One-step block methods for direct solving of linear boundary value Dirichlet and Neumann type problems

In this research, the block methods have been used to solve the second order linear boundary value problems of Dirichlet and Neumann type. Mathematical problems which involve higher order ordinary differential equations were likely to be reduced into the system of first order equations. However, these block methods will solve the problems directly without reducing it into the first order equations using constant step size. There are three methods that have been used in this research which are two point one-step block method, three point one-step block method and four point one-step block method. Each of these methods will be implemented to solve the second order linear boundary value problems with two different types of boundary conditions i.e. Dirichlet and Neumann type. Those three methods will be implemented together with the linear shooting technique to construct the numerical solution. The stability for each method will be presented. Numerical results of the methods are compared with the existing methods. As a conclusion, the proposed block methods can give better and comparable accuracy with the advantage of less costly. Thus, the proposed block methods are suitable to solve the second order linear boundary value problems of Dirichlet and Neumann type directly