The Existence of Noise Terms for Systems of Partial Differential and Integral Equations with ( HPM ) Method

In this paper we develop a framework for necessary condition for the existence of noise terms for systems of partial differential and integral equations with ( HPM ) method. We show that the noise terms are conditional and are generated for inhomogeneous equations if specific criteria is justified. And to illustrate the capability and reliability of this method We numerically test our approach for a variety of systems of inhomogeneous problems.


Introduction
Homotopy perturbation method ( HPM ) is useful and powerful method for solving linear and nonlinear differential equations. A brief discussion of Homotopy perturbation method will be emphasized, the complete details of the method are found in [3][4][5]. The Homotopy perturbation method goal is to find the solution of linear and nonlinear, partial differential equations and integral equations with dependence on small parameter. In this method the solution is considered as sum of an infinite series which, rapidly convergence to an accurate solutions. Systems of partial differential equations and integral equations were formally derived to describe nonlinear waves and arise in gas dynamics, water waves [6][7][8][9][10], flood waves in rivers, traffic flow and a wide range of biological and ecological systems. The noise terms phenomenon [11][12][13][14][15] gives a useful tool in that, if it appears, it gives a fast convergence of the solution by using two iterations only. It is important to note that these terms may appear for inhomogeneous problems, whereas homogeneous problems do not generate noise terms. It was formally shown that by canceling the noise terms that appear u 0 and u 1 from u 0 , even though u 1 contains further terms, the remaining noncancelled terms of u 0 may give the exact solution of the inhomogeneous problem. A complete and thorough study on noise terms can be found in details in [11][12][13][14][15].

The Noise Terms Phenomenon
The noise terms phenomenon [8][9][10][11][12] gives a useful tool in that, if it appears,it gives a fast convergence of the solution by using two iterations only.
It is significant to note that the noise terms may appear only for inhomogeneous problems.
The noise terms are defined as the identical terms, with opposite signs, that may appear in various components uk , k ≥1. It is important to note that these terms may appear for inhomogeneous problems, whereas homogeneous problems do not generate noise terms. It was formally shown that by canceling the noise terms that appear in u 0 and u 1 from u 0 , even though u 1 contains further terms, the remaining noncancelled terms of u 0 may give the exact solution of the inhomogeneous problem. This can be justified through substitution. Therefore, it is necessary to verify that the noncancelled terms of u 0 satisfy the PDE under discussion. A necessary condition for the generation of the noise terms for inhomogeneous problems is that the zeroth component u 0 must contain the exact solution u among other terms. A complete and thorough study on noise terms can be found in details in [8][9][10][11][12]. To give a clear overview of the content of this work, several illustrative examples of systems of partial differential and integral equations, have been selected to demonstrate the efficiency of the method and to confirm the necessary condition needed for the generation of the noise terms.

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The Existence of Noise Terms for Systems of Partial Differential and Integral Equations with ( HPM ) Method (1) With initial data (2) Where N 1 , N 2 ,…N n are nonlinear operators and g 1 ,g 2 ,..g n are inhomogeneous terms. To solve system (1) by homotopy perturbation method, we construct the following homotopies: Consider the solution of the system (3) as the following U 1 = U 10 + p U 11 +p 2 U 12 +…, U 2 = U 20 + P U 21 +p 2 U 22 +….
Equating the coefficients of the terms with the identical powers of p leads to Where M ij , i=1,2,…n, j= 0,1,2,…n-1, are terms that are obtained with equating the coefficients The approximate solutions of (1) can be obtained by letting p tend to one

Systems of Integral Equations
In what follows we will examine the noise terms phenomenon by studying a system of inhomogeneous integral equations. Example1. We first consider the inhomogeneous systems: Suppose the solution of Eq.(4) and Eq.(5) has the following form U= U 0 + pU 1 + p 2 U 2 +…..
Using the series (4) and (5) for the linear terms u(x) and v(x) and for the nonlinear terms u 2 v and v 2 u, and by Substituting Eq.(6) and Eq.(7) into Eq.(4) and Eq.(5) and equating the terms with identical powers of p leads to An important remark can be made here in that although the problem is an inhomogeneous problem, the noise terms did not appear between various component.

⋮
We can easily observe the noise terms ± x in the components U 0 and U 1 , and the noise terms ∓ x in the component V 0 and V 1 . By canceling these terms from U 0 and V 0 and by justifying that the remaining terms in the zeroth components justify the inhomogeneous problems , we have: Other noise terms between other component vanish in the limit.

System of Partial Differential Equations
Example 3.consider the system of inhomogeneous partial differential equations With the initial data u(x,0) = , v(x,0) = − .