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The aim of this work is to propose a numerical method for solving different types of partial and ordinary differential equations. The equations share the same common property for their solutions to become infinite (blow up behaviour) or to become null (extinction behaviour) in finite time. This type of equations is solved using a sliced time computing technique, combined with rescaling both the variable time and the solution of the differential system. The main criterion under which the slice of time is defined, consists in imposing that the rescaled solution should not be greater than a preset cut off value. Another selection criterion for the method is based on the invariance and similarity conditions, enforced on the rescaled model in each of the time slices

Le but de cette these est la construction, l'etude et la mise en oeuvre d'une methode de resolution numerique pour differents types d'equations differentielles ordinaires ou partielles, dont la caracteristique commune est la possibilite pour la solution de devenir infinie (comportement explosif oscillatoire ou non-oscillatoire) ou nulle (comportment extinctif) au bout d'un temps fini. Notre approche consiste a resoudre ce type d'equations par tranches de calcul avec un re-dimensionnement de la variable temps et de la solution de l'equation. Le critere principal definissant les tranches de temps est le non-depassement de la solution re-dimensionnee d'un seuil de calcul bien choisi. Un autre critere de la methode se base sur les concepts d'invariance ou de similarite imposee aux solutions re-dimensionnees dans chacune des tranches de calcul