tau^{2-\alpha}.)$ Proposed theoretical calculations are confirmed by numerical experiments on test problem.