One-Step Third-Derivative Block Method with Two-Hybrid Points for Solving Non-linear Dirichlet Second Order Boundary Value Problems

Abstract

The introduction of higher derivative in the development of numerical methods for solving second order boundary value problems of ordinary differential equations has been explored by few researchers. Taking the advantages of both hybrid block methods and the presence of higher derivative in deriving numerical methods, this study proposes a new one-step hybrid block method using two-hybrid (off-step) points for solving directly initial value problems of second order ordinary differential equations with the introduction of the third derivative. To derive this method, the approximate power series solution is interpolated at \( \left\{ {x_{n} ,x_{{n + \frac{1}{3}}} } \right\} \) while its second and third derivatives are collocated at all points \( \left\{ {x_{n} ,x_{{n + \frac{1}{3}}} ,x_{{n + \frac{2}{3}}} , x_{n + 1} } \right\} \) on the given interval. The basic properties such as zero stability, order, consistency and convergence are also investigated in this study. In order to solve non-linear Dirichlet second-order boundary value problems, we first convert them to their equivalent initial value problems by using non-linear shooting method. Then the proposed method is employed to solve the resultant initial value problems. The numerical results indicate that the new derived method outperforms the existing methods in solving the same problems.