Solutions

## 1. Introduction

Fins are extensively used to enhance the heat transfer between a solid surface and its convective, radiative, or convective radiative surface. Finned surfaces are widely used, for instance, for cooling electric transformers, the cylinders of air-craft engines, and other heat transfer equipment. In many applications various heat transfer modes, such as convection, nucleate boiling, transition boiling, and film boiling, the heat transfer coefficient is no longer uniform. A fin with an insulated end has been studied by many investigators [Sen, S. Trinh(1986)]; and [Unal (1987)]. Most of them are immersed in the investigation of single boiling mode on an extended surface. Under these circumstances very recently, [Chang (2005)] applied standard Adomian decomposition method for all possible types of heat transfer modes to investigate a straight fin governed by a power-law-type temperature dependent heat transfer coefficient using 13 terms. [Liu (1995)] found that Adomian method could not always satisfy all its boundaries conditions leading to boundaries errors.

The governing equations for the temperature distribution along the surfaces are nonlinear. In consequence, exact analytic solutions of such nonlinear problems are not available in general and scientists use some approximation techniques to approximate the solutions of nonlinear equations as a series solution such as perturbation method; see [Van Dyke M. (1975)], and Nayfeh A.H. (1973)], and homotopy perturbation method; see [He J. H. (1999, (2000), and(2003)].

In this chapter, we applied HPM to solve the linear and nonlinear equations of heat transfer by conduction in one-dimensional in two slabs of different material and thickness L.

## 2. The perturbation method

Many physics and engineering problems can be modelled by differential equations. However, it is difficult to obtain closed-form solutions for them, especially for nonlinear ones. In most cases, only approximate solutions (either analytical ones or numerical ones) can be expected. Perturbation method is one of the well-known methods for solving nonlinear problems analytically.

In general, the perturbation method is valid only for weakly nonlinear problems[Nayfeh (2000)]. For example, consider the following heat transfer problem governed by the nonlinear ordinary differential equation, see [Abbasbandy (2006)]:

where ε > 0 is a physical parameter, the prime denotes differentiation with respect to the time t. Although the closed-form solution of u(t) is unknown, it is easy to get the exact result *u(t)* into such a perturbation series

Substituting the above expression into (1) and equating the coefficients of the like powers of ε, to get the following linear differential equations

Solving the above equations one by one, one has

Thus, we obtain

which gives at t = 0 the derivative

Obviously, the above series is divergent for

To overcome the restrictions of perturbation techniques, some non-perturbation techniques are proposed, such as the Lyapunov's artificial small parameter method [Lyapunov A.M. (1992)], the δ-expansion method [Karmishin et al(1990)], the homotopy perturbation method [He H., J.,(1998)], and the variational iteration method (VIM), [He H., J.,(1999)],. Using these non-perturbation methods, one can indeed obtain approximations even if there are no small/large physical parameters. However, the convergence of solution series is not guaranteed. For example, by means of the HPM, we obtain the same and exact approximation of Eq.(1), as the perturbation result in Eq.(9), that is divergent for ε > 1, as shown in Fig.1. ; For details, see [Abbasbandy (2006)]. This example shows the importance of the convergence of solution series for all possible physical parameters. From physical points of view, the convergence of solution series is much more important than whether or not the used analytic method itself is independent of small/large physical parameters. If one does not keep this in mind, some useless results might be obtained. For example, let us consider the following linear differential equation [Ganji et al (2007)]:

Its exact solution reads

By means of the homotopy perturbation method, [Ganji et al (2007)] wrote the original equation in the following form:

subject to the initial condition

where

which gives the solution. For *p* = 1, and substitute (15) into the original equation (13) and initial condition in (14), then equating the coefficients of the like powers of *p*, one can get governing equations and the initial conditions for

and the 5th-order approximation reads

However, for any given

where *δ(t)* monotonously increases very quickly:

In fact, it is easy to find that the HPM series solution (16) is divergent for all x and t except t = 0 which however corresponds to the given initial condition in (11). In other words, the convergence radius of the HPM solution series (17) is zero. It should be emphasized that, the variational iteration method (VIM) obtained exactly the same result as (17) by the 6th iteration see; [He H. J., (1999)], and [Ganji et al (2007)]. This example illustrates that both of the HPM and the VIM might give divergent approximations. Thus, it is very important to ensure the convergence of solution series obtained.

## 3. Outline of Homotopy Perturbation Method (HPM)

The homotopy analysis method (HAM) has been proposed by Liao in his PhD dissertation in [Liao (1992)]. Liao introduced the so-called auxiliary parameter in [Liao (1997a)] to construct the following two-parameter family of equation:

where *h*, and thus one can always get a convergent series solution by means of choosing a proper value of *h*. Using the definition of Taylor series with respect to the embedding parameter *p* (which is a power series of *p* ), [Liao (1997b)] gave general equations for high-order approximations.

