Solving Fourth Order Parabolic PDE with Variable Coefficients Using Aboodh Transform Homotopy Perturbation Method
Khalid Suliman Aboodh
Department of Mathematics, Faculty of Science & technology, Omdurman Islamic University, Khartoum, Sudan
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To cite this article:
Khalid Suliman Aboodh. Solving Fourth Order Parabolic PDE with Variable Coefficients Using Aboodh Transform Homotopy Perturbation Method. Pure and Applied Mathematics Journal. Vol. 4, No. 5, 2015, pp. 219-224. doi: 10.11648/j.pamj.20150405.13
Abstract: Here, a new method called Aboodh transform homotopy perturbation method (ATHPM) is used to solve one dimensional fourth order parabolic linear partial differential equations with variable coefficients. The proposed method is a combination of the new integral transform "Aboodh transform" and the homotopy perturbation method. Some cases of one dimensional fourth order parabolic linear partial differential equations are solved to illustrate ability and reliability of mixture of Aboodh transform and homotopy perturbation method. We have compared the obtained analytical solution with the available Aboodh decomposition solution and homotopy perturbation method solution which is found to be exactly same. The results obtained reveal that the combination of Aboodh transform and homotopy perturbation method is quite capable, practically well appropriate for use in such problems.
Keywords: Aboodh Transform, Homotopy Perturbation Method, Linear Partial Differential Equation
1. Introduction
Many problems of physical interest are described by linear partial differential equations with initial and boundary conditions. One of them is fourth order parabolic partial differential equations with variable coefficients; these equations arise in the transverse vibration problems [1]. In recent years, many research workers have paid attention to find the solution of these equations by using various methods. Among these are the variational iteration method [Biazar and Ghazvini (2007)], Adomian decomposition method [Wazwaz (2001) and Biazar et al (2007)], homotopy perturbation method [Mehdi Dehghan and Jalil Manafian (2008)], homotopy analysis method [Najeeb Alam Khan, Asmat Ara, Muhammad Afzal and Azam Khan (2010)] and Laplace decomposition algorithm [Majid Khan, Muhammad Asif Gondal and Yasir Khan (2011)]
Prem Kiran G. Bhadane, and V. H. Pradhan (2013)]. In this paper we use coupling of new integral transform "Aboodh transform" and homotopy perturbation method. This method is a useful technique for solving linear and nonlinear differential equations. The main aim of this paper is to consider the effectiveness of the Aboodh transform homotopy perturbation method in solving higher order linear partial differential equations with variable coefficients. This method provides the solution in a rapid convergent series which leads the solution in a closed form.
2. Aboodh Transform Homotopy Perturbation Method [1, 2, 3, 4]
Let us consider the a one dimensional linear nonhomogeneous fourth order parabolic partial differential equation with variable coefficients as
(1)
where is a variable coefficient, with initial conditions
and
and boundary conditions as
,
(3)
Making use the Aboodh transform on both sides of
Eq. (1), we obtain
(4)
and, Aboodh transform of partial derivative [3]
With the aid of this property, Eq. (3) as redas
(5)
By using of initial conditions in Eq. (5), we have
or
(6)
Where
Applying Aboodh inverse on both sides of Eq. (6), admits to
(7)
Where , represents the term
Arising from the source term and the prescribed initial conditions.
By apply the homotopy perturbation method.
(8)
By substituting Eq. (8) into Eq. (7), we get
(9)
This is the coupling of the Aboodh transform and the homotopy perturbation method. collectiy the coefficient of like powers of , we have
And so on
In general recursive relation is given by
Then the solution can be expressed as
(10)
3. Application
In this section, to demonstrate the effectiveness of the proposal method we consider homogeneous and non homogeneous one dimensional fourth order linear partial differential equations with initial and boundary conditions.
Example 1.
Let us first consider fourth order homogeneous partial differential equation as [1,5]
(11)
With initial conditions
and (12)
and boundary conditions
(13)
Applying Aboodh transform to Eq. (11), admits to
Using initial conditions from Eq. (12), we have
By used Aboodh inverse on both sides of above Eq. (15),
To solve Eq.(12) using homotopy perturbation method,
we assume that
inserty Eq ( 17) of u(x, t) into Eq. (16), we get
Collecting the coefficient of like powers of in Eq. (18) we have
and so on in the same manner the rest of the components of iteration formula can be obtained and thus solution can be written in closed form as
which is an exact solution of Eq. (11) and can be verified through substitution
Example 2.
A second instructive model is the fourth order homogeneous partial differential equation as [5]
with the initial conditions
and
and boundary conditions
Applying Aboodh transform to Eq. (20), we get
Using initial conditions from Eq. (20), we get
Now taking Aboodh inverse on both sides of above
Eq. (24), we have
Now, we apply the homotopy perturbation method.
Putting this value of u(x, t) into Eq. (25), we get
collecty the coefficient of like powers of p , in Eq. (26) the following approximations are obtained
and so on in the same manner the rest of the components of iteration formula can be obtained and thus solution can be written in closed form as
Which is an exact solution of Eq. (20) and can be
Verified through substitution
Example 3.
in this case we consider the fourth order nonhomogeneous partial differential equation as [1,5]
with the following initial conditions
and
and boundary conditions
Applying Aboodh transform to Eq. (28), we get
Using initial conditions from Eq. (29), we get
now taking Aboodh inverse on both sides of above Eq. (31),we have
Now, we apply the homotopy perturbation method.
Putting this value of u(x, t) into Eq. (32), we get
Here, we assume that
Can be divided into the sum of two parts namely
and therefore we get [6]
Under this assumption, we propose a slight variation only in the components. The variation we propose is that only the partbe assigned to thewhere as the remaining part, be combined with the other terms to define.
In view of these, we formulate the modified recursive algorithm as follows
And so on in the same manner the rest of the components of iteration formula can be obtained for Thus solution can be written in closed form as
Which is an exact solution of Eq. (28) and can be verified through substitution
4. Conclusion
The aim of this paper is to show the applicability of the mixture of new integral transform "Aboodh transform" with the homotopy perturbation method to solve one dimensional fourth order homogeneous and nonhomogeneous linear partial differential equations with variable coefficients. This combination of two methods successfully worked to give very reliable and exact solutions to the equation. This proposed method provides an analytical approximation in a rapidly convergent sequence with in exclusive manner computed terms. Its rapid convergence shows that the method is trustworthy and introduces a significant improvement in solving linear partial differential equations over existing methods.
Finally it's worthwhile to mention that the proposed method can be applied to other linear and nonlinear partial differential equations arising in mathematical physics. This aim task in future.
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