Inverse problem for a multi-term time-fractional diffusion equation with the Caputo derivatives

TL;DR

This paper addresses the inverse source problem for a multi-term time-fractional diffusion equation with Caputo derivatives, providing conditions for existence, uniqueness, and regularity of solutions using spectral methods and asymptotic analysis.

Contribution

It introduces novel asymptotic expansions for the multinomial Mittag-Leffler function, enabling analysis of solution existence, uniqueness, and regularity in a complex fractional diffusion setting.

Findings

Derived uniform lower bounds for the solution's characteristic denominator

Established sufficient conditions for classical solution existence

Proved uniqueness under natural data assumptions

Abstract

This paper investigates an inverse source problem for a multi-term time-fractional diffusion equation with Caputo derivatives. The source term is separable as \(f(x)g(t)\), with the unknown spatial component \(f(x)\) reconstructed from an overdetermination condition at interior time \(t_0 \in (0, T]\). The elliptic part is governed by a self-adjoint positive differential operator \(A(x, D)\) of order \(m \ge 2\). The solution features a spectral representation using the multinomial Mittag-Leffler function, for which we derive novel precise asymptotic expansions. These asymptotics provide a uniform lower bound for the solution's characteristic denominator, enabling sufficient conditions for the existence of a classical solution. Uniqueness of the reconstructed source holds under natural assumptions on the data and \(g(t)\). Despite the problem's ill-posedness, high-regularity classical solutions are achievable under suitable structural conditions.