Fractional Telegraph equation with the Riemann-Liouville derivative

TL;DR

This paper investigates a fractional telegraph equation involving the Riemann-Liouville derivative, establishing existence, uniqueness, and stability of solutions, and extends the approach to more general elliptic operators.

Contribution

It provides a rigorous analysis of the fractional telegraph equation with Riemann-Liouville derivatives, including existence, uniqueness, stability, and a method adaptable to general elliptic operators.

Findings

Proved existence and uniqueness of solutions.

Derived stability inequalities.

Extended method to general elliptic operators.

Abstract

The Telegraph equation $(\partial_{t}^{\rho })^{2}u(x,t)+2\alpha \partial_{t}^{\rho }u(x,t)-u_{xx}(x,t)=f(x,t)$, where $0<t\leq T$ and $0<\rho<1$, with the Riemann-Liouville derivative is considered. Existence and uniqueness theorem for the solution to the problem under consideration is proved. Inequalities of stability are obtained. The applied method allows us to study a similar problem by taking instead of $d^2/dx^2$ an arbitrary elliptic differential operator $A(x, D)$, having a compact inverse.