Approximate Solution Using Elementary Functions of Mixed Problem with Boundary Conditions of the Second Kind for One-Dimensional Wave Equation

Abstract

The paper considers a mixed problem with boundary conditions of the second kind for a one-dimensional wave equation. The solution to this problem is written in integral form using the Green’s function. For practical use, this solution is of little use, since, firstly, the Green’s function is a trigonometric series and, therefore, its calculation presents certain difficulties, secondly, it is necessary to calculate approximately the five integrals with the Green’s function included in the solution of the problem, and, thirdly, it is extremely difficult to estimate the error of the approximate calculation of the solution.  In this work, these difficulties are overcome, namely, simple expression for the Green’s function  is found in terms of a periodic piecewise linear function, the integrals included in the approximate solution are calculated using periodic piecewise linear, piecewise quadratic and piecewise cubic functions, and, finally,  a  simple and efficient estimate of the approximation error is obtained. The error estimate is linear in the grid steps of the problem and uniform in the spatial variable at any fixed point in time.  Thus, an approximate solution of the problem with an arbitrarily small error is effectively expressed in terms of elementary functions.   An example of solving the problem by the proposed method is given, and graphs of the exact and approximate solutions are plotted.