Analytical Solutions and Well-Posedness of the One-Dimensional Wave Equation

Abstract

The one-dimensional wave equation constitutes a fundamental mathematical framework characterizing wave propagation across diverse continuous media. Classical physics often assumes idealized homogeneous boundaries; however, empirical phenomena frequently exhibit non-zero, time-variant constraints. This paper delineates the resolution of the one-dimensional wave equation incorporating functional, non-homogeneous boundary and initial conditions via a comprehensive multi-method analytical approach. Specifically, the classical Fourier Method of Separation of Variables (enhanced by the theoretical lifting technique), D'Alembert's Method of Characteristics, and the Laplace Transform method are evaluated. For each methodology, the mathematical well-posedness—encompassing rigorous theorems of existence, uniqueness, and metric stability—is explicitly proven through algebraic analysis. Furthermore, the paper provides a crucial comparative discussion detailing strict applicability constraints and the scenarios in which specific methods perform well or fail. Finally, explicit computational simulations that natively leverage dynamic functional boundary conditions are visualized in MATLAB.