Initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary elliptic differential operator

TL;DR

This paper investigates an initial-boundary value problem for a time-fractional subdiffusion equation involving an arbitrary elliptic operator, establishing conditions for the existence, uniqueness, and convergence of classical solutions.

Contribution

It provides a rigorous proof of solution existence and uniqueness using Fourier methods for a broad class of elliptic operators in fractional diffusion equations.

Findings

Proved existence and uniqueness of classical solutions.

Established sufficient and necessary conditions for Fourier series convergence.

Applied Fourier method to fractional subdiffusion equations with arbitrary elliptic operators.

Abstract

An initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary order elliptic differential operator is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. In the case of an initial-boundary value problem on N-dimensional torus, one can easily see that these conditions are not only sufficient, but also necessary.