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

✍ Scribed by Nabil R. Nassif; Noha Makhoul-Karam; Yeran Soukiassian

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 754 KB
• Volume: 227
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2008.07.020

✦ Synopsis

a b s t r a c t 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→T - b |y(t)| = lim t→T - b |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 cutoff 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.