- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 324869, 10 pages

http://dx.doi.org/10.1155/2013/324869

## Solution of Boundary Layer Problems with Heat Transfer by Optimal Homotopy Asymptotic Method

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan

Received 17 June 2013; Accepted 7 August 2013

Academic Editor: Carlo Bianca

Copyright © 2013 H. Ullah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Application of Optimal Homotopy Asymptotic Method (OHAM), a new analytic approximate technique for treatment of Falkner-Skan equations with heat transfer, has been applied in this work. To see the efficiency of the method, we consider Falkner-Skan equations with heat transfer. It provides us with a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature as finite difference (N. S. Asaithambi, 1997) and shooting method (Cebeci and Keller, 1971). The obtained solutions show that OHAM is effective, simpler, easier, and explicit.

#### 1. Introduction

Most of the problems in engineering sciences are nonlinear, particularly some of heat transfer problems. Limited analytic methods are presented for the solution of such problems in the literature. Therefore, the researcher’s profound attention is to hunt some new analytic methods for the solution of the problems. Most of the methods like Adomian Decomposition Method (ADM) [1], Variational Iteration Method (VIM) [2], Differential Transform Method (DTM) [3], Radial basis function [4] and Homotopy Perturbation Method (HPM) [5], are used for the solution of weakly nonlinear problems, and limited for strongly nonlinear problems. For the solution of the strongly nonlinear problems the perturbation methods were studied [6–8]. These methods comprise a small parameter which cannot be found easily. To overcome this issue, some new analytic methods such as Artificial Parameters Method [9], Homotopy Analysis Method (HAM) [10], and Homotopy Perturbation Method (HPM) [5] were introduced. These methods pooled the homotopy with the perturbation techniques. Recently, Marinca et al. introduced Optimal Homotopy Asymptotic Method (OHAM) [11–15] for the solution of nonlinear problems which made the perturbation methods independent of the assumption of small parameters.

The Falkner-Skan equation has been considered in the last forty years due to its importance in the boundary layer theory. The boundary layer theory plays a vital role in the diverse area of engineering and scientific applications. The solution of the Falkner-Skan equation has been studied numerically first by Hartree [16]. Smith and Cebeci [17, 18] solved this equation by shooting method. Maksyn [19] solved the Falkner-Skan equation by analytic approximation. Asithambi [20–22] found its solution by finite differences, Liao [23] applied homotopy analysis to solve Falkner-Skan equation, and recently Vera [24] found its solution by Fourier series. An important case is the Blasius equation. This problem was solved by Rosales and Valencia [25] using Fourier series. Boyd [26] found the solution of Falkner-Skan equation by numerical method. An enormous amount of research work has been invested in the study of nonlinear boundary value problems [27–38]. In this paper, we will deal with the Falkner-Skan equations with heat transfer, a nonlinear boundary value problem [24] in different forms.

The motivation of this paper is to enhance OHAM for the solution of Falkner-Skan equation with heat transfer. In [11–15], OHAM has been proved to be useful for obtaining an approximate solution of nonlinear boundary value problems. In this work, we have proved that OHAM is also useful and reliable for the solution of the Falkner-Skan equation with heat transfer, hence showing its validity and great potential for the solution of transient physical phenomenon in science and engineering.

In the succeeding section, the basic idea of OHAM [11–15] is formulated for the solution of boundary value problems arising in heat transfer. In Section 3, the effectiveness of the enhanced formulation of OHAM for Falkner-Skan equation with heat transfer has been studied. Two special cases [24] of Falkner-Skan equation with heat transfer problems have been analyzed.

#### 2. Basic Mathematical Theory of OHAM

Let us consider the following differential equation: along with boundary conditions of the form where is the linear operator, is an unknown function, is a known function, is a nonlinear differential operator, and is a boundary operator.

According to OHAM, one can construct an optimal homotopy : which satisfies where is an embedding parameter, is an unknown function, and is a nonzero auxiliary function. The auxiliary function is nonzero for and . Equation (3) is the structure of OHAM homotopy.

It is defined that respectively. Thus, as varies from to, the solution varies from to , where is obtained from (1) and (2) for as follows:

Next, we choose auxiliary function in the form where are constants and can be found latter.

