Representation of Green’s functions of the wave equation on a segment in finite terms

Abstract

Solutions of initial-boundary value problems on the excitation of oscillations of a finite segment by an instantaneous point sourse are investigated. Solutions to these problems, called Green's functions of the equation of oscillations on a segment, are known in the form of infinite Fourier series or series in terms of Heaviside functions. A. N. Krylov's method of accelerating the convergence of Fourier series for several types of boundary conditions not only accelerates the convergence, but allows one to compose expressions for Green's functions in finite terms. In this paper, finite expressions of Green's functions are given in the form of elementary functions of a real variable. Four different formulations of boundary conditions are considered, including the periodicity conditions.

Keywords

• equation of oscillations on a segment
• green’s function
• representation in finite terms
• boundary conditions
• a. n. krylov’s method