Solutions of the main boundary value problems for the time-fractional telegraph equation by the Green function method

TL;DR

This paper develops explicit Green function solutions for the inhomogeneous time-fractional telegraph equation with Caputo derivatives, establishing existence, uniqueness, and explicit solution formulas for boundary value problems.

Contribution

It provides a general representation of solutions using Green functions and explicitly constructs fundamental solutions for the fractional telegraph equation.

Findings

Explicit Green functions are derived for the fractional telegraph equation.

Existence and uniqueness theorems are proved for the Cauchy problem.

Solutions are explicitly constructed in terms of Green functions.

Abstract

The inhomogenous time-fractional telegraph equation with Caputo derevatives with constant coefficients is considered. For considered equation the general representation of regular solution in rectangular domain is obtained, and the fundamental solution is presented. Using this representation and the properties of fundamental solution, the Cauchy problem and the basic problems in half-strip and rectangular domains are studied. For Cauchy problem the theorems of existence and uniqueness of solution are proved, and the explicit form of solution is constructed. The solutions of the investigated problems are constructed in terms of the appropriate Green functions, which are also constructed an explicit form.