An initial-boundary problem for a mixed fractional wave equation

TL;DR

This work studies the unique solvability of an initial-boundary value problem for a mixed time-fractional wave equation on a rectangle, featuring time derivatives of orders $1<\alpha_1<2$, $0<\alpha_2<1$, and $1<\beta<2$. The authors apply a spectral expansion using the eigenbasis $\{\sin(n\pi x)\}$ to reduce the PDE to decoupled time-fractional ODEs for coefficients $T_n(t)$, solvable via Caputo derivatives and multi-variable Mittag-Leffler functions. They provide explicit representations for $T_n(t)$, determine the unknown coefficients from transmission and boundary data, and establish uniqueness under a nonvanishing determinant condition $\Delta_n$, along with uniform convergence of the resulting series for $u(t,x)$. The results contribute to the well-posedness theory of mixed-type fractional wave IBVPs and offer a concrete analytical framework for analyzing similar problems with piecewise fractional dynamics and Dirichlet boundaries.

Abstract

We aim to prove a unique solvability of an initial-boundary value problem (IBVP) for a time-fractional wave equation in a rectangular domain. We exploit the spectral expansion method as the main tool and used the solution to Cauchy problems for fractional-order differential equations. Moreover, we apply certain properties of the Mittag-Leffler-type functions of single and two variables to prove the uniform convergence of the solution to the considered problem, represented in the form of infinite series.