### 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 |
---|---|

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 |

### Fingerprint

### 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

**Computation of blowing-up solutions for second-order differential equations using re-scaling techniques.** / Nassif, Nabil R.; Makhoul-Karam, Noha; Soukiassian, Yeran.

Research output: Contribution to journal › Article

*Journal of Computational and Applied Mathematics*, vol. 227, no. 1, pp. 185-195. https://doi.org/10.1016/j.cam.2008.07.020

}

TY - JOUR

T1 - Computation of blowing-up solutions for second-order differential equations using re-scaling techniques

AU - Nassif, Nabil R.

AU - Makhoul-Karam, Noha

AU - Soukiassian, Yeran

PY - 2009/5/1

Y1 - 2009/5/1

N2 - 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.

AB - 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.

KW - Blow-up behavior

KW - End-of-slice condition

KW - Re-scaling

KW - Second-order differential equations

KW - Similarity

KW - Time-slices

UR - http://www.scopus.com/inward/record.url?scp=61849134785&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=61849134785&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2008.07.020

DO - 10.1016/j.cam.2008.07.020

M3 - Article

AN - SCOPUS:61849134785

VL - 227

SP - 185

EP - 195

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1

ER -