### 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, T_{b}] with lim_{t → Tb-} | y (t) | = lim_{t → 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 language | English |
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Pages (from-to) | 185-195 |

Number of pages | 11 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 227 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 May 2009 |

Externally published | Yes |

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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

### Cite this

*Journal of Computational and Applied Mathematics*,

*227*(1), 185-195. https://doi.org/10.1016/j.cam.2008.07.020