Analytical solutions of (2+1)-dimensional Burgers’ equation with damping term by HPM,ADM and DTM

We have employed Analytical solutions of (2 + 1)-dimensional Burgers’ equation with damping term by HPM,ADM and DTM as.

In the past few decades, traditional integral transform methods such as Fourier and Laplace transforms have commonly been used to solve engineering problems. These methods transform differential equations into algebraic equations which are easier to deal with. However, these integral transform methods are more complex and difficult when applying to nonlinear problems.
The HPM, proposed first by He[2,3], for solving the differential and integral equations, linear and nonlinear, has been the subject of extensive analytical and numerical studies. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which is easily solved. This method, which does not require a small parameter in an equation, has a significant advantage in that it provides an analytical approximate solution to a wide range of nonlinear problems in applied sciences.

Homotopy Perturbation Method (HPM)
To describe the HPM, consider the following general nonlinear differential equation under the boundary condition Where A is a general differential operator, B is a boundary operator,f(r)is a known analytic function and ∂Ω is a boundary of the domain Ω.The operator A can be divided into two parts L and N, Where L is a linear opertor while N is a nonlinear operator.Then Equation (3)can be rewritten as Using the homotopy tecnique, we construct a homotopy: Where p [0, 1] is an embedding parameter, u 0 is the initial approximation of Equation (3) which satisfies the boundary conditions.Obviously,considering Equation (6),we will have changing the process of p from zero to unity is just that V(r,p)from u 0 (r) to u(r).In topology,this is called the deformation also A(V)-f(r) and L(u)are called as homotopy .The homotopy perturbation method uses the homotopy parameter p as an expanding parameter [23][24][25]to obtain p →1 results the appropriate solution of equation (3)as A comparision of like powers of p gives the solutions of various orders series (9)is convergent for most of the cases.However convergence rate depends on the nonlinear,N (V ) He [25]suggested the following opinions: 1.The second derivative of N(V)with respect to v must be small as the parameter p may be relatively large.
2.The norm of L −1 ∂N ∂u must be smaller than one so that the series converges. In this section, we describe the above method by the following example to validate the efficiency of the HPM.

Example:1
Consider the (2+1)-dimensional Burgers' equation with daming as, under the initial condition Applying the homotopy perturbation method to Equation (10),we have In the view of HPM, we use the homotopy parameter p to expand the solution The approximate solution can be obtained by taking p=1 in equation (13)as Now substituing equation (12)into equation (11)and equating the terms with identical powers of p, we obtain the series of linear equations, which can be easily solved.First few linear equations are given as Using the initial condition (11), the soution of Equation (15)is given by Then the solution of Equation(16)will be Also, We can find the solution of Equation (17)by using the following formula etc.therefore, from equation (18),the approximate solution of equation (10)is given as Hence the exact solution can be expressed as u(x, y, t) = (x + y)e t + (n! + 1)(e t − 1), provided that 0 ≤ t < 1

Adomain Decomposition Method(ADM)
Consider the following linear operatpor and their inverse operators: Using the above notations,Equation(1)becomes Operating the inverse operators L −1 t to equation (25)and using the initial condition gives The decomposition method consists of representing the solution u(x,y,t)by the decomposition series The nonlinear term u x u y is represented by a series of the so called Adomain polynomials, given by The component u q (x, y, t) of the solution u(x,y,t)is determined in a recursive manner.Replacing the decomposition series (27)and (26)gives According to ADM the zero-th component u 0 (x, y, t) is identified from the initial or boundary cnditions and from the source terms.The remaining components of u(x, y, t) are determined in a recursion manner as follows Where thhe adomain polynomials for the nonlinear term u x u y are derived from the following recursive First few Adomian polynomials are given as using equation(31)for the adomain polynomials A k ,we get and so on.Then the q-th term, u q can be determined from Knowing the components of u, the analytical solution follows immediately.

Conclusion
1.In this work,homotopy perturbation methhod,DTM and adomian decomposition method have been successfully applied for solving (2+1)-dimensional Burgers' equation with damping term.
2.The solutions obtained by these methods are an infinite power series for an appropriate initial condition, which can,in turn be expressed in a closed form, the exact solution.
3.The results reveal that the methods are very effective, convenient and quite accurate mathematical tools for solving the (2+1)-dimensional Burgers' equation equation with damping.
4.The solution is calculated in the form of the convergent power series with easily computable components.
5.These method, which can be used without any need to complex computations.