DOI: https://doi.org/10.15588/1607-3274-2019-1-6

METHOD OF TWO-SIDED APPROXIMATIONS OF THE SOLUTION OF THE FIRST BOUNDARY VALUE PROBLEM FOR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON THE GREEN’S FUNCTION USE

M. V. Sidorov

Abstract

Context. The questions of constructing a two-sided iterative process for finding a positive solution of the first boundary value problem for an ordinary second-order differential equation on the basis of the method of Green’s functions are considered. The object of the study is the first boundary value problem for an ordinary second-order differential equation The purpose of the paper is to develop a method of twosided approximations of the problem solution by using the methods of the nonlinear operators theory in semi-ordered spaces.
Method. With the Green’s function help the original nonlinear boundary value problem for an ordinary differential equation is replaced by an equivalent integral equation, considered in the space of continuous functions, which is semi-ordered by means of the cone of nonnegative functions. The integral equation is represented as a nonlinear operator equation with a heterotone operator. For this equation a strongly invariant conic segment, the ends of which serve as initial approximations for two iterative sequences, is sought. The first of the sequences,
monotonically increasing, approximates the exact solution of the problem from below, and the second one, monotonically decreasing, approximates
it from above. Two conditions for the existence of a unique positive solution of the boundary value problem under consideration and two-sided convergence of successive approximations to it are given. General recommendations on the construction of a strongly invariant conic segment are also given. The developed method has a simple computational implementation and a posteriori error estimate, convenient for use in practice.
Results. The developed method was programmed and investigated in solving test problems. The results of the computational experiment are illustrated graphically and with the help of tables.
Conclusions. The conducted experiments have confirmed the efficiency and effectiveness of the developed method and allow to recommend it for use in practice for solving the problems of mathematical modeling of nonlinear processes. The prospects for further research may include the development of two-sided methods for solving problems for partial differential equations and non-stationary problems using semi-discrete methods (for example, the Rothe’s method of lines).

Keywords

nonlinear boundary value problem for an ordinary differential equation; positive solution; strongly invariant conical segment; heterotone operator; two-sided approximations; Green’s function.

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