New Approach for Solving Three Dimensional Space Partial Differential Equation

This paper presents a new transform method to solve partial differential equations, for finding suitable accurate solutions in a wider domain. It can be used to solve the problems without resorting to the frequency domain. The new transform is combined with the homotopy perturbation method in order to solve three dimensional second order partial differential equations with initial condition, and the convergence of the solution to the exact form is proved. The implementation of the suggested method demonstrates the usefulness in finding exact solutions. The practical implications show the effectiveness of approach and it is easily implemented in finding exact solutions. Finally, all algorithms in this paper are implemented in MATLAB version 7.12.


Introduction:
Many phenomena that arise in mathematical physics and engineering fields can be described by partial differential equations (PDEs). In physics for example, the heat flow and the wave propagation phenomena are well described by PDEs (1,2). So, it is a useful tool for describing natural phenomena of science and engineering models. A PDE is called linear if the power of the dependent variable and each partial derivative contained in the equation is one and the coefficients of the dependent variable and the coefficients of each partial derivative are constants or independent variables. However, if any of these conditions is not satisfied, the equation is called nonlinear. Most of engineering problems are nonlinear, and it is difficult to solve them analytically. The importance of obtaining the exact or approximate solution of nonlinear PDEs in physics and mathematics is still a significant problem that needs new methods to get exact or approximate solutions. Various powerful mathematical methods have been proposed for obtaining exact and approximate analytic solutions. Some of the classic analytic methods are perturbation techniques (3) and Hirota's bilinear method (4). Department  In recent years, many research workers have paid attention to study the solutions of nonlinear PDEs by using various methods. Among these are the Adomian decomposition method (ADM) (5), the tanh method, the homotopy perturbation method (HPM), the homotopy analysis method (HAM) (6), the differential transform method, Laplace decomposition method (7,8), and the variational iteration method (VIM) (9,10).
In this research, we will use the new method based on couple new transform with HPM which we will call the new transform homotopy perturbation method (NTHPM) to solve three dimensions second order partial differential equation of the form: (1) with initial condition (IC): ( , , , 0) = ( , , ); α is constant.
This method provides an effective and efficient way of solving a wide range of non-linear PDEs. The advantage of this method is its capability of combining two powerful methods for obtaining exact solutions for non-linear partial differential equations. In this research we consider a method in solve non-linear three dimensional space PDEs.

New Transform
New transform was introduced by Luma and Alaa (11) to solve differential equations and engineering problems. part from other advantages of new transform (NT) over other integral transforms such as accuracy and simplicity illustrated in (12), it consists of a very interesting fact about this transform.
The new transform of a function f(t), defined by Here some basic properties of the NT are introduced: If a, b are constants, f(t) and g(t) are functions having NT, then 1.

Solving Linear PDE by Suggested Method
To illustrate the ideas of suggested method new transform homotopy perturbation method (NTHPM) firstly rewrite the PDE (1) as following: ( ) + ( ) − = 0 (7) According to the classical perturbation technique, the solution of the equation (7) can be written as a power series of embedding parameter p, as follow: The convergence of series (8) at p =1 is discussed and proved in (13)(14)(15), which satisfies the differential equation (4). The final step is determining the parts u n (n= 0,1, 2,…), to get the solution u( , , , ).
Here, we couple the NT with HPM as follow: Taking the NT (with respect to the variable t) for the equation (7) to get: Now by using the differentiation property of NT (property 4) and equation (4), (9) becomes: Taking the inverse of the NT on both sides of equation (11), to get: ( , , , ) = ( , , ) Then substituting equation (8) into equation (12) to obtain: By comparing the coefficient of powers of p in both sides of the equation (13) we have: 0 = ( , , )

Illustrative Application
Here, the suggested method NTHPM will be used to solve the 3DS-PDE with initial condition as the following:

Problem 1
Consider the following 3DS -PDE where a, b and c are constants and is any coefficient. According to the equation (14) the powers series of p as following: This gives an exact solution of the problem.
Taking the NT (with respect to the variable t) for the equation (

Illustrative Example
In this section, the suggested method will be used to solve equation (

The Convergence of the Solution for Nonlinear Case
Now, we must prove the convergence of solution of equation (19) to the exact solution when we used the NTHPM, the solution is given in equation (34)

Theorem (2) (Convergence Theorem)
If the series (34) which was calculated by NTHPM is convergent then the limit point converges to the exact solution of moisture content equation (19).
Suppose that equation (34) converge, then we called the limit point as

Conclusion:
We employed the combination of new transform suggested by Luma and Alaa with HPM method to get a closed form solution of the three dimensional space PDE linear and nonlinear. The new method is free of unnecessary mathematical complexities. Although the problem considered has no exact solution, the accuracy, efficiency, and reliability of the new method are guaranteed. The convergence of obtained solution to the exact solution by using NTHPM is proved.