ON SOME NUMERICAL INTEGRATION CURVES FOR PDE IN NEIGHBORHOOD OF "BUTTERFLY" CATASTROPHE POINT

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Abstract

We consider a three-dimensional nonlinear wave equation with the source term smoothly changing over time and space due to a small parameter. The behavior of solutions of this PDE near the typical “butterfly” catastrophe point is studied. In the framework of matched asymptotic expansions method we derive a nonlinear ODE of the second order depending on three parameters to search for the special solution describing the rapid restructuring of the solution of the PDE in a small neighborhood of the catastrophe point, matching with expansion in a more outer layer. Numerical integration curves of the equation for the leading term of the inner asymptotic expansion are obtained.
Original languageEnglish
Pages (from-to)127-140
Number of pages14
JournalURAL MATHEMATICAL JOURNAL
Volume2
Issue number2(3)
DOIs
Publication statusPublished - 2016

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Catastrophe
Numerical integration
Nonlinear ODE
Curve
Matched Asymptotic Expansions
Nonlinear Wave Equation
Behavior of Solutions
Source Terms
Small Parameter
Asymptotic Expansion
Three-dimensional
Term
Framework

GRNTI

  • 27.00.00 MATHEMATICS

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title = "ON SOME NUMERICAL INTEGRATION CURVES FOR PDE IN NEIGHBORHOOD OF {"}BUTTERFLY{"} CATASTROPHE POINT",
abstract = "We consider a three-dimensional nonlinear wave equation with the source term smoothly changing over time and space due to a small parameter. The behavior of solutions of this PDE near the typical “butterfly” catastrophe point is studied. In the framework of matched asymptotic expansions method we derive a nonlinear ODE of the second order depending on three parameters to search for the special solution describing the rapid restructuring of the solution of the PDE in a small neighborhood of the catastrophe point, matching with expansion in a more outer layer. Numerical integration curves of the equation for the leading term of the inner asymptotic expansion are obtained.",
author = "Khachay, {Oleg Y.} and Nosov, {Pavel A.}",
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language = "English",
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publisher = "Институт математики и механики им. Н.Н. Красовского УрО РАН",
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ON SOME NUMERICAL INTEGRATION CURVES FOR PDE IN NEIGHBORHOOD OF "BUTTERFLY" CATASTROPHE POINT. / Khachay, Oleg Y.; Nosov, Pavel A.

In: URAL MATHEMATICAL JOURNAL, Vol. 2, No. 2(3), 2016, p. 127-140.

Research output: Contribution to journalArticleResearchpeer-review

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AB - We consider a three-dimensional nonlinear wave equation with the source term smoothly changing over time and space due to a small parameter. The behavior of solutions of this PDE near the typical “butterfly” catastrophe point is studied. In the framework of matched asymptotic expansions method we derive a nonlinear ODE of the second order depending on three parameters to search for the special solution describing the rapid restructuring of the solution of the PDE in a small neighborhood of the catastrophe point, matching with expansion in a more outer layer. Numerical integration curves of the equation for the leading term of the inner asymptotic expansion are obtained.

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DO - 10.15826/umj.2016.2.011

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