To obtain an approximate solution, we expand by Taylor’s series about in the following form:

Now substituting (8) into (1) and (2) and equating the coefficient of like powers of , we obtain the zeroth-order problem given by (6), the first- and second-order problems given by (9)–(11), respectively, and the general governing equations for given by (11): where is the coefficient of in the expansion series of about the embedding parameter as follows:

It should be underscored that the for are governed by the linear equations with linear boundary conditions that come from the original problem, which can be easily solved.

It has been observed that the convergence of the series (8) depends upon the auxiliary constants . If it is convergent at , one has

Substituting (13) into (1), it results in the following expression for residual:

If , then is the exact solution of the problem. Generally it does not happen, especially in nonlinear problems.

For the determinations of auxiliary constants, , there are different methods like Galerkin’s method, Ritz method, least squares method, and collocation method. One can apply the method of least squares as follows: where and are two values, depending on the nature of the given problem.

The auxiliary constants can be optimally found from

The th order approximate solution can be obtained bythese constants. The constants can also be determined by another method as follows:

The convergence of OHAM is directly proportional to the number of optimal constants which is determined by (16).

It is easy to observe [13] that the Homotopy Perturbation Method (HPM) proposed by He [4] is a special case of (3) when , and on the other hand, the Homotopy Analysis Method (HAM) proposed by Liao [11] is another special case of (3) when where is chosen from “-curves” [12].

#### 3. Application of OHAM to Falkner-Skan Equations with Heat Transfer

To demonstrate the effectiveness of OHAM formulation, two models are studied.

*Model 1 (see [23]). * When an incompressible fluid passes in the vicinity of solid boundaries, the Navier-Stokes equations may be reduced drastically into the boundary layer equations:
where is the free stream velocity, and are velocity components in - and -directions, and is the kinematic viscosity. In case of two-dimensional flow, the incompressible boundary layer flow over a wedge, when the free stream velocity is of the form , is the following similarity transformation:

Using (19) into (18), we obtained the Falkner-Skan equation
along with boundary conditions

According to (1), we have

The boundary conditions are

Applying the method formulation mentioned in Section 2 leads to the following.*Zeroth-Order Problem. *Consider
from which we obtain
*First-Order Problem. *Consider

Its solution is
*Second-Order Problem. *Consider
with BC
whose solution is

Adding (25), (27), and (29), we obtain
where

For the computation of the constants and applying the method of least square mentioned in (14)–(16), we get

Putting these values in (30), we obtained the approximate solution of the form

Now consider the energy equation of an incompressible fluid which passes through the vicinity of the solid boundaries:

Upon using the transformation in (19) into (37), we obtained
with boundary conditions
where “” is the thermal diffusivity and is the prandtl number.

Applying the method formulated in Section 2 leads to the following.*Zeroth-Order Problem. *Consider

Its solution is
*First-Order Problem. *Consider

whose solution is

*Second-Order Problem. *Consider

We obtain the following solution:

Adding (41), (43), and (45), we obtain where

Using (46) in (38) and applying the method of least square, we obtain

Substituting these values in (46) for and , we obtain

*Model 2 (see [23]). *In case of in (52), we obtain the Blasius equation which is the famous equation of fluid dynamics and represent the problem of an incompressible fluid that passes through on a semi-infinite flat plate.

One has

According to (1), we define the operators
where and represent the second and third derivatives of with respect to .

Applying the method formulated in Section 2 leads to the following.

*Zeroth-Order Problem. *Consider

Its solution is

*First-Order Problem*. Consider

whose solution is

*Second-Order Problem. *Consider
We obtain the following solution:

*Third-Order Problem*. Consider
Its solution is
From (53), (57), (59), and (61), we obtain
where
Using (62) in (50) and applying the technique as discussed in (14)–(16), we obtain
Substituting these values in (62), we have

#### 4. Results and Discussions

The formulation presented in Section 2 provides highly accurate solutions for the problems demonstrated in Section 3. We have used Mathematica 7 for most of our computational work. In Table 1, we have presented the initial slope for different values of obtained by OHAM. Table 1 shows that the results obtained by OHAM are in excellent agreement with the results found in the literature. Table 1 shows a benchmark for the initial slope found by different authors. It is found that the method present in this work is very good and provides the same values that optimized numerical methods with which it is compared. Figure 1 shows the variation of the function against for different values of with OHAM, while Figure 2 shows the variation of the function with respect to for different values of . Table 2 shows the solution of the Blasius equation obtained by the present method. In order to verify the accuracy of the present method, we have compared the results obtained by OHAM to the results available in the literature and found an excellent agreement. Figures 2 and 3 show the variation of with respect to for the Blasius equation which is identical to results in the literature [41].