[He J. H. (1999)] followed Dr. Liao’s early idea of Homotopy Perturbation Method (HPM) when he constructed the one-parameter family of equation:

where Eq.(20) represented special case of Eq.(19) for convergent solution of (HAM) at

With boundary conditions

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

The operator *A* can be generally divided into linear and nonlinear parts, say *L* and *N*. Therefore (21) can be

written as

[He (1999)] constructed a homotopy

where

In topology, *p* monotonically increases from zero to unit as the trivial problem *p*, as follows:

when

The series (28) is convergent for most cases, and the rate of convergence depends on

## 4. Application of Homotopy Perturbation Method HPM

An analytic method for strongly nonlinear problems, namely the homotopy analysis method (HAM) was proposed by Liao in 1992, six years earlier than the homotopy perturbation method by [He H., J.,(1998)], and the variational iteration method by [He H., J.,(1999)]. Different from perturbation techniques, the HAM is valid if a nonlinear problem contains small/large physical parameters.

More importantly, unlike all other analytic techniques, the HAM provides us with a simple way to adjust and control the convergence radius of solution series. Thus, one can always get accurate approximations by means of the HAM. In the next section, HPM is applied to solve the linear and nonlinear equations of heat transfer by conduction in one-dimensional in a slab of thickness (L). [Anwar (2010)] solved the linear and non-linear heat transfer equations by means of HPM.

### 4.1. Non-Linear Heat transfer equation

Consider the heat transfer equation by conduction in one-dimensional in a slab of thickness L. The governing equation describing the temperature distribution is:

Where the two faces are maintained at uniform temperatures *k* is assumed to vary linearly with temperature, that is:

where

where

The problem is formulated by using (19) as:

Where the Linear operator:

from Eq.(31), the initial guess is:

and the linear operator:

and the nonlinear operator of

where

and the convergence of series (40) is convergent at

For the s-th- order problems, if we first differentiate Eq.(32) s times with respect to p then divide by s! and setting p = 0 we obtain:

Where:

The general solutions of (42) can be written as:

where

The linear non-homogeneous (42) is solved for the order s = 1, 2, 3,..., for s=1, (42) becomes:

Then

the solution of (47) gives :

For s = 2, Eq.(42) becomes:

Solution of (49) gives :

Then, solution of (31) is:

Results of

X | |||||

0.1 | 0.912375 | 0.91008 | 0.9045 | 0.876375 | 0.828 |

0.2 | 0.824 | 0.82304 | 0.816 | 0.776 | 0.704 |

0.3 | 0.734125 | 0.73696 | 0.7315 | 0.692125 | 0.616 |

0.4 | 0.642 | 0.64992 | 0.648 | 0.618 | 0.552 |

0.5 | 0.546875 | 0.56 | 0.5625 | 0.546875 | 0.5 |

0.6 | 0.448 | 0.46528 | 0.472 | 0.472 | 0.448 |

0.7 | 0.3446249 | 0.36384 | 0.3735 | 0.386625 | 0.384 |

0.8 | 0.2359999 | 0.2537599 | 0.2639999 | 0.2839999 | 0.2959999 |

0.9 | 0.1213749 | 0.1331199 | 0.1404999 | 0.1573749 | 0.1719999 |

Special computer program was used as special case, the temperature distribution along a road of length (L = 1 m) when

X | |||||

0.0 | 100 | 100 | 100 | 100 | 100 |

0.1 | 95.61875 | 95.504 | 95.225 | 93.81875 | 91.4 |

0.2 | 91.2 | 91.152 | 90.8 | 88.8 | 85.2 |

0.3 | 86.70625 | 86.848 | 86.575 | 84.60625 | 80.8 |

0.4 | 82.1 | 82.496 | 82.4 | 80.9 | 77.6 |

0.5 | 77.34375 | 78 | 78.125 | 77.34375 | 75 |

0.6 | 72.4 | 73.264 | 73.6 | 73.6 | 72.4 |

0.7 | 67.23125 | 68.192 | 68.675 | 69.33125 | 69.2 |

0.8 | 61.8 | 62.688 | 63.2 | 64.2 | 64.8 |

0.9 | 56.06874 | 56.656 | 57.025 | 57.86874 | 58.6 |

1.0 | 50 | 50 | 50 | 50 | 50 |

### 4.2. Linear Heat transfer equation

In this section we consider the linear one-dimensional equation of heat transfer by conduction (diffusion equation) [Anderson (1984)]:

for initial condition

and boundary condition

Then:

For zeroth order of p:

Then

For first order of p:

For second order of p:

Using equation (56) for other orders of p, we can obtain the following results:

