Subscribe Now Subscribe Today
Research Article
 

Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method



M. Fazeli, S.A. Zahedi and N. Tolou
 
Facebook Twitter Digg Reddit Linkedin StumbleUpon E-mail
ABSTRACT

In this study, a powerful analytical method, termed homotopy perturbation method is utilized for finding explicit solutions of non-linear fourth-order parabolic equations. In order to manifest the capability of proposed approach, five illustrating examples have been presented and solved. The obtained solutions, in comparison to those of exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate solutions for these kinds of nonlinear differential equations.

Services
Related Articles in ASCI
Search in Google Scholar
View Citation
Report Citation

 
  How to cite this article:

M. Fazeli, S.A. Zahedi and N. Tolou, 2008. Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method. Journal of Applied Sciences, 8: 2619-2624.

DOI: 10.3923/jas.2008.2619.2624

URL: https://scialert.net/abstract/?doi=jas.2008.2619.2624
 

INTRODUCTION

Most scientific problems are inherently nonlinear as their functional equations are nonlinear such as parabolic equations. The nonlinear parabolic equations arise in various fields of mechanics, physics, statistics and material science, for instance: transverse vibrations of uniform flexible beams (Arshad et al., 2005), optimization of the trade-off between noise removal and edge preservation that may minimize a cost functional (You and Kaveh, 2000), the epitaxial growth of nanoscale thin films (Belinda et al., 2003) and waves of the steady natural convection in a vertical fluid layer (Tang and Christov, 2007). We know that most of these types of equations do not have analytical solution as these are functioning nonlinear. Therefore, those should be solved using numerical techniques, although; perturbation methods had been used for the analytical solutions of some especial cases before. In the numerical methods, stability and convergence should be considered so as to avoid divergence or inappropriate results. In the perturbation methods, we should exert the small parameter in the equation. Therefore, finding the small parameter and exerting it into the equation are difficulties of this method. Since there are some limitations with the common perturbation method, also because the basis of the common perturbation method is upon the existence of a small parameter, developing this method for different applications is very difficult.

Indeed, many different methods have been recently introduced some ways to eliminate the small parameter, such as the variational iteration method (He, 1998, 1999, 2000; Khatami et al., 2008), the homotopy perturbation method (He, 2003, 2005, 2006; Tolou et al., 2008) and the Exp-function method (He and Wu, 2006; Mahmoudi et al., 2008). In this letter, we will apply He`s homotopy perturbation method in order to drive the explicit solution of fourth-order parabolic equations analytically. In order to evaluate the benefits of proposed method, five illustrating examples have been used. Wazwaz (2001, 2002) obtained exact solution of these equations and (Biazar and Ghazvini, 2007) previously used variational iteration method to solve these equations. This research is motivated to extend these works by implementing homotopy perturbation method that shall be more efficient. The results of homotopy perturbation method have been compared and verified with those of exact solutions from the study of Wazwaz (2001, 2002).

The fourth-order parabolic partial differential equation with variable coefficients reads as following (Wazwaz, 2001, 2002):

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(1)

where, μ (x, y, z), λ (x, y, z) and η (x, y, z) are positive.

Subject to the initial conditions (Wazwaz, 2002):

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method

And the boundary conditions (Wazwaz, 2002):

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method

where, the functions fi, gi, ki, hi, Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method, i = 0.1 are continuous.

THE BASIC IDEA OF HPM

To illustrate the basic idea of this method, we consider the following nonlinear differential equation:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(2)

Considering the boundary conditions of:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(3)

where, A is a general differential operator, B a boundary operator, f (r) a known analytical function and Γ is the boundary of the domain Ω.

