Solution of the Modified Equal Width Equation, its Variant and Non-homogeneous Burgers’ Equation by RDT Method

The modified equal width equation and its variant are investigated. We have presented a reduced differential transform method to solve the MEW equation, its variant and non-homogeneous Burgers’ equation. This method is an alternative approach to overcome the demerit of complex calculation of differential transform method, capable of reducing the size of calculation and easily overcoming the difficulty of the perturbation technique or Adomain polynomials. The approximate analytical solutions of the equations are calculated in the form of series with easily computable components. Numerical results are derived and the obtained results are found in good agreement with the exact solutions.


Introduction
Nonlinear wave phenomena plays a major role in sciences such as fluid mechanics, plasma physics, solid state physics, optical fibers, chemical kinetics and geochemistry. The phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equations [1]. Because of the complexity of the nonlinear wave equations, there is no unified method to find the solutions of these equations. Several methods, analytical and numerical, such as the homogeneous balance method [2], Hirota bilinear method [3], Variational iteration method [4], the decomposition method [5], Backlund transformation and inverse scattering method [6], Homotopy Perturbation method [7], the sine-cosine methods [8], the Tan hyperbolic function method [9] and the Reduced Differential transform method [10] are used to solve the partial differential equations governing the nonlinear waves. The MEW equation, which is related to the RLW equation, has solitary wave solutions with both positive and negative amplitudes, all of which have the same width [11,12]. The MEW equation is a nonlinear wave equation with cubic nonlinearity with a pulse like solitary wave solution. The present work is motivated by the intention to extend the use of earlier method in the literature to solve the MEW equation, its nonlinear variant and the non-homogeneous Burgers' equation. In this paper, we solve the above problems by the Reduced Differential transform method [10]. The main advantage of the method is the fact that it provides its user with an analytical approximation, in many cases an exact solution, in a rapidly convergent sequence with elegantly computed terms. The structure of this paper is organized as follows: In section 2, we begin with some basic definitions and explain the reduced differential transformation method. In section 3, we apply this method to solve the above three problems.

Analysis of the Method
The basic definitions in the Reduced Differential transform method [10] are as follows:

Definition
If function u(x; t) is analytic and k-times continuously differentiable with respect to time t and space x in the domain of interest, then let (1) where the function ( ) is the transformed function of the original function ( , ) . The differential inverse transform of ( ) is defined as follows Then combining these two equations we write (3) From the above definitions, it can be found that the concept of the Reduced Differential transform method is derived from the Taylor's series expansion.
We write the gas dynamics equation in standard form with initial condition where ( ) = ( , ) , ≡ 3 2 is a linear operator which has mixed partial derivatives and ( ) is a nonlinear term. According to RDTM, the iteration formula is � ( )� are the reduced differential transform transformations of the functions ( ( , )) and ( ( , )), respectively. Definition 2.1 implies that the initial approximation 0 ( ) is given by the initial condition, that is Substituting 0 ( ) in the iteration formula (6), we obtain the values of ( ) . Then the differential inverse transformation of the set of values [ ( )] =0 gives approximation solution as

Applications
Now, we will solve three problems using the RDTM.
The MEW equation, which is related to the RLW equation, has solitary waves solutions with both positive and negative amplitudes, all of which have the same width. The MEW equation is a nonlinear wave equation with cubic nonlinearity with a pulse like solitary wave solution.
We consider above equation with the initial condition ( , 0) = sech . (10) We know that the exact solution for above problem is By RDTM, the transformed form of equation (9) is Using the initial condition (11), we obtain 0 ( ) = sech , 1 ( ) = 3 (sech ) 3  Therefore, the exact solution of (9) with the initial condition (10)

The Variant of Modified Equal Width (MEW) Equation
Consider the variant of modified equal width (MEW) equation Here = ( , ). We know that the exact solution for above problem is By RDTM, the transformed form of equation (13) is  Therefore, the exact solution of (13) with the initial condition (14) is given by

Burgers' Equation
Consider the non-homogeneous Burgers' equation Using the initial condition (18), we obtain the initial approximation as 0 ( ) = 2.
Using the definition of RDTM, first few terms and are as follows

Conclusions
The main goal of the present work is to implement the Reduced Differential transform method [10], and more importantly to overcome the computational difficulty of Differential Transform method. The aims of this work were achieved, and travelling wave solutions were formally obtained as a power series. The transformation technique was used to demonstrate that our analysis is applicable to a variety of nonlinear evolution equations.
The main advantage of the method is the fact that it provides its user with an analytical approximation, in many cases an exact solution, in a rapidly convergent sequence with elegantly computed terms. Its small size of computation in comparison with the computational size required in other numerical methods, and its rapid convergence show that the method is reliable and introduces a significant improvement in solving the MEW equation and Burgers' equation over existing methods. As the method is usually tedious, we have used the software package "Mathematica" to calculate the series obtained from the RDTM.