It is obvious that

Fig.5 and Fig.6 represent the HPM solution T(x, t) for

X | t=0 | t = 0.1 | t = 0.2 | t = 0.3 | t = 0.4 |

0 | 0 | 0 | 0 | 0 | 0 |

0.05 | 0.3091373 | 0.2082383 | 0.1402717 | 9.448861E-2 | 6.364859E-2 |

0.1 | 0.5879898 | 0.3960766 | 0.2668017 | 0.1797206 | 0.1210618 |

0.15 | 0.8092399 | 0.5451131 | 0.3671944 | 0.2473463 | 0.1666153 |

0.2 | 0.9512127 | 0.6407476 | 0.4316148 | 0.2907406 | 0.1958462 |

0.25 | 0.9999998 | 0.6736112 | 0.4537521 | 0.3056525 | 0.205891 |

0.3 | 0.9508218 | 0.6404843 | 0.4314375 | 0.2906211 | 0.1957657 |

0.35 | 0.8084964 | 0.5446123 | 0.366857 | 0.247119 | 0.1664622 |

0.4 | 0.5869664 | 0.3953872 | 0.2663373 | 0.1794078 | 0.1208511 |

0.45 | 0.3079342 | 0.207428 | 0.1397258 | 0.0941209 | 0.0634009 |

0.5 | 0 | 0 | 0 | 0 | 0 |

0.55 | -0.3103399 | -0.2090485 | -0.1408174 | -0.0948562 | -6.389621E-2 |

0.60 | -0.5890125 | -0.3967655 | -0.2672657 | -0.1800332 | -0.1212724 |

0.65 | -0.8099824 | -0.5456133 | -0.3675313 | -0.2475732 | -0.1667681 |

0.70 | -0.9516023 | -0.64101 | -0.4317916 | -0.2908597 | -0.1959264 |

0.75 | -0.9999982 | -0.6736101 | -0.4537514 | -0.3056521 | -0.2058907 |

0.80 | -0.9504291 | -0.6402198 | -0.4312593 | -0.2905011 | -0.1956849 |

0.85 | -0.8077511 | -0.5441103 | -0.3665189 | -0.2468912 | -0.1663087 |

0.90 | -0.5859417 | -0.3946969 | -0.2658723 | -0.1790946 | -0.1206401 |

0.95 | -0.3067303 | -0.206617 | -0.1391795 | -9.37529E-2 | -6.315302E-2 |

1 | 0 | 0 | 0 | 0 | 0 |

X | t=0 | t = 0.1 | t = 0.2 | t = 0.3 | t = 0.4 |

0 | 0 | 0 | 0 | 0 | 0 |

0.05 | 0.3091373 | 0.2537208 | 0.2082383 | 0.1709092 | 0.1402717 |

0.1 | 0.5879898 | 0.4825858 | 0.3960766 | 0.3250752 | 0.2668017 |

0.15 | 0.8092399 | 0.6641742 | 0.5451131 | 0.4473952 | 0.3671944 |

0.2 | 0.9512127 | 0.7806966 | 0.6407476 | 0.5258861 | 0.4316148 |

0.25 | 0.9999998 | 0.8207381 | 0.6736112 | 0.5528585 | 0.4537521 |

0.3 | 0.9508218 | 0.7803758 | 0.6404843 | 0.52567 | 0.4314375 |

0.35 | 0.8084964 | 0.6635639 | 0.5446123 | 0.4469841 | 0.366857 |

0.4 | 0.5869664 | 0.4817458 | 0.3953872 | 0.3245094 | 0.2663373 |

0.45 | 0.3079342 | 0.2527334 | 0.207428 | 0.1702441 | 0.1397258 |

0.5 | 0 | 0 | 0 | 0 | 0 |

0.55 | -0.3103399 | -0.2547078 | -0.2090485 | -0.1715741 | -0.1408174 |

0.60 | -0.5890125 | -0.4834251 | -0.3967655 | -0.3256406 | -0.2672657 |

0.65 | -0.8099824 | -0.6647836 | -0.5456133 | -0.4478057 | -0.3675313 |

0.70 | -0.9516023 | -0.7810164 | -0.64101 | -0.5261015 | -0.4317916 |

0.75 | -0.9999982 | -0.8207368 | -0.6736101 | -0.5528576 | -0.4537514 |

0.80 | -0.9504291 | -0.7800536 | -0.6402198 | -0.5254529 | -0.4312593 |

0.85 | -0.8077511 | -0.6629522 | -0.5441103 | -0.4465721 | -0.3665189 |

0.90 | -0.5859417 | -0.4809047 | -0.3946969 | -0.3239429 | -0.2658723 |

0.95 | -0.3067303 | -0.2517453 | -0.206617 | -0.1695785 | -0.1391795 |

1 | 0 | 0 | 0 | 0 | 0 |

## 5. Discussion

For example.1, clearly, for

## 6. Conclusion

Homotopy Perturbation Method HPM is applied to solve the linear and nonlinear partial differential equation. Two numerical simulations are presented to illustrate and confirm the theoretical results. The two problems are about heat transfer by conduction in two slabs. Results obtained by the homotopy perturbation method are presented in tables and figures. Results are compared with those studied by the generalized approximation method by [Sajida et al (2008)]. Homotopy Perturbation Method is considered as effective method in solving partial differential equation.