Non-linear blow-up problems for systems of ODEs and PDEs: Non-local transformations, numerical and exact solutions

Abstract

In Cauchy problems with blow-up solutions there exists a singular point whose position is unknown a priori (for this reason, the application of standard fixed-step numerical methods for solving such problems can lead to significant errors). In this paper, we describe a method for numerical integration of blow-up problems for non-linear systems of coupled ordinary differential equations of the first order (xm)t′=fm(t,x1,…,xn), m=1,…,n, based on the introduction a new non-local independent variable ξ, which is related to the original variables t and x1,…,xn by the equation ξt′=g(t,x1,…,xn,ξ). With a suitable choice of the regularizing function g, the proposed method leads to equivalent problems whose solutions are represented in parametric form and do not have blowing-up singular points; therefore, the transformed problems admit the use of standard numerical methods with a fixed stepsize in ξ. Several test problems are formulated for systems of ordinary differential equations that have monotonic and non-monotonic blow-up solutions, which are expressed in elementary functions. Comparison of exact and numerical solutions of test problems showed the high efficiency of numerical methods based on non-local transformations of a special kind. The qualitative features of numerical integration of blow-up problems for single ODEs of higher orders with the use of non-local transformations are described. The efficiency of various regularizing functions is compared. It is shown that non-local transformations in combination with the method of lines can be successfully used to integrate initial–boundary value problems, described by non-linear parabolic and hyperbolic PDEs, that have blow-up solutions. We consider test problems (admitting exact solutions) for nonlinear partial differential equations such as equations of the heat-conduction type and Klein–Gordon type equations, in which the blowing-up occurs both in an isolated point of space x=x∗, and on the entire range of variation of the space variable 0≤x≤1. The results of numerical integration of test problems, obtained when approximating PDEs by systems with a different number of coupled ODEs, are compared with exact solutions.