Mixture of a New Integral Transform and Homotopy Perturbation Method for Solving Nonlinear Partial Differential Equations

In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. The proposed method introduces also He’s polynomials [1]. The analytical results of examples are calculated in terms of convergent series with easily computed components [2].


Introduction
A new integral transform is derived from the classical Fourier integral.A new integral transform [3] was introduced by Artion Kashuri and Associate Professor Akli Fundo to facilitate the process of solving ordinary and partial differential equations in the time domain.Some integral transform method such as Laplace, Fourier, Sumudu and Elzaki transforms methods, are used to solve general nonlinear non-homogenous partial differential equation with initial conditions and use fullness of these integral transform lies in their ability to transform differential equations into algebraic equations which allows simple and systematic solution procedures.Non-linear phenomena, that appear in many areas of scientific fields such as solid state physics, plasma physics, fluid mechanics, population models and chemical kinetics, can be modeled by nonlinear differential equations.The importance of obtaining the exact or approximate solutions of nonlinear partial differential equations in physics and mathematics is still a significant problem that needs new methods to discover exact or approximate solutions.Also a new integral transform and some of its fundamental properties are used to solve general nonlinear non-homogenous partial differential equation with initial conditions.A new integral transform is defined for functions of exponential order.We consider functions in the set F de-fined by: , , 0, such that e , if 1 0, F , the constant For a given function in the set M must be finite number may be finite or infinite.
1 2 A new integral transform denoted by the operator , k k   K  is defined by the integral equation: A new integral transform was applied to partial differential equations, ordinary differential equations, system of ordinary and partial differential equations and integral equations.A new integral transform is a powerful tool for solving some differential equations.In this paper, we combined a new integral transform and homotopy perturbation method     f method.For any function t

 
, we assume that a integral Equation (2) exist.
Definition 1.1.[4] Given a function and employ the notation,    exist and are continuous on and are constant coefficients.Then, 0, Proof.It is easy to verify that the right-hand side of first equation is a continuous function on whose a new integral transform is Hence, So we have proof the theorem.
Proof.We assume that   , x t is piecewise con-tinuous and it is of exponential order.To obtain a new integral transform of partial derivatives we use integration by parts as follows: we have: We can easily extend this result to the partial derivative by using mathematical induction to get (c). (d) Above we have used the Leibniz's rule to find that: Above we have used the Leibniz's rule to find that:  partial We can easily extend this result to the derivative by using mathematical induction to get (f) So w ave p f the th rem. e h roo eo

Homotopy Perturbation Method (HPM)
e-The homotopy perturbation method is considered as sp cial case of homotopy analysis method.Let X and Y be the topological spaces.If and g x .To explain the homotopy perturbatio ethod, onsider a general equation of the type, n m we c where L is any differential operator.We define a convex hom topy where   G u is a functional operator with whi ,p The   co aces an implicitly defined curve starting point HPM uses the embeddi parameter p as nding pa ng an expa rameter [1,5] and write the solution as a power series: If 1 p  then (8) corresponds to (7) and becomes the approximate The embedding parameter monotonically increases fr .It is well known that the series (8) is convermost of the cases and also the rate of convergence is depending on   L u .We assume that (9) has a unique solution.The co isons of like powers of p give solutions of various orders.

(HPM)
where D is linear differential operator of order two, R is linear differential operator of less order t D , N han is the general nonlinear differential operator and   , g x is the source term.Taking a new integral transform both sides of equation ( 10) we get: Using the differentiation property of a transform by theorem (1.3) and above initial conditions, w new integral e have: Applying the inverse new integral transform on sides of Equation ( 12) we find: , , , , where   , G x t represents the term arising source term and the prescribed initial conditions.Now, ly the ho and the nonlinear term can be decomp where from the we app motopy perturbation method.
    0 , , H u are the so-called He's p that represents the nonlinear terms and are given by the : Substituting Equations ( 14) and (15) in Equation we get: This is the coupling of a new integral transform and the homotopy perturbation method [5].Comparing the co ma  efficients of like powers of p, the following approxitions are obtained: , , he method is capable of reducing the volume of the computational w pared to the classical methods while still maintaining the hi thod    Then the solution is : ork as comgh accuracy of the numerical result and no requirement to complicated calculations.The size reduction amounts to an improvement of the performance of the approach.

Application of the mixture of a new integral transform and homotopy perturbation me
HPM for solving e this section we t the so  nonlinear partial diff rential equations.In apply the homotopy perturbation method and a new integral transform method in order to ge lution procedure of this.
The following examples illustrate the use of this new mixture method in solving certain initial value problems described by nonlinear partial differential equations.
Example 3.1.Consider the following non-homogenous nonlinear partial differential equation with initial conditions: The inverse new integral transform implies that, , 1 Now applying the homotopy perturbation method in Equation ( 22) we get: , , The first few components of

 
n H u are given by: Comparing the coefficients of the same powers of p , we get: Then the solution is: Consider the following non-homogenous nonlinear partial differential equation with initial conditi s: By applying a new integral transform of Equation ( 27) subject to the initial conditions (28) we have: The inverse new integral transform implies that, in Equation (30) we get: Now applying the homotopy perturbation method  w H u are He's polynomials that represents the onlinear terms.Then, n , , , , The first few c s of

 
n omponent H u are given by: Comparing the coefficients of the sam powers of p , we get: , Then the solution is: Now applying the homotopy Equation (38) we get:   n H u are He's polynomials that represents nonlinear terms.Then, the p u x t pu x t p u x t u x t pu x t p u x t where, The first few components of

 
n u are given by: Then the solution is: Consider the following homogenous nonlinear partia differential equation with initial condition: new integral transform of Eq ion (43) subject to the initial condition (44) we have: The inverse new integral transform  Now applying the homotopy pertur Equation (46) we get: Then the solution is:

Conclusion
In this paper, we mixture a new integral transform an homotopy perturbation method to solve nonlinear partial differential equations.The solution of four nonlinear partial differential equations with initial conditions is pr ethod and simple calculation of H orth mentioning that the method is capable of reducing the volume of the computational work as compared to the classical methods while still maintaining the high accuracy of the numerical result and no requirement to complicated calculations.The size reduction amounts to an improvement of t of the approach.
Mixture of a New Integral Transform anda general nonlinear non-homogenous partial differential equation with initial conditions of the form: 20) orm subject to the initial conditions (20) we have: By applying a new integral transf of Equation (19) He's polynomials that represents the nonlinear terms.Then,