Sliced-time computations with re-scaling for blowing-up solutions to initial value differential equations

Abstract

In this paper, we present a new approach to simulate time-dependent initial value differential equations which solutions have a common property of blowing-up in a finite time. For that purpose, we introduce the concept of "sliced-time computations", whereby, a sequence of time intervals (slices) {[Tn-1, Tn]|n ≥ 1} is defined on the basis of a change of variables (re-scaling), allowing the generation of computational models that share symbolically or numerically "similarity" criteria. One of these properties is to impose that the re-scaled solution computed on each slice do not exceed a well-defined cut-off value (or threshold) S. In this work we provide fundamental elements of the method, illustrated on a scalar ordinary differential equation y′ = f(y) where f(y) verifies ∫ 0∞ ∫(y)dy < ∞. Numerical results on various ordinary and partial differential equations are available in [7], some of which will be presented in this paper. © Springer-Verlag Berlin Heidelberg 2005.