The Solution of Porous Medium Equation by Aboodh Homotopy Perturbation Method
Mohand M. AbdelrahimMahgoub^{1,}^{ 2}, Abdelilah K. Hassan Sedeeg^{1,}^{ 3}
^{1}Mathematics Department Faculty of Sciences and Arts Almikwah, Albaha University, Albaha, Saudi Arabia
^{2}Mathematics Department Faculty of Sciences-Omdurman Islamic University, Khartoum, Sudan
^{3}Mathematics Department Faculty of Education- Holy Quran and Islamic Sciences University, Khartoum, Sudan
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To cite this article:
Mohand M. Abdelrahim Mahgoub, Abdelilah K. Hassan Sedeeg. The Solution of Porous Medium Equationby Aboodh Homotopy Perturbation Method. American Journal of Applied Mathematics. Vol. 4, No. 5, 2016, pp. 217-221. doi: 10.11648/j.ajam.20160405.13
Received: August 26, 2016; Accepted: September 3, 2016; Published: October 9, 2016
Abstract: In this paper, the homotopy perturbation method (HPM) and Aboodh transform are employed to obtain analytical solution of the porous medium equation. The proposed method (ATHPM) is an elegant combination of the new integral transform "Aboodh Transform" and the homotopy perturbation method. The porous medium equations have importance in engineering and sciences and constitute a good model for many systems in various fields. The results tell us that the proposed method is more efficient and easier to handle when is compared with existing other methods in such partial differential equations.
Keywords: Aboodh Transform, Homotopy Perturbation Method, Non Linear Partial Differential Equation, Porous Medium Equation
1. Introduction
The heat equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. The nonlinear heat equation usually called the Porous Medium Equation [5-7] (PME).
(1)
where is a rational number.
This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer, or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. All of these reasons support the interest of its study both for the mathematician and the scientist. In the resent years, many researchers mainly had paid attention to studying the solution of nonlinear partial differential equations by using various methods. Among these are the variational iteration method [Biazar and Ghazvini (2007)] [11], Adomian decomposition method [Wazwaz (2001) and Biazar et al (2007)], homotopy perturbation method [Mehdi Dehghan and JalilManafian (2008)], homotopy analysis method [Najeeb Alam Khan, Asmat Ara, Muhammad Afzal and Azam Khan (2010)], Elzaki Homotopy Perturbation Method [8-10], and Laplace decomposition algorithm [Majid Khan, Muhammad Asif Gondal and Yasir Khan (2011)]. Motivated and inspired by the on-going research inthese areas, we consider a new method, which is called Aboodh transform homotopy perturbation method (ATHPM) [1-4]. This method is suggested by combining the homotopy perturbation method and Aboodh transform. This method is a useful technique for solving linear and nonlinear differential equations. This article considers the effectiveness of the homotopy perturbation Aboodh transform method in solving. The porous medium equation.
2. Aboodh Transform Homotopy Perturbation Method
The general form of nonlinear non homogeneous partial differential equation can be considered as the follow:
(2)
with the following initial conditions
Where D is the second order linear differential operator , is the linear differential operator of less order than D, N represents the general non-linear differential operator and is the source term.
Ting Aboodh transform (denoted throughout this paper by ) on both sides of Eq. (2), to get:
(3)
Using the differentiation property of Aboodh transform and above initial conditions, we have:
(4)
Operating with the Aboodh inverse on both sides of Eq. (4) gives:
(5)
Where represents the term arising from the source term and the prescribed initial condition.
Now, we apply the homotopy perturbation method
(6)
And the nonlinear term can be decomposed as:
(7)
Where are He’s polynomial and given by:
(8)
Substituting Eqs. (6) and (7) in Eq. (5) we get:
(9)
Which is the coupling of the Aboodh transform and the homotopy perturbation method using He’s polynomials. Comparing the coefficient of like powers of , the following approximations are obtained:
... ... ... ... ... ... ... ... ...
Then the solution is;
(10)
3. Applications
In this section we apply the homotopy perturbation Aboodh transform method for solving the porous medium equation.
Example 1:
Let us take in equation (1), we get
(11)
With the initial condition;
(12)
Exact solution [1] of this equation is with the values of arbitrary constants taken as and , thusthe solution becomes .
We can find solution by applying the Aboodh transform of both sides of Eq. (11) subject to the initial condition
(13)
Using the differential property of Aboodh transform Eq. (13) can be written as:
(14)
Using initial condition Eq. (12), Eq. (14) can be written as:
(15)
The inverse Aboodh transform implies that:
(16)
Now, we apply the homotopy perturbation method, we get:
(17)
Where are He's polynomial s. The first two components of He's polynomials are given by
Comparing the coefficient of like powers of, the following approximations are obtained;
Proceeding in similar manner we can obtain further values, substituting above values in equation (10), we get solution in the form of a series
(18)
This is the solution of (11) and which is exactly the exact solution given above.
Example 2:
Let us take in equation (1), we get
(19)
With the initial condition as
(20)
Exact solution [1] of this equation is with the values of arbitrary constants taken as and , thusthe solution becomes .
We can find solution by applying the Aboodh transform of both sides of Eq. (19) subject to the initial condition
(21)
Using the differential property of Aboodh transform Eq.(21) can be written as:
(22)
Using initial condition Eq. (20), Eq. (22) can be written as:
(23)
The inverse Aboodh transform implies that:
(24)
Now, we apply the homotopy perturbation method, we get:
(25)
Where are He's polynomial s. The first two components of He's polynomials are given by
Comparing the coefficient of like powers of in Eq.(25), the following approximations are obtained;
and so on we will found that for .
Substituting above values in equation (10) we get solution in the form of a series
(26)
This is the solution of Eq. (19) and which is exactly the exact solution given above.
Example 3:
Let us take in equation (1), we
(27)
With the initial condition as
(28)
We can find solution by applying the Aboodh transform of both sides of Eq. (27) subject to the initial condition
(29)
Using the differential property of Aboodh transform Eq. (21) can be written as:
(30)
Using initial condition Eq. (20), Eq. (22) can be written as:
(31)
The inverse Aboodh transform implies that:
(32)
Now, we apply the homotopy perturbation method, we get:
(33)
Where are He's polynomial s. The first two components of He's polynomials are given by
(34)
Comparing the coefficient of like powers of in Eq. (25), the following approximations are obtained;
(35)
On substituting these terms in equation (10) we obtained solution in the form of a series
(36)
This result can be verified through substitution.
4. Conclusion
In this paper, the ATHPM has been successfully applied to find the solution of the analytical solution for porous medium equation. The method is reliable and easy to use. The results show that the ATHPM is a powerful and efficient technique in finding exact and approximate solutions of the porous medium equation. the method is trustworthy and introduces a significant improvement in solving nonlinear partial differential equations over existing methods.
References