#### 5. Conclusion

In this work, we have seen the effectiveness of OHAM [11–15] to Falkner-Skan, Energy and Blasius equations. By applying the basic idea of OHAM to Falken-Skan, Energy and Blasius equations, we found that it is simpler in applicability and, more convenient to control convergence and involved less computational overhead. Therefore, OHAM shows its validity and great potential for the solution Falken-Skan, Energy and Blasius equations, with heat transfer problems arising in science and engineering.

#### References

- S. H. Chowdhury, “A comparison between the modified homotopy perturbation method and adomian decomposition method for solving nonlinear heat transfer equations,”
*Journal of Applied Sciences*, vol. 11, no. 7, pp. 1416–1420, 2011. View at Publisher · View at Google Scholar · View at Scopus - D. D. Ganji, G. A. Afrouzi, and R. A. Talarposhti, “Application of variational iteration method and homotopy-perturbation method for nonlinear heat diffusion and heat transfer equations,”
*Physics Letters A*, vol. 368, no. 6, pp. 450–457, 2007. View at Publisher · View at Google Scholar · View at Scopus - H. Yaghoobi and M. Torabi, “The application of differential transformation method to nonlinear equations arising in heat transfer,”
*International Communications in Heat and Mass Transfer*, vol. 38, no. 6, pp. 815–820, 2011. View at Publisher · View at Google Scholar · View at Scopus - D. D. Ganji, “The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer,”
*Physics Letters A*, vol. 355, no. 4-5, pp. 337–341, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. H. He, “Homotopy perturbation technique,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 178, pp. 257–262, 1999. View at Google Scholar - R. Bellman,
*Perturbation Techniques in Mathematics, Physics, and Engineering*, Holt, Rinehart and Winston, New York, NY, USA, 1964. View at MathSciNet - J. D. Cole,
*Perturbation Methods in Applied Mathematics*, Blaisdell Publishing Co. Ginn and Co, Waltham, Mass, USA, 1968. View at MathSciNet - R. E. O'Malley Jr.,
*Introduction to Singular Perturbations*, Academic Press, New York, NY, USA, 1974. View at MathSciNet - G. L. Liu, “New research direction in singular perturbation theory: artificial parameter approach and inverse perturbation technique,” in
*Proceedings of the 7th Conference of 7th Modern Mathematics and Mechanics*, 1997. - S. J. Liao,
*The proposed homotopy analysis technique for the solution of nonlinear problems [Ph.D. thesis]*, Shanghai Jiao Tong University, 1992. - N. Herişanu and V. Marinca, “Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia,”
*Meccanica*, vol. 45, no. 6, pp. 847–855, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - V. Marinca and N. Herişanu, “Application of Optimal Homotopy Asymptotic Method for solving nonlinear equations arising in heat transfer,”
*International Communications in Heat and Mass Transfer*, vol. 35, no. 6, pp. 710–715, 2008. View at Publisher · View at Google Scholar · View at Scopus - V. Marinca, N. Herişanu, C. Bota, and B. Marinca, “An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate,”
*Applied Mathematics Letters*, vol. 22, no. 2, pp. 245–251, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Marinca, N. Herisanu, and I. Nemes, “A new analytic approach to nonlinear vibration of an electrical machine,”
*Proceedings of the Romanian Academy*, vol. 9, pp. 229–236, 2008. View at Google Scholar - V. Marinca and N. Herişanu, “Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method,”
*Journal of Sound and Vibration*, vol. 329, no. 9, pp. 1450–1459, 2010. View at Publisher · View at Google Scholar · View at Scopus - D. R. Hartree, “On an equation occurring in Falkner-Skan’s approximate treatment of the equations boundary layer,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 33, pp. 233–239, 1921. View at Google Scholar - A. M. O. Smith, “Improved solution of the Falkner-Skan equation boundary layer equation,”
*Sherman M. Fairchild Fund Paper*FF-10, 1954. View at Google Scholar - T. Cebeci and H. B. Keller, “Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equation,”
*Journal of Computational Physics*, vol. 7, no. 2, pp. 289–300, 1971. View at Google Scholar · View at Scopus - D. Meksyn,
*New Methods in Laminar Boundary Layer Theory*, Pergamon Press, 1961. - A. Asaithambi, “Numerical solution of the Falkner-Skan equation using piecewise linear functions,”
*Applied Mathematics and Computation*, vol. 159, no. 1, pp. 267–273, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Asaithambi, “A finite-difference method for the Falkner-Skan equation,”
