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
DOI:
https://doi.org/10.15588/1607-3274-2019-1-6Keywords:
nonlinear boundary value problem for an ordinary differential equation, positive solution, strongly invariant conical segment, heterotone operator, two-sided approximations, Green’s function.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).
References
Pao C. V. Nonlinear parabolic and elliptic equations. New
York, Plenum Press, 1992, 794 p. DOI: 10.1007/978-1-4615-
-3
Samarskij A. A., Mihajlov A. P. Matematicheskoe modelirovanie:
Idei. Metody. Primery. 2-e izd., ispr. Moscow, Fizmatlit,
, 320 p.
Myshkis A. D. Jelementy teorii matematicheskih modelej. 3-e
izd., ispr. Moscow, KomKniga, 2007, 192 p.
Frank-Kameneckij D. A. Osnovy makrokinetiki. Diffuzija i
teploperedacha v himicheskoj kinetike. Moscow, Intellekt,
, 408 p.
Samarskij A. A. Teorija raznostnyh shem. 3-e izd., ispr. Moscow,
Nauka, 1989, 616 p.
Chen G., Zhou J., Ni W.-M. Algorithms and visualization for
solutions of nonlinear elliptic equations, International Journal
of Bifurcation and Chaos in Applied Sciences and Engineering,
, Vol. 10, No. 7, pp. 1565–1612. DOI:
1142/S0218127400001006
Krasnosel’skij M. A. Polozhitel’nye reshenija operatornyh
uravnenij. Moscow, Fiz-matgiz, 1962, 394 p.
Opojcev V. I., Hurodze T. A. Nelinejnye operatory v prostranstvah
s konusom. Tbilisi, Izd-vo Tbilis. un-ta, 1984, 246 p.
Kolosov A. I., Kolosova S. V., Sidorov M. V. Konstruktivnoe
issledovanie kraevyh zadach dlja nelinejnyh differencial’nyh
uravnenij, Vіsnik Zaporіz’kogo nacіonal’nogo unіversitetu.
Serіja: fіziko-matematichnі nauki, 2012, No. 2, pp. 50–57.
Shuvar B. A., Kopach M. І., Mentins’kij S. M., Obshta A. F
Dvostoronnі nablizhenі metodi. Іvano-Frankovs’k, VDV CІT,
, 515 p.
Zajcev V. F., Poljanin A. D. Spravochnik po obyknovennym
differencial’nym uravnenijam. Moscow, Fizmatlit, 2001, 576 p.
Chaplygin S. A. Novyj metod priblizhennogo integrirovanija
differencial’nyh uravnenij. Moscow, Gostehizdat, 1950, 102 p.
Vulih B. Z. Vvedenie v teoriju poluuporjadochennyh prostranstv.
Moscow, GIFML, 1961, 260 p.
Amann H. Fixed Point Equations and Nonlinear Eigenvalue
Problems in Ordered Banach Spaces, SIAM Review, 1976,
Vol. 18, No. 4, pp. 620–709. DOI: 10.1137/1018114
Kolosov A. I. Nelinejnye kraevye zadachi so svobodnoj granicej
dlja obyknovennyh differencial’nyh urav-nenij matematicheskoj
fiziki: dis. … doktora fiz.-mat. nauk : 01.01.03 / Kolosov Anatolij
Ivanovich. Mosow, 1991, 267 p.
Kurpel’ N. S., Shuvar B. A. Dvustoronnie operatornye neravenstva
i ih primenenie. Kyiv, Nauk. dumka, 1980, 268 p.
Zhao Z. Uniqueness of positive solutions for singular nonlinear
second-order boundary-value problems, Nonlinear Analysis:
Theory, Methods & Applications, 1994, Vol. 23, No. 6, pp. 755–
DOI: 10.1016/0362-546X(94)90217-8.
O’Regan D. Singular second order boundary value problems,
Nonlinear Analysis: Theory, Methods & Applications, 1990,
Vol. 15, No. 12, pp. 1097–1109. DOI: 10.1016/0362-
X(90)90046-J.
Rus M.-D. The method of monotone iterations for mixed monotone
operators : Ph.D. Thesis Summary. Cluj-Napoca, 2010,
p.
Voronenko M. D., Sidorov M. V. Konstruktivne doslіdzhennja
nelіnіjnij krajovih zadach dlja zvichajnih diferencіal’nih
rіvnjan’, Radiojelektronika i informatika, 2018, No. 1 (80),
pp. 48–54.
Downloads
How to Cite
Issue
Section
License
Copyright (c) 2019 M. V. Sidorov
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Creative Commons Licensing Notifications in the Copyright Notices
The journal allows the authors to hold the copyright without restrictions and to retain publishing rights without restrictions.
The journal allows readers to read, download, copy, distribute, print, search, or link to the full texts of its articles.
The journal allows to reuse and remixing of its content, in accordance with a Creative Commons license СС BY -SA.
Authors who publish with this journal agree to the following terms:
-
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License CC BY-SA that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
-
Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
-
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
