A computational approach to solving some applied rigid second-order problems

Abstract

When a differential equation exhibits chaos, stiffness, damping and/or oscillation in its solution component, such a differential equation is termed rigid. Over the years, solving such problems has posed a serious challenge to researchers. A new computational approach was employed in this research by proposing a Two-Point Hybrid Block Method (TPHBM) for the solutions of some applied rigid second order problems. The method was constructed via the interpolation and collocation techniques within a two-step integration interval [xn, xn+2]. The procedure for the construction of the method involved the interpolation of the basis function at two selected points, collocation at all points and the incorporation of four off-grid points. This is aimed at improving the accuracy of the method, circumventing the Dahlquist-barrier as well as optimizing the order of the method while maintaining a low step-number. The paper further analyzed some basic properties of the method. Rigid second order problems like Bessel, Duffing and Kepler problems were solved using the TPHBM and the results generated showed that it outperformed other existing methods.