*Applied Mathematics and Computation*, vol. 92, no. 2-3, pp. 135–141, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. S. Asaithambi, “A numerical method for the solution of the Falkner-Skan equation,”
*Applied Mathematics and Computation*, vol. 81, no. 2-3, pp. 259–264, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-J. Liao, “A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate,”
*Journal of Fluid Mechanics*, vol. 385, pp. 101–128, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Rosales-Vera and A. Valencia, “Solutions of Falkner-Skan equation with heat transfer by Fourier series,”
*International Communications in Heat and Mass Transfer*, vol. 37, no. 7, pp. 761–765, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Rosales and A. Valencia, “A note on solution of blasius equation by Fourier series,”
*Advances in Applied Mechanics*, vol. 6, pp. 33–38, 2009. View at Google Scholar - J. P. Boyd, “The Blasius function in the complex plane,”
*Experimental Mathematics*, vol. 8, no. 4, pp. 381–394, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Liao, “On the homotopy analysis method for nonlinear problems,”
*Applied Mathematics and Computation*, vol. 147, no. 2, pp. 499–513, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Idrees, S. Islam, and A. M. Tirnizi, “Application of optimal homotopy asymptotic method of the Korteqag-de-Varies equation,”
*Computers and Mathematics with Applications*, vol. 63, pp. 695–707, 2012. View at Google Scholar - S. Haq, M. Idrees, and S. Islam, “Application of optimal homotopy asymptotic method to eight order initial and boundary value problem,”
*Applied Mathematics and Computation*, vol. 4, pp. 73–80, 2010. View at Google Scholar - S. Iqbal, M. Idrees, A. M. Siddiqui, and A. R. Ansari, “Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method,”
*Applied Mathematics and Computation*, vol. 216, no. 10, pp. 2898–2909, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Haq, M. Idrees, and S. Islam, “Application of optimal homotopy asymptotic method to eight order boundary value problem,”
*Journal of Computational and Applied Mathematics*, vol. 2, pp. 73–80, 2010. View at Google Scholar - M. Idrees, S. Haq, and S. Islam, “Application of optimal hommotopy asymptotic method to fourth order boundary value problem,”
*World Applied Sciences Journal*, vol. 9, pp. 131–137, 2010. View at Google Scholar - M. Idrees, S. Islam, S. Haq, and S. Islam, “Application of the optimal homotopy asymptotic method to squeezing flow,”
*Computers & Mathematics with Applications*, vol. 59, no. 12, pp. 3858–3866, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Idrees, S. Haq, and S. Islam, “Application of optimal homotopy asymptotic method to sixth order boundary value problems,”
*World Applied Sciences Journal*, vol. 9, pp. 138–143, 2010. View at Google Scholar - H. Ullah, S. Islam, M. Idrees, and M. Arif, “Application of optimal homotopy asymptotic method to heat transfer problems,”
*Research Journal of Recent Sciences*. In press. - J. Ali, S. Islam, S. Islam, and G. Zaman, “The solution of multipoint boundary value problems by the optimal homotopy asymptotic method,”
*Computers & Mathematics with Applications*, vol. 59, no. 6, pp. 2000–2006, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Ali, S. Islam, H. Khan, and S. I. Ali Shah, “The optimal homotopy asymptotic method for the solution of higher-order boundary value problems in finite domains,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 401217, 14 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus - R. Nawaz, H. Ullah, S. Islam, and M. Idrees, “Application of optimal homotopy asymptotic method to Burger’s equations,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 387478, 8 pages, 2013. View at Publisher · View at Google Scholar - A. A. Salama, “Higher-order method for solving free boundary-value problems,”
*Numerical Heat Transfer B*, vol. 45, no. 4, pp. 385–394, 2004. View at Publisher · View at Google Scholar · View at Scopus - J. Zhang and B. Chen, “An iterative method for solving the Falkner-Skan equation,”
*Applied Mathematics and Computation*, vol. 210, no. 1, pp. 215–222, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Abbasbandy, “A numerical solution of Blasius equation by Adomian's decomposition method and comparison with homotopy perturbation method,”
*Chaos, Solitons and Fractals*, vol. 31, no. 1, pp. 257–260, 2007. View at Publisher · View at Google Scholar · View at Scopus