Numerical Solution of Linear Dirichlet Two-Point Boundary Value Problems Using Block Method

Abstract

This paper presents a direct two-point block one-step method for solving linear Dirichlet boundary value problems (BVPs) directly. The block method is formulated using Lagrange interpolating polynomial. Mathematical problems which involve higher order ordinary differential equations (ODEs) were likely to be reduced into the system of first order equations in order to solve it. However, this method will solve the second order linear Dirichlet BVPs directly without reducing it to the system of first order equations. The direct solution of the linear Dirichlet BVPs will be calculated at the two-points simultaneously using constant step size. This method will be used together with the linear shooting technique to construct the numerical solution. The implementation is based on the predictor and corrector formulas in the PE(CE) r mode. Numerical results are given to show the efficiency and performance of this method compared to the existing methods.

An Accurate Implicit Quarter Step First Derivative Block Hybrid Method (AIQSFDBHM) for Solving Ordinary Differential Equations

“…Kumleng, et al [7] construct a family of continuous block A-stable third derivative for numerical integration of (1.1). Other authors who have done considerable work on the numerical solution of (1.1) case include [8,9,10], to predictor-corrector methods [11,12,13], and then block methods [14,15,16,17].…”

Computational approaches to initial-boundary value problems with Neumann boundary conditions

“…However, solutions of these problems are valid only in onedirectional domain; that is, either in time or space. Moreover, most of the existing methods are constructed for problems with Dirichlet boundary conditions [14][15][16][17][18], and very few methods with Neumann boundary conditions [19][20][21][22][23][24] due to their difficulties in dealing with. In [25], Adomian suggested a modified method for various PDEs with initial and boundary conditions.…”

Numerical Solution of First Order Initial Value Problems Using a Self-Starting Implicit Two-Step Obrechkoff-Type Block Method

“…Numerous numerical approaches have been proposed by scholars for the numerical approximation of initial value problems. These methods range from discrete schemes (Lambert, 1973;Butcher, 2008;Fatunla, 1988), to predictor-corrector methods (Kayode and Adeyeye, 2011;Adesanya et al, 2008;Awoyemi and Idowu, 2005) and then block methods (Omar and Kuboye, 2015;Hasni et al, 2013;Areo and Adeniyi, 2013).…”