On the Initial Boundary Value Problem to the Time-Fractional Wave Equation with Acoustic Boundary Conditions

TL;DR

This paper investigates the well-posedness of a time-fractional wave equation with acoustic boundary conditions in bounded domains, employing the Faedo-Galerkin method and extending classical ODE solution techniques.

Contribution

It establishes well-posedness for the initial boundary value problem of a time-fractional wave equation with acoustic boundary conditions using an extended Faedo-Galerkin approach.

Findings

Proves well-posedness of the fractional wave equation with acoustic boundary conditions.

Develops an extended Faedo-Galerkin method for fractional differential systems.

Solves a generalized system of time-fractional ODEs extending classical theorems.

Abstract

This paper is concerned with the study of the well-posedeness for the initial boundary value problem to the time-fractional wave equation with acoustic boundary conditions. The problem is considered in a bounded and connected domain $\Omega \subset {\mathbb{R}^{n}}$, $n \geq 2$, which includes simply connected regions. The boundary of $\Omega$ is made up of two disjoint pieces $\Gamma_{0}$ and $\Gamma_{1}.$ Homogeneous Dirichlet conditions are enforced on $\Gamma_0$, while acoustic boundary conditions are considered on $\Gamma_1$. To establish our main result, we employ the Faedo-Galerkin method and successfully solve a general system of time-fractional ordinary differential equations which extends the scope of the classical Picard-Lindel\"of theorem.