An Inverse Source Problem For a Time-Fractional Mixed Wave-Diffusion-Wave Equation in a Cylindrical Domain

TL;DR

This paper studies an inverse source problem for a fractional wave-diffusion-wave equation with variable order in a cylindrical domain, using Fourier-Bessel series and separation of variables to prove solution existence.

Contribution

It introduces a novel approach combining variable-order fractional derivatives and Fourier-Bessel series for inverse problems in cylindrical geometries.

Findings

Established the existence of solutions for the inverse source problem.

Analyzed the uniform convergence of the Fourier-Bessel series solution.

Abstract

This paper addresses the inverse source problem for a mixed-type fractional wave-diffusion-wave equation posed in a cylindrical domain. The governing equation involves a time-dependent variable-order fractional derivative, which enables the model to effectively capture temporal transitions between wave-like and diffusive behaviors. The solution is constructed in the form of a Fourier-Bessel series. By employing the method of separation of variables together with fundamental properties of Bessel functions, we analyze the uniform convergence of the resulting infinite series. This analysis ultimately leads to a rigorous proof of the existence of a solution.