A Higher-order Block Method for Numerical Approximation of Third-order Boundary Value Problems in ODEs

Abstract

In recent times, numerical approximation of 3rd-order boundary value problems (BVPs) has attracted great attention due to its wide applications in solving problems arising from sciences and engineering. Hence, A higher-order block method is constructed for the direct solution of 3rd-order linear and non-linear BVPs. The approach of interpolation and collocation is adopted in the derivation. Power series approximate solution is interpolated at the points required to suitably handle both linear and non-linear third-order BVPs while the collocation was done at all the multiderivative points. The three sets of discrete schemes together with their first, and second derivatives formed the required higher-order block method (HBM) which is applied to standard third-order BVPs. The HBM is self-starting since it doesn’t need any separate predictor or starting values. The investigation of the convergence analysis of the HBM is completely examined and discussed. The improving tactics are fully considered and discussed which resulted in better performance of the HBM. Three numerical examples were presented to show the performance and the strength of the HBM over other numerical methods. The comparison of the HBM errors and other existing work in the literature was also shown in curves.

Keywords

• Convergence analysis, Linear & nonlinear problems, Ordinary differential equations, Power series basic function, Third-order boundary value problems