Inverse source non-local problem for mixed type equation with Caputo fractional differential operator

TL;DR

This paper investigates the unique solvability of an inverse-source problem involving a time-fractional mixed type equation with integral transmitting conditions, using series expansion and properties of Mittag-Leffler functions.

Contribution

It introduces a novel approach for solving inverse-source problems with fractional derivatives and integral conditions, including new properties of Mittag-Leffler functions for simplifying proofs.

Findings

Proved uniqueness and existence of solutions under regularity conditions.

Analyzed influence of transmitting conditions on solvability.

Established new properties of Mittag-Leffler functions.

Abstract

In the present work, we discuss a unique solvability of an inverse-source problem with integral transmitting condition for time-fractional mixed type equation in a rectangular domain, where the unknown source term depends on space variable only. The method of solution based on a series expansion using bi-orthogonal basis of space corresponding to a nonself-adjoint boundary value problem. Under certain regularity conditions on the given data, we prove the uniqueness and existence of the solution for the given problem. Influence of transmitting condition on the solvability of the problem is shown as well. Precisely, two different cases were considered; a case of full integral form ($0<\gamma<1$) and a special case ($\gamma=1$) of transmitting condition. In order to simplify the bulky expressions appearing in the proof of the main result, we have established a new property of the recently introduced Mittag-Leffler type function of two variables (see Lemma 2.1).