Approximate methods for solving nonlinear initial boundary-value problems based on special constructions of series

Abstract

New constructions of several series with recurrently calculated coefficients are described for a wide class of nonlinear partial differential equations. The coefficients are calculated without applying a truncation procedure. The properties of basic functions in such series are explored as well as their analytic representation and approximate capabilities. The series constructed are used to solve periodical Cauchy problems for quasilinear dissipative systems and initial boundary-value problems for several types of nonlinear equations. New applications of characteristic series to the solution of the Goursat problem for a nonlinear hyperbolic equation and to the solution of the initial boundary-value problem for a nonlinear equation describing filtration of a gas in a porous soil are given. The series convergence theorems are formulated, the estimates of coefficients are given, and numerical results are presented.Series with the coefficients calculated recurrently are now rather rarely used to solve nonlinear partial differential equations. Either difference approaches or finite element methods prevail. Occasionally, direct statistical modelling is also used. However, while using these methods, we are led to overcome a number of obstacles (in particular, very fine grids or a large number of basic functions are needed) and to perform large-scale computations. Perhaps, for special classes of equations, only the so-called 'saturation-free methods' [1,6] allow the main difficulties to be resolved and quite accurate approximate solutions to be obtained economically. Good use is made of these methods for linear elliptic equations. They have also gained widespread acceptance in solving a number of nonlinear hydrodynamics problems [6,7]. Successful application of these approaches requires in-depth study of the problem being solved.One of the trends in developing economic methods for obtaining highly accurate solutions for a wide class of nonlinear differential equations is the use of series with the coefficients calculated recurrently. Unlike well-known Fourier methods for linear equations, in which the coefficients are also calculated recurrently, in the nonlinear case the variables cannot be usually separated and the superposition principle is inapplicable. If the solutions are represented by Fourier series, one or another truncation procedure is usually used to derive the final system of equations, from which the coefficients are to be determined. Thus, when increasing the number of basic functions, one has to repeatedly calculate the coefficients, which are usually found from a nonlinear system of equations. Asymptotic expansion methods lead to a successive determination of the expansion coefficients from linear systems of equations (probably, except for the zeroeth coefficient), but they require that the equation should contain a small or a large parameter.In what follows, for quite a wide class of nonlinear equations we will describe several constructions of series, which allow the series coefficients to be calculate...

On application of Fourier method for one class of equations describing nonlinear oscillations

“…where N ≥ J. If we substitute (6) into (1), take into account the assumptions (4), (5) and equate the expressions at the same X s (x), s = 1, N, we obtain the system of nonlinear differential equations for the coefficients z s (ε, t)…”

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Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

“…Sidorov and his followers (see, for example, refs. [12,13]). Statements of boundary value problems on the initiation of filtration waves, as well as methods for constructing the solutions that have HDW-type, in the class of analytical functions, are proposed in [12,13].…”

“…[12,13]). Statements of boundary value problems on the initiation of filtration waves, as well as methods for constructing the solutions that have HDW-type, in the class of analytical functions, are proposed in [12,13]. The authors of this article also belong to the scientific school of A. F. Sidorov.…”

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