Publication:
Non-linear blow-up problems for systems of ODEs and PDEs: Non-local transformations, numerical and exact solutions

Дата
2019
Авторы
Shingareva, I. K.
Polyanin, A. D.
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Научные группы
Организационные подразделения
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Институт интеллектуальных кибернетических систем
Цель ИИКС и стратегия развития - это подготовка кадров, способных противостоять современным угрозам и вызовам, обладающих знаниями и компетенциями в области кибернетики, информационной и финансовой безопасности для решения задач разработки базового программного обеспечения, повышения защищенности критически важных информационных систем и противодействия отмыванию денег, полученных преступным путем, и финансированию терроризма.
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Аннотация
© 2019 Elsevier Ltd 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 ,…,x n ), m=1,…,n, based on the introduction a new non-local independent variable ξ which is related to the original variables t and x 1 ,…,x n by the equation ξ t ′ =g(t,x 1 ,…,x n ,ξ). 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 ξ. 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≤x≤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.
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Shingareva, I. K. Non-linear blow-up problems for systems of ODEs and PDEs: Non-local transformations, numerical and exact solutions / Shingareva, I.K., Polyanin, A.D. // International Journal of Non-Linear Mechanics. - 2019. - 10.1016/j.ijnonlinmec.2019.01.012
URI
https://www.doi.org/10.1016/j.ijnonlinmec.2019.01.012
 https://www.scopus.com/record/display.uri?eid=2-s2.0-85061273549&origin=resultslist
 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=Alerting&SrcApp=Alerting&DestApp=WOS_CPL&DestLinkType=FullRecord&UT=WOS:000463312600003
 https://openrepository.mephi.ru/handle/123456789/16568
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