Initial-boundary value problem for a subdiffusion equation with the Caputo derivative

TL;DR

This paper studies a subdiffusion equation with Caputo derivatives on an N-dimensional torus, using Fourier methods based on eigenfunction expansion, applicable to various domains and elliptic operators with fixed solution existence conditions.

Contribution

Introduces a Fourier-based method for solving time-fractional subdiffusion equations on arbitrary domains with variable coefficient elliptic operators, establishing solution existence conditions.

Findings

Solution existence conditions are strict and cannot be weakened.

Method applicable to arbitrary domains and variable coefficient elliptic operators.

Provides an explicit example illustrating the solution conditions.

Abstract

We investigate an initial-boundary value problem for a time-fractional subdiffusion equation with the Caputo derivatives on $N$-dimensional torus by the classical Fourier method. Since our solution is established on the eigenfunction expansion of elliptic operator, the method proposed in this article can be used to an arbitrary domain and an elliptic operator with variable coefficients. It should be noted that the conditions for the existence of a solution to the initial-boundary value problem found in the article cannot be weakened, and the article provides a corresponding example.