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.

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.


GOST Style Citations


1. Pao C. V. Nonlinear parabolic and elliptic equations /
C. V. Pao. – New York : Plenum Press, 1992. – 794 p. DOI:
10.1007/978-1-4615-3034-3
2. Самарский А. А. Математическое моделирование: Идеи.
Методы. Примеры / А. А. Самарский, А. П. Михайлов. – 2-е
изд., испр. – М. : Физматлит, 2001. – 320 с.
3. Мышкис А. Д. Элементы теории математических моделей /
А. Д. Мышкис. – 3-е изд., испр. – М. : КомКнига, 2007. –
192 с.
4. Франк-Каменецкий Д. А. Основы макрокинетики. Диффу-
зия и теплопередача в химической кинетике / Д. А. Франк-
Каменецкий. – М. : Интеллект, 2008. – 408 с.
5. Самарский А. А. Теория разностных схем / А. А. Самар-
ский. – 3-е изд., испр. – М. : Наука, 1989. – 616 с.
6. Chen G. Algorithms and visualization for solutions of nonlinear
elliptic equations / G. Chen, J. Zhou, W.-M. Ni // International
Journal of Bifurcation and Chaos in Applied Sciences and Engineering.
– 2000. – Vol. 10, № 7. – P. 1565–1612. DOI:
10.1142/S0218127400001006
7. Красносельский М. А. Положительные решения оператор-
ных уравнений / М. А. Красносельский. – М. : Физматгиз,
1962. – 394 с.
8. Опойцев В. И. Нелинейные операторы в пространствах с
конусом / В. И. Опойцев, Т. А. Хуродзе. – Тбилиси : Изд-во
Тбилис. ун-та, 1984. – 246 с.
9. Колосов А. И. Конструктивное исследование краевых задач
для нелинейных дифференциальных уравнений / А. И. Ко-
лосов, С. В. Колосова, М. В. Сидоров // Вісник Запорізького
національного університету. Серія: фізико-математичні
науки. – 2012. – № 2. – С. 50–57.
10. Двосторонні наближені методи / [Б. А. Шувар, М. І. Копач,
С. М. Ментинський, А. Ф. Обшта]. – Івано-Франковськ :
ВДВ ЦІТ, 2007. – 515 с.
11. Зайцев В. Ф. Справочник по обыкновенным дифференци-
альным уравнениям / В. Ф. Зайцев, А. Д. Полянин. – М. :
Физматлит, 2001 – 576 с.
12. Чаплыгин С. А. Новый метод приближенного интегрирова-
ния дифференциальных уравнений / С. А. Чаплыгин. – М. :
Гостехиздат, 1950. – 102 с.
13. Вулих Б. З. Введение в теорию полуупорядоченных про-
странств / Б. З. Вулих. – М. : ГИФМЛ, 1961. – 260 с.
14. Amann H. Fixed Point Equations and Nonlinear Eigenvalue
Problems in Ordered Banach Spaces / H. Amann // SIAM Review.
– 1976. – Vol. 18, № 4. – P. 620–709. DOI:
10.1137/1018114
15. Колосов А. И. Нелинейные краевые задачи со свободной
границей для обыкновенных дифференциальных уравнений
математической физики: дис. … доктора физ.-мат. наук :
01.01.03 / Колосов Анатолий Иванович. – Москва, 1991. –
267 с.
16. Курпель Н. С. Двусторонние операторные неравенства и их
применение / Н. С. Курпель, Б. А. Шувар. – К. : Наук. дум-
ка, 1980. – 268 с.
17. Zhao Z. Uniqueness of positive solutions for singular nonlinear
second-order boundary-value problems / Z. Zhao // Nonlinear
Analysis: Theory, Methods & Applications. – 1994. – Vol. 23,
№ 6. – P. 755–765. DOI: 10.1016/0362-546X(94)90217-8.
18. O’Regan D. Singular second order boundary value problems /
D. O’Regan // Nonlinear Analysis: Theory, Methods & Applications.
– 1990. – Vol. 15, № 12. – P. 1097–1109. DOI:
10.1016/0362-546X(90)90046-J.
19. Rus M.-D. The method of monotone iterations for mixed monotone
operators : Ph.D. Thesis Summary / M.-D. Rus. – Cluj-
Napoca, 2010. – 45 p.
20. Вороненко М. Д. Конструктивне дослідження нелінійний
крайових задач для звичайних диференціальних рівнянь /
М. Д. Вороненко, М. В. Сидоров // Радиоэлектроника и ин-
форматика. – 2018. – № 1 (80). – С. 48–54







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