A New Approach to Solve Nonlinear Partial Differential Equations

Nonlinear phenomena play a crucial role in applied mathematics and physics. Explicit solutions to the nonlinear equations are of fundamental importance. Various methods for obtaining explicit solution to nonlinear evolution equations have been proposed. In this letter homotopy perturbation method (HPM) is employed for solving one-dimensional non-homogeneous parabolic partial differential equation with a variable coefficient and a system of nonlinear partial differential equations. The final results obtained by means of HPM, were compared with those results obtained from the exact solution and the Adomian Decomposition Method (ADM). The comparison shows a precise agreement between the results, and introduces this new method as an applicable one which it needs less computations and is much easier and more convenient than others, so it can be widely used in engineering too.


INTRODUCTION
Nonlinear phenomena play a crucial role in applied mathematics and physics. Explicit solutions to the nonlinear equations are of fundamental importance. Various methods for obtaining explicit solution to nonlinear evolution equations have been proposed. Hirota's dependent variable transformation, the inverse scattering transform, and the Bäcklund transformation are the most important ones. A common feature to all of these methods is that they are using the transformations to reduce the equation under study into simpler one, and then solve it. Unlike classical techniques, the nonlinear equations are solved easily and elegantly with out transforming or linearizing the equation by using the homotopy perturbation method (HPM). It provides an efficient explicit solution with high accuracy, minimal calculations, and avoidance of physically unrealistic assumptions.
The HPM was first proposed by He, and has been shown to solve a large class of nonlinear problems effectively, easily, and accurately with approximations converging rapidly to accurate solutions. The HPM was proposed to search for limit cycles or bifurcation curves of nonlinear equations.
In a heuristic example was given to illustrate the basic idea of the HPM and its advantages over the δmethod, the method was also applied to solve boundary value problems, and heat radiation equations [1,2,[3][4][5][6][7] .
In this study, we implemented the HPM for finding the approximate solutions of one-dimensional nonhomogeneous parabolic partial differential equation with a variable coefficient and a system of nonlinear partial differential equations [8,9] .
These two equations are as fallows: = and the boundary conditions are: Which it is easily can be seen that it has an exact solution of ( , ) exp( ) sin u x t x x t = + − [8]. For the second equation, let's consider the system of nonlinear partial differential equations [9] :

MATERIALS AND METHODS
Homotopy-perturbation method: To explain this method, let us consider the following function: Where A, B, ( ) f r and Γ are a general differential operator, a boundary operator, a known analytical function and the boundary of the domain Ω , respectively. Generally speaking, the operator A can be divided in to a linear part L and a nonlinear part N. Eq. (5) can therefore, be written as: By the homotopy technique, we construct a homotopy ( , ) : is an embedding parameter, while 0 u is an initial approximation of Eq. (5), which satisfies the boundary conditions. Obviously, from Eqs. (8) and (9) we will have: The changing process of p from zero to unity is just that of ( , ) v r p from 0 u to ( ) u r .In topology, this is called According to the HPM, we can first use the embedding parameter p as a "small parameter", and assume that the solutions of Eqs. (8) and (9) can be written as a power series in p: Setting 1 p = results in the approximate solution of Eq.
The combination of the perturbation method and the homotopy method is called the HPM, which eliminates the drawbacks of the traditional perturbation methods while keeping all its advantages. The series (13) is convergent for most cases. However, the convergent rate depends on the nonlinear operator ( ) A v . Moreover, the following suggestions were made by He: (1) The second derivative of ( ) N v with respect to v must be small because the parameter may be relatively large, i.e. 1 p → .
Substituting Eq. (12) in to Eq. (14) and rearranging the resultant equation based on powers of p-terms, one has: With the following conditions: , (0, ) sin( ), (1, ) 1 , In order to obtain the unknowns, we should solve Eqs.
So we have: In the same manner, the rest of components were obtained using the maple package.
According to the HPM, we can conclude that And the initial approximations are as fallows: Solving equations (28) through (35), and using the initial conditions (Eqs. (26)), one can find the fallowing results: . v x y t yt = According to the HPM, we can conclude that  (3) they compared with the results of the Adomian Decomposition Method [9] which it is another approximate method for solving nonlinear partial differential equations. It is obvious that; considering more power of p leads us to the more accurate results.

CONCLUSION
In this letter, the homotopy perturbation method (HPM) was used for finding the approximate solutions of one-dimensional non-homogeneous parabolic partial differential equation with a variable coefficient and a system of nonlinear partial differential equations. In can be concluded that the HPM is very powerful and efficient technique in finding exact solutions for wide classes of problems. It is worth pointing out that the HPM presents a rapid convergence for the solutions.
The obtained solutions are compared with the exact solutions and the ADM [9] . the two solved examples show that the results of the present method are in excellent agreement with those obtained by the exact solution and ADM. The HPM has got many merits and much more advantages than the ADM. This method is to overcome the difficulties arising in calculation of Adomian Polynomials. Also the HPM does not require small parameters in the equation, so that the limitations of the traditional perturbation methods can be eliminated, and also the calculations in the HPM are simple and straightforward. The reliability of the method and the reduction in the size of computational domain gives this method a wider applicability. The results show that the HPM is a powerful mathematical tool for solving systems of nonlinear partial differential equations having wide applications in engineering.