Numerical solution of Boundary Value Problems by Piecewise Analysis Method

In this paper, we use an efficient numerical algorithm for solving two point fourth-order linear  and nonlinear boundary value problems, which is based on the homotopy analysis method (HAM), namely, the piecewise – homotopy analysis method ( P-HAM).The method contains an auxiliary parameter that provides a powerful tool to analysis strongly linear and nonlinear ( without linearization ) problems directly. Numerical examples are presented to compare the results obtained with some existing results found in literatures. Results obtained by the RHAM performed better in terms of accuracy achieved. Keywords:            Piecewise-homotopy analysis, perturbation, Adomain decomposition method, Variational Iteration, Boundary Value Problems

Variational Iteration Method for Fifth-Order Boundary Value Problems Using He's Polynomials

We apply the variational iteration method using He's polynomials (VIMHP) for solving the fifth-order boundary value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods and is mainly due to Ghorbani (2007). The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discritization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method

Solutions of Tenth and Ninth-order Boundary Value Problems by Modified Variational Iteration Method

In this paper, we apply the modified variational iteration method (MVIM) for solving the ninth and tenth-order boundary value problems. The proposed modification is made by introducing He’s polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using the Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method

An Algorithm of Interaction Coordination in Multilevel Control of Nonlinear Systems

This paper proposes a coordination algorithm for a multilevel control of a large-scale dynamical system. The system considered consists of weakly interconnected nonlinear subsystems and the performance index is quadratic in states and controls. According to the variational principle, the optimal control is given by solving a nonlinear two-point boundary-value problem, of which analytical solution is generally impossible. The present technique is to solve the overall problem, first by solving decomposed problems of the subsystems, and secondly by coordinating interactions among the subsystems. Since each subsystem problem is a linear two-point boundary-value problem, it is relatively easy to solve. The present idea of coordination is to adjust directly the interaction variables by an iteration without using the conventional Lagrange multiplier. A sufficient condition for convergence of the iteration algorithm is presented in the paper. The algorithm is computationally simple and the convergence is quite rapid for the problem of weakly coupled systems with small nonlinearities. The effectiveness of the method is illustrated in two examples

Modified Sumudu Transform Analytical Approximate Methods For Solving Boundary Value Problems

In this study, emphasis is placed on analytical approximate methods. These methods include the combination of the Sumudu transform (ST) with the homotopy perturbation method (HPM), namely the Sumudu transform homotopy perturbation method (STHPM), the combination of the ST with the variational iteration method (VIM), namely the Sumudu transform variational iteration method (STVIM) and finally, the combination of the ST with the homotopy analysis method (HAM), namely the Sumudu transform homotopy analysis method (STHAM). Although these standard methods have been successfully used in solving various types of differential equations, they still suffer from the weakness in choosing the initial guess. In addition, they require an infinite number of iterations which negatively affect the accuracy and convergence of the solutions. The main objective of this thesis is to modify, apply and analyze these methods to handle the difficulties and drawbacks and find the analytical approximate solutions for some cases of linear and nonlinear ordinary differential equations (ODEs). These cases include second-order two-point boundary value problems (BVPs), singular and systems of second-order two-point BVPs. For the proposed methods, the trial function was employed as an initial approximation to provide more accurate approximate solutions for the considered problems. In addition, for the STVIM method, a new algorithm has been proposed to solve various kinds of linear and nonlinear second-order two-point BVPs. In this algorithm, the convolution theorem has been used to find an optimal Lagrange multiplier