A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations

A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point...
Ausführliche Beschreibung

A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval ( − ∞ , ln ⁡ a ] . On some finite irregular grid { t n ∈ ( − ∞ , ln ⁡ a ] , n = 0 , 1 , . . . , N , t N = ln ⁡ a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. Ausführliche Beschreibung