Falkner Block Methods for solving second-order ODEs

Abstract

This article presents a two-step Falkner block method for the numerical solution of second-order initial value problems (IVPs) in ordinary differential equations (ODEs). The scheme is derived using collocation and interpolation techniques, the proposed Falkner block method offers a novel solution to second-order IVPs in ODEs. It eliminates the transformation of the second-order
ODEs into a system of first-order equations, approximating the solution through polynomial interpolation at strategically chosen collocation points, which enhances accuracy and computational efficiency. Utilizing appropriate basis functions for interpolation allows the Falkner block method to capture the solution’s features, effectively enabling higher-order approximations. This collo-
cation framework ensures the residuals align with the original differential equation, promoting a stable numerical scheme. Additionally, the method facilitates simultaneous calculations of multiple values, improving computational speed and accommodating diverse initial conditions. Comparative studies demonstrate that the Falkner block method outperforms traditional numerical methods and other hybrid codes in global accuracy and convergence