METHOD OF SOLVING THE CAUCHY PROBLEM FOR THE WAVE EQUATION IN ONE SPACE DIMENSION BY REDUCING THE ORDER OF DIFFERENTIATIONS

Abstract

In this article, we present method of solving the Cauchy problem for the wave equation in one space dimension by reducing the order of differentiations. To obtain its solution, we use the solution of the Cauchy problem for a linear partial differential equation of the first order with constant coefficients. These methods for solving hyperbolic partial differential equations can be computationally cumbersome and involve much more elementary transformations, but with their help we can quite simply find the class of functions for which we obtain a given solution. In this way, it is possible to transform many types of partial differential equations with constant coefficients, in which differentiation of some given order is carried out in both variables, to an equation whose partial derivative order is one less. The main point of the method for obtaining a solution is the choice of the correct sequence of application of elementary transformations and the simplest operations of differential and integral calculus on the initial given equation.