Boundary value problem for the time-fractional telegraph equation with Caputo derivatives

TL;DR

This paper establishes a Green formula for Caputo fractional derivatives, derives integral representations for solutions of the time-fractional telegraph equation, and constructs unique solutions for initial-boundary value problems, advancing fractional PDE analysis.

Contribution

It introduces a Green formula for Caputo derivatives and provides a method to obtain integral solutions for the time-fractional telegraph equation, enabling analysis of more general fractional PDEs.

Findings

Green formula for Caputo derivatives proved

Integral representation of solutions derived

Unique solutions for initial-boundary value problems constructed

Abstract

In this paper the Green formula for the operator of fractional differentiation in Caputo sense is proved. By using this formula the integral representation of all regular in a rectangular domains solutions is obtained in the form of the Green formula for operator generating the time-fractional telegraph equation. The unique solutions of the initial-boundary value problem with boundary conditions of first kind is constructed. The proposed approach can be used to study the more general evolution FPDE as well as ODE with Caputo derivatives.