# Inverse source problems for degenerate time-fractional PDE

## TL;DR

This paper studies inverse source problems for degenerate time-fractional PDEs in rectangular domains, providing series solutions, convergence, uniqueness, and new estimates for special functions involved.

## Contribution

It introduces novel series solutions for two types of degenerate time-fractional PDE inverse problems and establishes new estimates for the generalized Mittag-Leffler function.

## Key findings

- Solutions expressed as Fourier-Legendre and Fourier-Sine series.
- Proved convergence and uniqueness of solutions.
- Derived new estimates for the generalized Mittag-Leffler function.

## Abstract

In this paper, we investigate two inverse source problems for degenerate time-fractional partial differential equation in rectangular domains. The first problem involves a space-degenerate partial differential equation and the second one involves a time-degenerate partial differential equation. Solutions to both problem are expressed in series expansions. For the first problem, we obtained solutions in the form of Fourier-Legendre series. Convergence and uniqueness of solutions have been discussed. Solutions to the second problem are expressed in the form of Fourier-Sine series and they involve a generalized Mittag- Leffler type function. Moreover, we have established a new estimate for this generalized Mittag-Leffler type function. The obtained results are illustrated by providing example solutions using certain given data at the initial and final time.