Correct Solution of the Mixed Boundary Value Problem for the n-Dimensional Wave Equation

Abstract

This research work investigates the well-posedness of a mixed boundary value problem for the $ n $-dimensional wave equation subject to Cauchy and non-homogeneous Neumann conditions within the framework of Sobolev spaces. The primary scientific novelty of this study lies in the construction of the analytical solution for an $ n $-dimensional physical process governed by a source term given in the separable form $ f(x, t) = P_n(t) \cdot F(x, t) $. Here, $ P_n(t) $ represents an $ n $th degree algebraic polynomial, while $ F(x, t) $ denotes a Fourier series. The uniqueness of the solution $ u(x, t) $ is rigorously sought within the appropriate functional space $ W^{2, 1}_2(\Omega_T) $. The initial state data are taken from the Sobolev spaces $ W^1_2(\Omega) $ and $ L_2(\Omega) $, whereas the non-homogeneous Neumann boundary condition is prescribed by the function $ g(x, t) $.