Computation of blowing-up solutions for second-order differential equations using re-scaling techniques

Nabil R. Nassif, Noha Makhoul-Karam, Yeran Soukiassian

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8 Citations (Scopus)

Abstract

This paper presents a new technique to solve efficiently initial value ordinary differential equations of the second-order which solutions tend to have a very unstable behavior. This phenomenon has been proved by Souplet et al. in [P. Souplet, Critical exponents, special large-time behavior and oscillatory blow-up in nonlinear ode's, Differential and Integral Equations 11 (1998) 147-167; P. Souplet, Etude des solutions globales de certaines équations différentielles ordinaires du second ordre non-linéaires, Comptes Rendus de I'Academie des Sciences Paris Série I 313 (1991) 365-370; P. Souplet, Existence of exceptional growing-up solutions for a class of nonlinear second order ordinary differential equations, Asymptotic Analysis 11 (1995) 185-207; P. Souplet, M. Jazar, M. Balabane, Oscillatory blow-up in nonlinear second order ode's: The critical case, Discrete And Continuous dynamical systems 9 (3) (2003)] for the ordinary differential equation y - b | y |q - 1 y + | y |p - 1 y = 0, t > 0, p > 0, q > 0, whereby the time interval of existence of the solution is finite [0, Tb] with limt → Tb- | y (t) | = limt → Tb- | y (t) | = ∞. The blow-up of the solution and its derivatives is handled numerically using a re-scaling technique and a time-slices approach that controls the growth of the re-scaled variable through a cut-off value S. The re-scaled models on each time slice obey a criterion of mathematical and computational similarity. We conduct numerical experiments that confirm the accuracy of our re-scaled algorithms.

Original languageEnglish
Pages (from-to)185-195
Number of pages11
JournalJournal of Computational and Applied Mathematics
Volume227
Issue number1
DOIs
Publication statusPublished - 1 May 2009
Externally publishedYes

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Keywords

  • Blow-up behavior
  • End-of-slice condition
  • Re-scaling
  • Second-order differential equations
  • Similarity
  • Time-slices

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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