Magdy A. El-Tawil

Solving Riccati differential equation using Adomian's decomposition method

In this paper, we suggest a method to solve the matrix Riccati differential equation. The suggested method, which we called multistage Adomians decomposition method (MADM), can be considered as an extension of the Adomians decomposition method (ADM) which is very efficient in solving a variety of differential and algebraic equations. The solution is introduced in a recursive form which can be used to obtain the solution for the whole time horizon. Comparisons are made between MADM and the exact solution and further between MADM and different numerical methods.

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.

The approximate solutions of some stochastic differential equations using transformations

In this paper, the transformation technique together with a numerical technique is used to obtain an approximate probability density function (pdf) for the solution process of some stochastic differential equations. The approach is applied on some first order and second order differential equations with making comparisons with the exact solution if possible.

A new technique of using homotopy analysis method for second order nonlinear differential equations

Abstract In this paper, a new technique of homotopy analysis method (nHAM) is proposed for solving second order nonlinear differential equations. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM). The proposed provides an approximate solution by rewriting the second order nonlinear differential equation in the form of two first order differential equations. The solution of these two differential equations is obtained as a power series solution. This scheme is tested on four non-linear exactly solvable differential equations. Three of the examples are initial value problems and the fourth is boundary value problem. The results demonstrate reliability and efficiency of the algorithm developed.

A proposed technique of SFEM on solving ordinary random differential equation

The stochastic finite element (SFEM) is a new method to solve random differential equation using the known deterministic finite element technique (FEM) adapted to stochastic techniques for solving stochastic differential equations. In this paper, a proposed technique of FEM and random variable transformation (RVT) is used to solve a differential equation with random excitation. The technique shows high accuracy when solving a case study compared with the exact solution.

The homotopy Wiener-Hermite expansion and perturbation technique (WHEP)

The Wiener-Hermite expansion linked with perturbation technique (WHEP) was used to solve perturbed non-linear stochastic differential equations. In this article, the homotopy perturbation method is used instead of the conventional perturbation methods which generalizes the WHEP technique such that it can be applied on non-linear stochastic differential equations without the necessity of the presence of the small parameter. The technique is called homotopy WHEP and is demonstrated through many non-linear problems.

Using FEM-RVT technique for solving a randomly excited ordinary differential equation with a random operator

The technique of stochastic finite element (SFEM) which is the finite element technique FEM adapted to stochastic problems can be re-described to use random variable transformation technique RVT. A new FEM-RVT technique was successfully used in solving stochastic problems with random excitation [M. El-Tawil, W. El-Tahhan, A. Hussein, A proposed technique of SFEM on solving ordinary random differential equation, J. Appl. Math. Comput. 161 (2005) 35-47]. In this paper, the technique is adapted to solve a randomly excited differential equation with a random operator. The technique shows high accuracy when solving a case study compared with the exact solution. Finally a problem with unknown exact solution is solved using this technique.

Using Homotopy WHEP technique for solving a stochastic nonlinear diffusion equation

In this paper, the diffusion equation under square and cubic nonlinearities and stochastic nonhomogeneity is solved using the Homotopy WHEP technique. The homotopy perturbation method is introduced in the WHEP technique to deal with non-perturbative systems. The new technique is then used to solve the nonlinear diffusion equation by making comparisons with Homotopy perturbation method (HPM). The method of analysis is illustrated through case studies and figures.