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

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 978-1-5225-2713-8.ch004.m01, surface area 978-1-5225-2713-8.ch004.m02, density 978-1-5225-2713-8.ch004.m03, specific heat 978-1-5225-2713-8.ch004.m04 emissivity 978-1-5225-2713-8.ch004.m05 and the initial temperature 978-1-5225-2713-8.ch004.m06. At 978-1-5225-2713-8.ch004.m07, the system is exposed to an environment with convective heat transfer with the coefficient of 978-1-5225-2713-8.ch004.m08 and the temperature 978-1-5225-2713-8.ch004.m09. The system also loses heat through radiation and the effective sink temperature is 978-1-5225-2713-8.ch004.m10. The cooling equation and the initial conditions are as follows:

978-1-5225-2713-8.ch004.m11
(1)
978-1-5225-2713-8.ch004.m12
(2)

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

978-1-5225-2713-8.ch004.m13
(3)

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

978-1-5225-2713-8.ch004.m14
(4)
978-1-5225-2713-8.ch004.m15
(5)

For simplicity, we assume the case 978-1-5225-2713-8.ch004.m16. So we have:

978-1-5225-2713-8.ch004.m17
(6)
978-1-5225-2713-8.ch004.m18
(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:

978-1-5225-2713-8.ch004.m19
(8)

Its stationary conditions can be obtained as follows:

978-1-5225-2713-8.ch004.m20
(9)

The Lagrangian multiplier can therefore be identified as:

978-1-5225-2713-8.ch004.m21
(10)

As a result, we obtain the following iteration formula:

978-1-5225-2713-8.ch004.m22
(11)

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

978-1-5225-2713-8.ch004.m23
(12)

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

978-1-5225-2713-8.ch004.m24
(13)

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

978-1-5225-2713-8.ch004.m25
(14)

In the same way, we obtain 978-1-5225-2713-8.ch004.m26 as follows:

978-1-5225-2713-8.ch004.m27
(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:

978-1-5225-2713-8.ch004.m28
16) where 978-1-5225-2713-8.ch004.m29is the linear part of the equation and 978-1-5225-2713-8.ch004.m30is the initial approximation (Rajabi, Ganji & Taherian, 2006).We consider 978-1-5225-2713-8.ch004.m31 as:

978-1-5225-2713-8.ch004.m32
(17)

Substituting Eq. (17) into Eq. (16) and rearranging based on powers of 978-1-5225-2713-8.ch004.m33terms, we have:

978-1-5225-2713-8.ch004.m34
(18)
978-1-5225-2713-8.ch004.m35
(19) and
978-1-5225-2713-8.ch004.m36
(20)
978-1-5225-2713-8.ch004.m37
(21) and finally,

978-1-5225-2713-8.ch004.m38
(22)
978-1-5225-2713-8.ch004.m39
(23)

Solving Eqs. (18)- (23) results in 978-1-5225-2713-8.ch004.m40 When 978-1-5225-2713-8.ch004.m41, we have:

978-1-5225-2713-8.ch004.m42
(24)

Perturbation Method

For very small 978-1-5225-2713-8.ch004.m43, let us assume a regular perturbation expansion and calculate the first three terms (Nayfeh, 1973), thus we assume:

978-1-5225-2713-8.ch004.m44
(25)

Substituting Eq. (25) into Eq. (6) and after expansion and rearranging based on coefficients of 978-1-5225-2713-8.ch004.m45 terms, we have:

978-1-5225-2713-8.ch004.m46
(26)
978-1-5225-2713-8.ch004.m47
(27)
978-1-5225-2713-8.ch004.m48
(28)

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

978-1-5225-2713-8.ch004.m49
(29)

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

978-1-5225-2713-8.ch004.m50
(30)

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