The operator A can be divided into two parts of L and N, where L is the linear part, while N is a nonlinear one. Eq. 2 can, therefore, be rewritten as:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(4)

By the homotopy technique, we construct a homotopy as Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Methodwhich satisfies:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(5)

where, p ε [0, 1] is an embedding parameter and u0 is an initial approximation of Eq. 4 which satisfies the boundary conditions. Obviously, considering Eq. 4 we will have:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(6)

The changing process of p from zero to unity is just that of v (r, p) from u0 (r) to u (r). In topology, this is called deformation and L(v)οL(u0) and A (v)οf (r) are called homotopy. According to HPM, we can first use the embedding parameter p as small parameter and assume that the solution of Eq. 5 can be written as a power series in p:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(7)

Setting p = 1 results in the approximate solution of Eq. 5:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(8)

The combination of the perturbation method and the homotopy method is called the HPM, which lacks the limitations of the traditional perturbation methods, although this technique has full advantages of the traditional perturbation techniques. The series (8) is convergent for most cases. However, the convergence rate depends on the nonlinear operator A(v).

IMPLIMENTATION OF HPM

In order to illustrate the solution procedure and to show the capability of the method, five examples of different kind of fourth-order non-linear parabolic partial differential equations is presented here.

Example one: Consider the following one dimensional, variable coefficient fourth-order parabolic partial differential equations (Wazwaz, 2002).

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(9)

Subject to the initial conditions:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(10)

And the boundary conditions:

Substituting Eq. 7 and 9 into Eq. 5, after some simplification and substitution and rearranging based on powers of p-terms we have:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(11a)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(11b)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(11c)

Accuracy of solution shall give rise as n in Eq. 7 and power of p increasing. Solving Eq. 11 subject to boundary conditions will result in:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(12a)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(12b)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(12c)

So on substituting Eq. 12 into Eq. 7 gives the approximate solution in the following form:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(13)

Driving Eq. 7 for n>2 gives

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(14)

While this is the same as the exact solution presented (Wazwaz, 2001, 2002).

Example two: Consider the following parabolic equation (Wazwaz, 2002).

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(15)

Subject to the following initial conditions

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method

And the boundary conditions of

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method

As stated in example one, Increasing the n in Eq. 7, give rise to the accuracy of the solution.

Substituting v(r) from Eq. 7 and 9 into Eq. 5, after some simplification and substitution and rearranging based on powers of p-terms up to second orders of p, we have:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(16a)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(16b)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(16c)

In the same manner, the rest of component can be obtained in order to obtain better approximation.

The solution of set of Eq. 16 gives:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(17a)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(17b)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(17c)

Substituting the Eq. 17a-c into Eq. 7 gives the approximate solution for n = 2 as:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(18)

Therefore if we continue,

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(19)

This is as the exact solution that has been obtained previously by Wazwaz (2001, 2002).

Example three: Now we solve the following one dimensional non-homogeneous fourth-order equation (Wazwaz, 2002).

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(20)

In the same manner to previous examples by implementing HPM to Eq. 20 we have:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(21a)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(21b)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(21c)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(21d)

Thus,

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(22a)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(22b)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(22c)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(22d)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(22e)

And so,

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(23)

Equation 23 is as the exact solution that has been obtained before (Wazwaz, 2002).

Example four: Consider the fourth-order parabolic equation in two space variables (Wazwaz, 2001).

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(24)

And the initial conditions are:

Also, boundary conditions are:

Afterwards, implementing HPM to Eq. 24 result in:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(25a)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(25b)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(25c)

Then, solving Eq. 25 we have:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(26a)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(26b)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(26c)

So on substituting Eq. 26 into Eq. 7 gives the approximate solution in the following form:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(27)

Thus,

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(28)

Equation 28 is the same as the result obtained previously (Wazwaz, 2001, 2002).

Example five: Finally, we solve the following partial differential equation in three space variables (Wazwaz, 2001).

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(29)

Subject to the initial conditions:

And the boundary conditions:

In the same manner of previous examples, after some manipulation and rearranging based on powers of p-terms we have:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(30a)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(30b)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(30c)

Solving Eq. 29 subject to initially condition give:

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(31a)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(31b)

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(31c)

So,

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(32)

It is known that as Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method as n → ∞; thus an exact solution is obtained which reads.

Image for - Explicit Solution of Non-Linear Fourth-Order Parabolic Equations via Homotopy Perturbation Method
(33)

And this is the exact solution (Wazwaz, 2001, 2002).

