Forward and inverse problems for a mixed-type equation with the Caputo fractional derivative and Dezin-type non-local condition

TL;DR

This paper investigates forward and inverse problems for a mixed-type PDE involving Caputo fractional derivatives and Dezin-type non-local conditions, establishing existence, uniqueness, and parameter dependency results.

Contribution

It introduces a novel analysis of mixed-type equations with fractional derivatives and non-local conditions, providing new existence and uniqueness results for both forward and inverse problems.

Findings

Existence and uniqueness of solutions for the forward problem are proven.

The solvability depends on the parameter in the Dezin-type condition.

Inverse problem solutions are unique under certain conditions on the separable right-hand side.

Abstract

This work is dedicated to the study of a mixed-type partial differential equation involving a Caputo fractional derivative in the time domain $t > 0$ and a classical parabolic equation in the domain $t < 0$, along with Dezin-type non-local boundary and gluing conditions. The forward and inverse problems are studied in detail. For the forward problem, the existence and uniqueness of solutions are established using the Fourier method, under appropriate assumptions on the initial data and the right-hand side. We also analyze the dependency of solvability on the parameter $\lambda$, from the Dezin-type condition. For the inverse problem, where the right-hand side is separable as $F(x,t) = f(x)g(t)$ (the unknown function is $f(x)$), the existence and uniqueness of a solution are proven under a certain condition on the function $g(t)$ (a constant sign is sufficient).