Assessment of Homotopy Perturbation and Variational Iteration Methods in Heat Transfer Equations

Assessment of Homotopy Perturbation and Variational Iteration Methods in Heat Transfer Equations

DOI: 10.4018/978-1-5225-2713-8.ch004
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Abstract

We explained VIM and HPM methods in Chapter 1 and Chapter 2. In this chapter, we compare these methods for nonlinear heat transfer problems.
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The Application Of Vim, Hpm, And Perturbation Method In Heat Transfer

In order to assess the accuracy of VIM for solving nonlinear equations and to compare it with HPM and perturbation method, we will consider the following examples.

Cooling of a Lumped System by Combined Convection and Radiation

Consider the problem of combined convective–radiative cooling of lumped system (Aziz & Na, 1984). Let the system have volume , surface area , density , specific heat emissivity and the initial temperature . At , the system is exposed to an environment with convective heat transfer with the coefficient of and the temperature . The system also loses heat through radiation and the effective sink temperature is . The cooling equation and the initial conditions are as follows:

(1)
(2)

To solve the equation, we do the following changes of parameters:

(3)

After the parameter change, the heat transfer equation will result in the following form:

(4)
(5)

For simplicity, we assume the case . So we have:

(6)
(7)

Eqs. (6) and (7) are solved though PM, HPM and VIM as follows:

Variational Iteration Method

To solve Eqs. (6) and (7), we can construct the following correction functional:

(8)

Its stationary conditions can be obtained as follows:

(9)

The Lagrangian multiplier can therefore be identified as:

(10)

As a result, we obtain the following iteration formula:

(11)

Now we start with an arbitrary initial approximation that satisfies the initial condition:

(12)

Using the above variational formula (11), we have:

(13)

Substituting Eq. (12) into Eq. (13) and after some simplifications, we have:

(14)

In the same way, we obtain as follows:

(15) and so on. In the same way the rest of the components of the iteration formula can be obtained.

Homotopy Perturbation Method

After separating the linear and nonlinear parts of the equation, we apply HPM to Eqs. (6) and (7), as follows:

16) where is the linear part of the equation and is the initial approximation (Rajabi, Ganji & Taherian, 2006).We consider as:

(17)

Substituting Eq. (17) into Eq. (16) and rearranging based on powers of terms, we have:

(18)
(19) and
(20)
(21) and finally,

(22)
(23)

Solving Eqs. (18)- (23) results in When , we have:

(24)

Perturbation Method

For very small , let us assume a regular perturbation expansion and calculate the first three terms (Nayfeh, 1973), thus we assume:

(25)

Substituting Eq. (25) into Eq. (6) and after expansion and rearranging based on coefficients of terms, we have:

(26)
(27)
(28)

The solutions of Eqs. (26)- (28) are:

(29)

The three-term expansion in Eq. (25) now becomes:

(30)

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