CONCLUSION

In this study, for the first time a kind of analytical method called, HPM has been successfully applied to find the solution of the parabolic equations. This method has been used for solving five examples of parabolic equations. The results show that this method provides excellent approximations to the solution of this nonlinear systems with high accuracy. It is worth pointing out that this method presents a rapid convergence for the solutions with out the difficulties that have been arisen in traditional analytical methods. As shown, the homotopy perturbation method doesn`t need a small parameter. Finally, it has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method.

REFERENCES

1:  Arshad, K., K. Islam and A. Tariq, 2005. Sextic spline solution for solving a fourth-order parabolic partial differential equation. Int. J. Comput. Math., 82: 871-879.
CrossRef  |  

2:  Belinda, B.K., S. Oliver and W. Michael, 2003. A fourth-order parabolic equation modeling epitaxial thin film growth. J. Math. Anal. Applied, 286: 459-490.
CrossRef  |  

3:  Biazar, J. and H. Ghazvini, 2007. He's variational iteration method for fourth-order parabolic equations. Comput. Math. Applic., 54: 1047-1054.
Direct Link  |  

4:  He, J.H., 1998. Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput. Methods Applied Mech. Eng., 167: 69-73.
CrossRef  |  

5:  He, J.H., 1999. Variational iteration method-a kind of non-linear analytical technique: Some examples. Int. J. Nonlinear Mech., 34: 699-708.
CrossRef  |  Direct Link  |  

6:  He, J.H., 2000. Variational iteration method for autonomous ordinary differential systems. Applied Math. Comput., 114: 115-123.
CrossRef  |  Direct Link  |  

7:  He, J.H., 2003. Homotopy perturbation method: A new nonlinear analytical technique. Applied Math. Comput., 135: 73-79.
CrossRef  |  Direct Link  |  

8:  He, J.H., 2005. Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci. Numer. Simul., 6: 207-208.
CrossRef  |  Direct Link  |  

9:  He, J.H., 2006. Homotopy perturbation method for solving boundary value problems. Phys. Lett. A, 350: 87-88.
CrossRef  |  

10:  He, J.H. and X.H. Wu, 2006. Exp-function method for nonlinear wave equations. Chaos Solitons Fractals, 30: 700-708.
CrossRef  |  

11:  Khatami, I., N. Tolou, J. Mahmoudi and M. Rezvani, 2008. Application of homotopy analysis method and variational iteration method for shock wave equation. J. Applied Sci., 8: 848-853.
CrossRef  |  Direct Link  |  

12:  Mahmoudi, J., N. Tolou, I. Khatami, A. Barari and D.D. Ganji, 2008. Explicit solution of nonlinear ZK-BBM wave equation using exp-function method. J. Applied Sci., 8: 258-363.
CrossRef  |  Direct Link  |  

13:  Tang, X.H. and C.I. Christov, 2007. Non-linear waves of the steady natural convection in a vertical fluid layer: A numerical approach. Math. Comput. Simul., 74: 203-213.
CrossRef  |  Direct Link  |  

14:  Tolou, N., I. Khatami, B. Jafari and D.D. Ganji, 2008. Analytical solution of nonlinear vibrating systems. Am. J. Applied Sci., 5: 1219-1224.
CrossRef  |  Direct Link  |  

15:  Wazwaz, A.M., 2001. Analytical treatment for variable coefficients fourth-order parabolic partial differential equations. Applied Math. Comput., 123: 219-227.
CrossRef  |  

16:  Wazwaz, A.M., 2002. Exact solutions for variable coefficients fourth-order parabolic partial differential equations in higher-dimensional spaces. Applied Math. Comput., 130: 415-424.
CrossRef  |  

17:  You, Y.L. and M. Kaveh, 2000. Fourth-order partial differential equations for noise removal. IEEE Trans. Image Process., 9: 1723-1730.
CrossRef  |  Direct Link  |  

©  2022 Science Alert. All Rights Reserved