On the reduction of some second-order nonlinear ODEs in physics and mechanics to first-order nonlinear integrodifferential and Abel's classes of equations

Sotiropoulou, AB; Panayotounakos, DE

dc.contributor.author	Sotiropoulou, AB	en
dc.contributor.author	Panayotounakos, DE	en
dc.date.accessioned	2014-03-01T01:19:23Z
dc.date.available	2014-03-01T01:19:23Z
dc.date.issued	2003	en
dc.identifier.issn	0167-8442	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/15452
dc.subject	Asymptotic solutions	en
dc.subject	Cubic nonlinearities	en
dc.subject	Nonlinear ordinary differential equations	en
dc.subject.classification	Engineering, Mechanical	en
dc.subject.classification	Mechanics	en
dc.subject.other	Mathematical transformations	en
dc.subject.other	Nonlinear systems	en
dc.subject.other	Ordinary differential equations	en
dc.subject.other	Stiffness	en
dc.subject.other	Nonlinear oscillations	en
dc.subject.other	Crack propagation	en
dc.subject.other	fracture mechanics	en
dc.subject.other	numerical method	en
dc.title	On the reduction of some second-order nonlinear ODEs in physics and mechanics to first-order nonlinear integrodifferential and Abel's classes of equations	en
heal.type	journalArticle	en
heal.identifier.primary	10.1016/j.tafmec.2003.09.001	en
heal.identifier.secondary	http://dx.doi.org/10.1016/j.tafmec.2003.09.001	en
heal.language	English	en
heal.publicationDate	2003	en
heal.abstract	Second-order ordinary differential equations (ODEs) with strong nonlinear stiffness terms (cubic nonlinearities) governing wave motions, dynamic crack propagations, nonlinear oscillations etc. in physics and nonlinear mechanics are analyzed. Selecting as guide line a second-order nonlinear ODE of the form of the forced Duffing equation and using admissible functional transformations it is possible to reduce it to an equivalent first-order nonlinear integrodifferential equation. The reduced equation is exact. In the limits of small or large values of the parameter characterizing this nonlinear problem, it is shown that further reductions lead to a nonlinear ODE of the Abet classes. Taking into account the known exact analytic solutions of this equivalent equation it is proved that there does not exist an exact analytic solution of this type of equations. However, in cases when convenient functional relations connecting all parameters of the corresponding null equation and the characteristics of the driving force exist, approximate analytic solutions to the problem under consideration are provided. (C) 2003 Elsevier Ltd. All rights reserved.	en
heal.publisher	ELSEVIER SCIENCE BV	en
heal.journalName	Theoretical and Applied Fracture Mechanics	en
dc.identifier.doi	10.1016/j.tafmec.2003.09.001	en
dc.identifier.isi	ISI:000186833200005	en
dc.identifier.volume	40	en
dc.identifier.issue	3	en
dc.identifier.spage	255	en
dc.identifier.epage	270	en