Abstract
One of the analytical methods of presenting solutions of nonlinear partial differential equations is the method of special series in powers of specially selected functions called basic functions. The coefficients of such series are found successively as solutions of linear differential equations. The basic functions can also contain arbitrary functions. By using such functional arbitrariness allows in some cases, to prove the global convergence of the corresponding constructed series, and also allows us to prove the solvability of the boundary value problem for the Korteweg-de Vries equation. In the paper for a nonlinear wave equation a theorem on the possibility of satisfying a given boundary condition using an arbitrary function contained in the basic function is proved.
Highlights
One of the analytical methods of presenting solutions of nonlinear partial differential equations is the method of special series in powers of specially selected functions called basic functions
The basic functions can contain arbitrary functions. By using such functional arbitrariness allows in some cases, to prove the global convergence of the corresponding constructed series, and allows us to prove the solvability of the boundary value problem for the Korteweg-de Vries equation
In the paper for a nonlinear wave equation a theorem on the possibility of satisfying a given boundary condition using an arbitrary function contained in the basic function is proved
Summary
Method of special series [1, 2, 3] is a method of representation of solutions of nonlinear partial differential equations in the form of series by the powers of one or several functions chosen in a special way, which allow the series coefficients to be calculated recurrently without applying any truncation procedures. In some cases it was possible to construct the basis functions that take into account presence of a known exact solutions, as well as the specific character of non-linear equations. These series were built and studied in [16, 17].
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