|
ИСТИНА |
Войти в систему Регистрация |
ИПМех РАН |
||
Non-linear blow-up problems for systems of ODEs and PDEs: Non-local transformations, numerical and exact solutionsстатья
Информация о цитировании статьи получена из
Web of Science,
Scopus
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 19 марта 2019 г.
- Авторы: Polyanin A.D., Shingareva I.K.
- Журнал: International Journal of Non-Linear Mechanics
- Том: 111
- Год издания: 2019
- Издательство: Pergamon Press Ltd.
- Местоположение издательства: United Kingdom
- Первая страница: 28
- Последняя страница: 41
- DOI: 10.1016/j.ijnonlinmec.2019.01.012
- Аннотация: In Cauchy problems with blow-up solutions there exists a singular point whose position is unknown a priori (for this reason, the application of standard fixed-step numerical methods for solving such problems can lead to significant errors). In this paper, we describe a method for numerical integration of blow-up problems for non-linear systems of coupled ordinary differential equations of the first order $(x_m)'_t=f_m(t,x_1,\dots,x_n)$, $m=1,\dots,n$, based on the introduction a new non-local independent variable $\xi$, which is related to the original variables $t$ and $x_1,\dots,x_n$ by the equation $\xi'_t=g(t,x_1,\dots,x_n,\xi)$. With a suitable choice of the regularizing function $g$, the proposed method leads to equivalent problems whose solutions are represented in parametric form and do not have blowing-up singular points; therefore, the transformed problems admit the use of standard numerical methods with a fixed stepsize in $\xi$. Several test problems are formulated for systems of ordinary differential equations that have monotonic and non-monotonic blow-up solutions, which are expressed in elementary functions. Comparison of exact and numerical solutions of test problems showed the high efficiency of numerical methods based on non-local transformations of a special kind. The qualitative features of numerical integration of blow-up problems for single ODEs of higher orders with the use of non-local transformations are described. The efficiency of various regularizing functions is compared. It is shown that non-local transformations in combination with the method of lines can be successfully used to integrate initial-boundary value problems, described by non-linear parabolic and hyperbolic PDEs, that have blow-up solutions. We consider test problems (admitting exact solutions) for nonlinear partial differential equations such as equations of the heat-conduction type and Klein--Gordon type equations, in which the blowing-up occurs both in an isolated point of space $x=x_*$, and on the entire range of variation of the space variable $0\le x\le 1$. The results of numerical integration of test problems, obtained when approximating PDEs by systems with a different number of coupled ODEs, are compared with exact solutions.
- Добавил в систему: Полянин Андрей Дмитриевич