Tamer A. Abassy

The Solution of KdV and mKdV Equations Using Adomian Padé Approximation

Adomian Decomposition method (ADM) is an approximate method, which can be adapted to solve nonlinear partial differential equations. In this paper, we solve the KdV and modified KdV (mKdV) equations using ADM-Pade technique, which gives the approximate solution with fast convergence rate and high accuracy in the case of solitary wave solution and closed form solution in the case of rational polynomial solution.

Modified variational iteration method for Boussinesq equation

This paper applies the modified variational iteration method to solve a class of nonlinear partial differential equations. Boussinesq equation is used as a case-study to illustrate the simplicity and effectiveness of the method. Comparison between variational iteration method and Adomian decomposition method is made.

Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms and the Padé technique

In this paper, the variational iteration method (VIM) is reintroduced with Laplace transforms and the Pade technique treatment to obtain closed form solutions of nonlinear equations. Some examples, including the coupled Burgers equation, compacton k(n,n) equation, Zakharov-Kuznetsov Zk(n,n) equation, and KdV and mKdV equations are given to show the effectiveness of the coupled VIM-Laplace-Pade and VIM-Pade techniques.

Improved Adomian decomposition method

In this paper a new treatment for the Adomian decomposition method (ADM) is introduced. The new treatment is called the improved Adomian decomposition method (IADM) which improves the results obtained from the known Adomian decomposition method. The improved Adomian decomposition method is applied for the analytic treatment of nonlinear initial value problems. The improved method accelerates the convergence of the series solution, and provides the exact power series solution. It solves the drawbacks in the standard Adomian decomposition method.

Modified variational iteration method (nonlinear homogeneous initial value problem)

The modified variational iteration method is applied for analytical treatment of nonlinear homogeneous initial value problem. The modified variational iteration method accelerates the convergence of the power series solution and reduces the size of work. A comparison between modified variational iteration method (MVIM) and variational iteration method (VIM) was made. The comparison enhances the use of the modified variational iteration method if we wish to obtain an approximate power series solution that converges faster to the closed form solution. The method is very simple and easy.

Improved Adomian decomposition method (solving nonlinear non-homogenous initial value problem)

The Adomian decomposition method (ADM) cannot deal generally with the non-homogenous differential equations. The method still needs improvements. This paper introduces a qualitative improvement in the method. The improved method is called the improved Adomian decomposition method (IADM). A comparison between ADM and IADM shows that the ADM gives results that is not accepted in some examples and cannot solve the other examples while IADM solves the problems effectively and efficiently.

A numerical study of adding an artificial dissipation term for solving the nonlinear dispersive equations K(n,n)

A linearized implicit finite difference method is devised for K(n,n). The stability and accuracy of the proposed methods are discussed. A compacton wave solution of the equation K(n,n) is used to examine the accuracy and efficiency of the proposed methods and study the effect of the added artificial dissipation term to solve the K(n,n) equation using finite difference method. The dynamics of waves having various initial wavepackets are discussed.