Block Nyström Method for Singular Differential Equations of the Lane–Emden Type and Problems with Highly Oscillatory Solutions

Abstract

Singular and highly oscillatory ordinary differential equations (ODEs) are of great scientific significance, since they frequently arise in practice. Hence, in this paper, we present a Block Nystrom method (BNM) that is applied to directly solve the Lane–Emden type equations which belong to a class of nonlinear singular ODEs. The application of the BNM is also extended to solve highly oscillatory ODEs directly without reducing the ODE into an equivalent first order system. The BNM is formulated from its continuous scheme which is constructed from an appropriate power series via collocation and interpolation techniques. The convergence and stability properties of the BNM are discussed. Accuracy and efficiency benefits of the method are demonstrated via several numerical examples.