Numerical integration of blow-up problems on the basis of non-local transformations and differential constraints

Abstract

Several new methods of numerical integration of Cauchy problems with blow-up solutions for nonlinear ordinary differential equations of the first-and secondorder are described. Solutions of such problems have singularities whose positions are unknown a priori (for this reason, the standard numerical methods for solving problems with blow-up solutions can lead to significant errors). The first * This article is a significantly extended and more general version of the article A.D. Polyanin and I.K. Shingareva, The use of differential and non-local transformations for numerical integration of non-linear blow-up problems,