Theoretical Analysis of the Wave Equation

Abstract

The main aim of this work is to study the problem of wave equation, first as a physical motivation and then with a careful analysis of the problem, emphasizing mathematical aspects of theoretical physics, enriching it with greater mathematical rigor, clarity of thought and cleaning of arguments and assumptions. This situation leads us to a problem in which value of a spatial variable or its derivative is specified by boundary conditions. In order to obtain solutions to these initial problems or boundary values, the Fourier resolution method is used, which consists in two steps. The first uses the separation of variables so that it can obtain eigenvalue problems for ordinary differential equations which are closely related to partial differential equations under study. On this step, we obtain a range of solutions of partial differential equation that satisfies part of the boundary conditions. The second step, called Fourier analysis, which main idea is using the solution of the problem as a series whose terms are products of these solutions by appropriately chosen coefficients.