Latest Developments in Nonlinear Sciences

This paper outlines a detailed study of some latest trends and developments in nonlinear sciences. The major focus of our study will be variational iteration (VIM) and its modifications, homotopy perturbation (HPM), parameter expansion and exp-function methods. The above mentioned schemes are highly accurate, extraordinary efficient, capable to cope with the versatility of the physical problems and are being used to solve a wide class of nonlinear problems. Several examples are given which reveal the justification of our claim

Variational Iteration Method for Solving Initial and Boundary Value Problems of Bratu-type

In this paper, we present a reliable framework to solve the initial and boundary value problems of Bratu-type which are widely applicable in fuel ignition of the combustion theory and heat transfer. The algorithm rests mainly on a relatively new technique, the variational iteration method. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm

On the Solution of the Vibration Equation by Means of the Homotopy Perturbation Method

In this paper, we present a reliable algorithm, the homotopy perturbation method, to solve the well-known vibration equation for very large membrane which is given initial conditions. By using initial value, the explicit solutions of the equation for different cases have been derived, which accelerate the rapid convergence of the series solution. Numerical results show that the homotopy perturbation method is easy to implement and accurate when applied to differential equations. Numerical results for different particular cases of the problem are presented graphically

Numerical Comparison of Methods for Hirota-Satsuma Model

This paper outlines the implementation of the modified decomposition method (MDM) to solve a very important physical model namely Hirota-Satsuma model which occurs quite often in applied sciences. Numerical results and comparisons with homotopy perturbation (HPM) and Adomian’s decomposition (ADM) methods explicitly reveal the complete reliability of the proposed MDM. It is observed that the suggested algorithm (MDM) is more user-friendly and is easier to implement compared to HPM and ADM

On Numerical Solutions of Two-Dimensional Boussinesq Equations by Using Adomian Decomposition and He\u27s Homotopy Perturbation Method

In this paper, we obtain the approximate solution for 2-dimensional Boussinesq equation with initial condition by Adomian\u27s decomposition and homotopy perturbation methods and numerical results are compared with exact solutions

Exact solitary-wave Special Solutions for the Nonlinear Dispersive K(m,n) Equations by Means of the Homotopy Analysis Method

In this paper, we study the nonlinear dispersive K(m,n) equations which exhibit solutions with solitary patterns. New exact solitary solutions are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the homotopy analysis method in K(m,n) equations. The nonlinear equations K(m,n) are studied for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of K(m,n) equations are established

Variational Iteration Method for Solving Telegraph Equations

In this paper, we apply the variational iteration method (VIM) for solving telegraph equations, which arise in the propagation of electrical signals along a telegraph line. The suggested algorithm is more efficient and easier to handle as compare to the decomposition method. Numerical results show the efficiency and accuracy of the proposed VIM

A Reliable Approach for Higher-order Integro-differential Equations

In this paper, we apply the variational iteration method (VIM) for solving higher-order integro differential equations by converting the problems into system of integral equations. The proposed technique is applied to the re-formulated system of integro-differential equations. Numerical results show the accuracy and efficiency of the suggested algorithm. The fact that the VIM solves nonlinear problems without calculating Adomian’s polynomials is a clear advantage of this technique over the decomposition method

Solving Higher Dimensional Initial Boundary Value Problems by Variational Iteration Decomposition Method

In this paper, we apply a relatively new technique which is called the variational iteration decomposition method (VIDM) by combining the traditional variational iteration and the decomposition methods for solving higher dimensional initial boundary value problems. The proposed method is an elegant combination of variational iteration and the decomposition methods. The analytical results of the problems have been obtained in terms of convergent series with easily computable components. The method is quite efficient and is practically well suited for use in these problems. Several examples are given to verify the accuracy and efficiency of the proposed technique