Three-Step Block Method for Solving Nonlinear Boundary Value Problems

Abstract

We propose a three-step block method of Adam’s type to solve nonlinear second-order two-point boundary value problems of Dirichlet type and Neumann type directly. We also extend this method to solve the system of second-order boundary value problems which have the same or different two boundary conditions. The method will be implemented in predictor corrector mode and obtain the approximate solutions at three points simultaneously using variable step size strategy. The proposed block method will be adapted with multiple shooting techniques via the three-step iterative method. The boundary value problem will be solved without reducing to first-order equations. The numerical results are presented to demonstrate the effectiveness of the proposed method.

Diagonal Block Method for Solving Two-Point Boundary Value Problems with Robin Boundary Conditions

“…Since the order of the proposed method is = 4 ≥ 1, the method is consistent. satisfies | | ≤ 1 and for those roots with | | = 1, the multiplicity must not exceed two as in Lambert [14] and See et al [15]. Rewrite (15) to (16) in matrix form as follows:…”

Solving higher order nonlinear ordinary differential equations with least squares support vector machines

“…subject to boundary conditions y(0) = 0, y(1) = 1. This problem is solved for x ∈ [0, 1] and the approximate solution obtained by the proposed method is compared with the solution obtained by MATLAB built-in solver bvp4c [11], [26] in Fig. 5 The exact solution and the approximate solution obtained by the proposed LS-SVM algorithm are compared for 11 equidistant training points in Table 10.…”