Representation of solutions of initial boundary value problems for nonlinear equations of mathematical physics by special series

Abstract

An analytical method is used to represent solutions of nonlinear partial differential equations. This method is called a method of special series. The essence of this method is in the expansion of solutions of nonlinear partial differential equations into series by the powers of special basic functions, which can also contain an arbitrary function. Such a choice of basic functions makes it possible to find the coefficients of series from a sequence of linear ordinary differential equations and to investigate convergence of these series. A class of boundary conditions that can be satisfied with the help of the proposed series is determined and a theorem on the series convergence to the solution of the initial-boundary value problem for a certain class of initial conditions on the example of a nonlinear filtration equation is proved. The proposed method of constructing solutions can be used to find solutions for a wider class of nonlinear equations of mathematical physics.