Averaging equations of mathematical physics with coefficients dependent on coordinates and time

Differential equations with variable coefficients describe the processes proceeding in inhomogeneous materials in which mechanical characteristics change either abruptly or continuously in the boundary area between the phases. One of the general approaches to solving equations with variable coefficients is the use of the averaging method, which implies some of the ways to represent the solution of the initial equation in terms of a solution of an equation with constant coefficients. In the present paper, an integral formula has been obtained which presents the solution of the original linear differential equation of the second order with the coefficients depending on the coordinates and time, through the solution of the same equation with constant coefficients (the concomitant equation). The kernel of the integral formula includes the Green function of the original equation and the difference of the coefficients of the original and concomitant equations. From the integral formula an equivalent representation of the solution of the initial equation in the form of a series of all possible derivatives of the solution of the concomitant equation is obtained. The coefficients of the series are called structure functions. They depend substantially on the form of the inhomogeneity and tend to zero as the coefficients of the original equation tend to the constant coefficients of the concomitant equation. A system of recurrence equations satisfied by the structural functions is written. Examples of calculation of the structure functions are given. © 2017 Begell House, Inc.