INTRODUCTION

Fins are extensively used to enhance the heat transfer between a solid surface and its convective, radiative, or convective radiative surface (Kern and Kraus, 1972). Finned surfaces are widely used, for instance, for cooling electric transformers, the cylinders of air-craft engines, and other heat transfer equipment. Finned surfaces are widely used, for instance, for cooling electric transformers, the cylinders of air-craft engines, and other heat transfer equipment. The temperature distribution of a straight rectangular fin with a power-law temperature dependent surface heat flux can be determined by the solutions of a one-dimensional steady state heat conduction equation which, in dimensionless form, is given by Chang (2005):

(1)

subject to the boundary conditions:

(2)

where, the axial distance x is measured from the fin tip,θ is the temperature, and N is the convective-conductive parameter of the fin. The values of n vary in a wide range between 4 and 5 depending on the mode of boiling (Liaw and Yeh, 1994a,b). For example, the exponent n may take the respective values -4, -0.25, 0, 2, and 3, depending on whether the fin is subject to transition boiling, laminar film boiling or condensation, convection, nucleate boiling, and radiation into free space at zero absolute temperature.

The approximate analytical solution to 1-2 was presented by Chang (2005) using the analytic Adomian Decomposition Method (ADM). Sometimes it is a very intricate problem to calculate the so-called Adomian polynomials involved in ADM. Another analytic method which has been shown to be much simpler than the ADM is called the Homotopy-perturbation Method (HPM), first developed by He (1998, 1999, 2000, 2003, 2004, 2006a,b). We note that based on HPM, Ghorbani and Saberi-Nadjafi (2007) and Ghorbani (2009) were able to overcome the difficulty in ADM through the so-called He polynomials. HPM yields rapidly conv ergent series solutions (He, 2006a; El-Latif, 2005; Noor and Mohyud-Din, 2007). Recently, the applicability of HPM was extended to singular second-order differential equations (Chowdhury and Hashim, 2007a,b), nonlinear population dynamics models (Chowdhury et al., 2007), general time-independent Emden-Fowler equations (Chowdhury and Hashim, 2009a), time-dependent Emden-Fowler type equations (Chowdhury and Hashim, 2007b), Klein-Gordon and sine-Gordon equations (Chowdhury and Hashim, 2009b). Very recently, Chowdhury et al. (2008) and Hashim and Chowdhury (2008) were the first to successfully apply the Multistage Homotopy-perturbation Method (MHPM) to the chaotic Lorenz system and a class of system of ODEs.

In this study, we present a proper procedure based on HPM for solving analytically problem 1-2. In doing so, we corrected the work of Ganji (2006).

MODIFIED HOMOTOPY-PERTURBATION METHOD

Since the HPM is now standard and for brevity, the reader is referred to He (1998, 1999, 2000, 2003, 2004, 2006a,b) for basic ideas of HPM. To illustrate the basic ideas of the Modified Homotopy-perturbation Method (MHPM), we consider the following general nonlinear differential equation (He, 2006a, b):

(3)

with boundary conditions:

(4)

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

The operator A generally divided into two parts L and N, where L is linear while N is nonlinear. Therefore, Eq. 4 can be written as follows:

(5)

We construct a homotopy y(r,p):Ω x [0,1]→ of Eq. 3 which satisfies:

(6)

which is equivalent to:

(7)

where pε[0,1] is an embedding parameter and Y₀ is an initial approximation which satisfies boundary conditions. It follows from Eq. 6 and 7 that:

(8)

Thus, the changing process of p from 0 to 1 is just that of y(r,p) from y₀(r) to y_r(p). In topology this called deformation and L(y)–L(0) and A(y)–f(r) are called homotopic. Here the embedding parameter is introduced much more naturally, unaffected by artificial factors; further it can be considered as a small for 0≤p≤1. So it is very natural to assume that the solution of Eq. 7 and 8 can be expressed as:

(9)

According to HPM, the approximate solution of Eq. 7 can be expressed as a series of the power of p, i.e.,

(10)

Now, we apply the above procedure as an algorithm for approximation the dynamics response in a sequence of time intervals (time step) [0,1),[0,t₂),[0,t₃),...[t_n–1 t_n) such that the initial condition in [t_p,t_p+1] is taken to be the condition at t_p. For practical computations, a finite number of terms in the series:

(11)

are used in a time step procedure just outlined.

Application of modified HPM: Here, we apply an alternative approach of HPM to find approximate analytical solution to 1-2. To do so, we first construct a homotopy y(r,p):Ω x [0,1]→ which satisfies:

(12)

Suppose the solution of Eq. 1 has the form:

(13)

and let us choose the initial approximation as:

(14)

where c is to be determined.

Substituting 13 into 12 and equating the terms with identical powers of p, we get the following system of linear differential equations:

Solving the above equations, we have:

etc.

According to Eq. 13 and the assumption p = 1, the six-term approximate solution to (1) is:

(15)

The complete solution is obtained once the constant c is determined by imposing the second boundary condition given by Eq. 2. Note that the value of c must lie in the interval (0,1) to represent the temperature at the fin tip (Chang, 2005).

To carry out the iterations in every subinterval of equal length Δt [0,t₁), [0,t₂), [0,t₃)...[t_n–1t), e would need to know the values of the following initial conditions, c = θ(t^*).

In general, we do not have these information at our clearance except at the initial point t^* = t₀ = 0 but we can obtain these values following the MHPM. We note that the 6-term approximations of θ denoted as . For practical computations, a finite number of terms in the series solution are used in a time step procedure just outlined.