Boundary value problems for the diffusion equation of the variable order in differential and difference settings

TL;DR

This paper investigates boundary value problems for variable order diffusion equations in differential and difference contexts, demonstrating the applicability of energy inequality methods and validating results through numerical tests.

Contribution

It extends classical energy inequality techniques to fractional and variable order diffusion equations in both differential and difference frameworks.

Findings

Energy inequalities are effective for a priori estimates in variable order diffusion problems.

Numerical calculations confirm the theoretical results.

Method applies to both differential and difference settings.

Abstract

Solutions of boundary value problems for a diffusion equation of fractional and variable order in differential and difference settings are studied. It is shown that the method of energy inequalities is applicable to obtaining a priori estimates for these problems exactly as in the classical case. The credibility of the obtained results is verified by performing numerical calculations for a test problem.