A difference method of solving the Steklov nonlocal boundary value problem of the second kind for the time-fractional diffusion equation

TL;DR

This paper develops and analyzes difference schemes for solving the time-fractional diffusion equation with nonlocal boundary conditions, establishing their stability and convergence through energy inequalities and numerical validation.

Contribution

It introduces a new difference method for the Steklov nonlocal boundary value problem of the second kind for the time-fractional diffusion equation, with proven stability and convergence.

Findings

The difference schemes are stable under certain conditions.

A priori estimates ensure convergence of the numerical solutions.

Numerical tests confirm the theoretical results.

Abstract

We consider difference schemes for the time-fractional diffusion equation with variable coefficients and nonlocal boundary conditions containing real parameters $\alpha$, $\beta$ and $\gamma$. By the method of energy inequalities, for the solution of the difference problem, we obtain a priori estimates, which imply the stability and convergence of these difference schemes. The obtained results are supported by the numerical calculations carried out for some test problems.