Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

TL;DR

This paper extends maximum principles to the one-dimensional time-fractional diffusion equation, constructs solutions via Fourier series, and analyzes their properties, including smoothness and asymptotics, for initial-boundary-value problems.

Contribution

It introduces a maximum principle for the time-fractional diffusion equation and constructs classical solutions using eigenfunction expansions.

Findings

Maximum principle extended to fractional PDEs

Classical solutions constructed via Fourier series

Solution properties including smoothness and asymptotic behavior

Abstract

In this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the one-dimensional time-fractional diffusion equation $$ (D_t^{\alpha} u)(t) = \frac{\partial}{\partial x}(p(x) \frac{\partial u}{\partial x}) -q(x)\, u + F(x,t),\ \ x\in (0,l),\ t\in (0,T) $$ the generalized solution to the initial-boundary-value problem with the Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of the solution are investigated including its smoothness and asymptotics for some special cases of the source function.