Solvability of a non-local problem with integral transmitting condition for mixed type equation with Caputo fractional derivative

TL;DR

This paper investigates the solvability of a non-local diffusion-wave problem involving Caputo fractional derivatives, establishing conditions for uniqueness and existence of solutions through energy methods and integral equation reduction.

Contribution

It introduces a new approach to prove uniqueness and existence for a non-local fractional diffusion-wave problem with integral transmitting conditions.

Findings

Uniqueness of solutions proven using modified energy integral method.

Existence established via reduction to Fredholm integral equations.

Results contribute to the mathematical theory of fractional differential equations.

Abstract

In the present paper, we discuss solvability questions of a non-local problem with integral form transmitting conditions for diffusion-wave equation with the Caputo fractional derivative in a domain bounded by smooth curves. The uniqueness of the solution of the formulated problem we prove using energy integral method with some modifications. The existence of solution will be proved by equivalent reduction of the studied problem into a system of second kind Fredholm integral equations.