In this paper, a predictor corrector two-point block method is proposed to solve the wellknown Blasius and Sakiadis flow numerically. The Blasius and Sakiadis flow will be modeled by a third order boundary value problem. The main motivation of this study is to provide a new method that can solve the higher order BVP directly without reducing it to a system of first order equation. Two approximate solutions will be obtained simultaneously in a single step by using predictor corrector two-point block method able to solve the third order boundary value problem directly. The proposed direct predictor corrector two-point block method will be adapted with multiple shooting techniques via a three-step iterative method. The advantage of the proposed code is that the multiple shooting will converge faster than the shooting method that has been implemented in other software. The developed code will automatically choose the guessing values in order to solve the given problems. Some numerical results are presented and a comparison to the existing methods has been included to show the performance of the proposed method for solving Blasius and Sakiadis flow.

In this paper, we propose an algorithm of two-point block method to solve the nonlinear system of third-order boundary value problems directly. The proposed method is presented in a simple form of Adams type and two approximate solutions will be obtained simultaneously with the block method using variable step size strategy. The method will be implemented with the multiple shooting technique via the three-step iterative method to generate the missing initial value. Most of the existence method will reduce the third-order boundary value problems to a system of first order equations where the systems of six equations need to be solved. The method we proposed in this paper will solve the third-order boundary value problems directly. Two numerical examples are given to illustrate the efficiency of the proposed method.

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.

In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. The problems of single-well, double-well and double-hump Van der Pol-Dufing equations are studied in this paper. The Chebyshev Neural Network (ChNN) model will be applied to obtain the numerical solutions of these types of equations for the first time. The hidden layer is eliminated by expanding the input pattern by Chebyshev polynomials which employs a single layer neural network. In order to modify the network parameters and to minimize the computed error function, a feed forward neural network model with error back propagation principle is used. The obtained numerical results form the ChNN model will be compared with the analytical solutions, namely Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM), Differential Transform Method (DTM) and exact. Comparisons of the solutions obtained with existing numerical results show that this method is a capable tool for solving this kind of nonlinear problems.

In this paper, we discussed and compared the computational complexity for two-point block method and one-point method of Adams type. The computational complexity for both methods is determined based on the number of arithmetic operations performed and expressed in O(n). These two methods will be used to solve two-point second order boundary value problem directly and implemented using variable step size strategy adapted with the multiple shooting technique via three-step iterative method. Two numerical examples will be tested. The results show that the computational complexity of these methods is reliable to estimate the cost of these methods in term of the execution time. We conclude that the twopoint block method has better computational performance compare to the one-point method as the total number of steps is larger.