Computation of blowing-up solutions for second-order differential equations using re-scaling techniques Academic Article uri icon


  • 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 etal. 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 quations diffrentielles ordinaires du second ordre non-linaires, Comptes Rendus de I'Academie des Sciences Paris Srie 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. 2008 Elsevier B.V. All rights reserved.

published proceedings

  • Journal of Computational and Applied Mathematics

author list (cited authors)

  • Nassif, N. R., Makhoul-Karam, N., & Soukiassian, Y.

citation count

  • 7

complete list of authors

  • Nassif, Nabil R||Makhoul-Karam, Noha||Soukiassian, Yeran

publication date

  • January 2009