The use of differential and non-local transformations for numerical integration of non-linear blow-up problems

Abstract

Two new methods of numerical integration of Cauchy problems for nonlinear ODEs of the first- and second-order, which have blow-up solutions are described. In such problems, the position of the singular point is not known in advance. The first method is based on obtaining an equivalent system of equations by applying a differential transformation, where the first derivative (given in the original equation) is chosen as a new independent variable, t=yx′. The second method is based on introducing a new auxiliary non-local variable of the form ξ=∫x0xg(x,y,yx′)dx with the subsequent transformation to the Cauchy problem for the corresponding system of coupled ODEs. Both methods lead to problems whose solutions are represented in parametric form and do not have blowing-up singular points; therefore the standard fixed-step numerical methods can be applied. The efficiency of the proposed methods is illustrated with a number of test problems that admit exact solutions. It is shown that the methods, based on special exp-type transformations (which are particular cases of the general non-local transformation), are more efficient than the method based on the hodograph transformation, the method of the arc-length transformation, and the method based on the differential transformation. The method, based on introducing a non-local variable, can be generalized to the nth-order ODEs and systems of coupled ODEs.