The theory of the numerical-analytic method: Achievements and new trends of development. VII

A new numerical-analytic algorithm for the investigation of periodic solutions of nonlinear periodic systems of differential equations dx dtA t x f t x / () (, ) = + in the critical case is developed.The problem of the existence of solutions and their approximate construction is studied. Estimates for the convergence of successive periodic approximations are obtained.The theory of periodic boundary-value problems is an extensively developed branch of the theory of ordinary differential equations. There are fairly many works devoted to the investigation of periodic boundary-value problems, which is explained, on the one hand, by their wide application in the study of various technical and other natural processes and, on the other hand, by the wide variety of related problems and the complexity of their investigation, especially in the case of nonlinear systems. Many methods have been developed for the investigation of periodic solutions of systems of differential and other equations. In this connection, without pretending to be exhaustive, we mention the monographs of Bogolyubov and Mitropol'skii [1], Samoilenko and N. Ronto [2 -4], Boichuk, Zhuravlev, and Samoilenko [5], Luchka [6], Malkin [7], Grebenikov and Ryabov [8], and Yakubovich and Starzhinskii [9], which also contain surveys of results and extensive bibliography on related questions.

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“…Substituting (18) in (11), we finally establish that system (7) has a k-parameter family of T-periodic solutions of the form…”

Section: Periodic Solutions Of Linear Periodic Systemsmentioning

confidence: 99%

Once again on the Samoilenko numerical-analytic method of successive periodic approximations

Korol

Perestyuk

2006

Ukr Math J

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“…It is well-known that, formally, the methods used in the the theory of boundary value problems for the investigation and approximate construction of solutions, can be characterised as analytic, functional-analytic, numerical and numerical-analytic ones (see, e. g. [1,4,7,16,20,23] and the references therein). Among the numericalanalytic methods we share out those which are based upon some types of successive approximations.…”

Section: Introductionmentioning

confidence: 99%

Periodic successive approximations and interval halving

Ront¹

2012

MMN

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“…; see, e. g., [10,23] for a bibliography). A survey of many papers devoted to the issues mentioned and published during the last three decades can be found in [11][12][13][14][15][16][17]. The approach indicated suggests one to replace the given boundary value problem by a family of Cauchy problems for a suitably perturbed differential c 2009 Miskolc University Press system containing some artificially introduced parameter´2 R n whose value is to be determined later.…”

Section: Introductionmentioning

confidence: 99%

Succesive approximation method for some linear boundary value problems for differential equations with a special type of argument deviation

Rontó¹

2009

MMN

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The theory of the numerical-analytic method: Achievements and new trends of development. VII

Cited by 11 publications

References 11 publications

Once again on the Samoilenko numerical-analytic method of successive periodic approximations

Once again on the Samoilenko numerical-analytic method of successive periodic approximations

Periodic successive approximations and interval halving

Succesive approximation method for some linear boundary value problems for differential equations with a special type of argument deviation

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