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image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Ukrainian Mathematic...arrow_drop_down
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
Ukrainian Mathematical Journal
Article . 1995 . Peer-reviewed
License: Springer TDM
Data sources: Crossref
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
zbMATH Open
Article . 1995
Data sources: zbMATH Open
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A method for solution of nonlinear ordinary differential equations

A method for solving of nonlinear ordinary differential equations
Authors: Perchik, E. L.;

A method for solution of nonlinear ordinary differential equations

Abstract

The author proposes a scheme for solving the problem \[ u''+2u'/x+\beta ^2u^\nu =0,\;u(0)=0, \quad u'(0)=0. \] After rather cumbersome manipulations he obtains the integral equation \[ u(x)=1+\lambda ^{-1}\int_{0}^{x}\Biggl[(c\lambda +\xi)\psi [a](\xi)-\varepsilon ^{-\mu }(c\xi)^{-1}\int_{0}^{\xi }\zeta ^2a(\zeta) d\zeta \Biggr]d\xi \] with \(a=-\varepsilon ^{\mu }\beta ^2u^{\nu-1 },\;\mu =1/(2-\nu),\;c,\lambda \) are parameters, and \(\psi[a](\xi)\) is a linear Volterra operator with complicated kernel. The author finds a ``natural'' Volterra equation \(u(x)=1-\int_{0}^{x}s^{-2}\int_{0}^{s}\beta ^2\xi ^2u^\nu (\xi) d\xi ds\) to be inefficient for applications since its kernel contains the singularity \(s^{-2}\). But the same singularity appears in \(\psi\) for the presence of the term \((\varepsilon ^\mu cx^2)^{-1}\int_{0}^{x}\xi ^2a(\xi) d\xi \). So, the advantages of the author's approach are not obvious.

Keywords

Volterra integral equations, Fredholm equation, Volterra equation, Emden equation, Fredholm integral equations, Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations, Nonlinear ordinary differential equations and systems, Poisson integral, Theoretical approximation of solutions to ordinary differential equations

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
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