Unique solvability of a non-local problem for mixed type equation with fractional derivative

TL;DR

This paper proves the unique solvability of a boundary value problem involving a mixed parabolic-hyperbolic equation with fractional derivatives, reduced to Volterra integral equations, advancing understanding of fractional PDEs with non-local conditions.

Contribution

It introduces a new approach to analyze mixed type equations with fractional derivatives and non-local boundary conditions, establishing conditions for unique solutions.

Findings

Reduction to Volterra integral equations facilitates solvability analysis.

Established conditions ensuring unique solutions for the non-local fractional PDE.

Extended the theory of mixed type equations with fractional derivatives.

Abstract

In this work we investigate a boundary problem with non-local conditions for mixed parabolic-hyperbolic type equation with three lines of type-changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part we use solution of the first boundary problem with appropriate Green's function and in hyperbolic parts we use corresponding solutions of the Cauchy problem.