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

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 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.