Comparison of HAM and HPM solutions in heat radiation equations

International Communications in Heat and Mass Transfer 36 (2009) 59–62

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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t

Comparison of HAM and HPM solutions in heat radiation equations☆ M. Sajid a,⁎, T. Hayat b a b

Theoretical Plasma Physics Division, PINSTECH, P.O. Nilore, Islamabad 44000, Pakistan Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan

a r t i c l e

i n f o

Available online 9 October 2008 Keywords: Homotopy analysis method Convergence Heat transfer Radiation equation

a b s t r a c t In this paper it is proved that the perturbation and homotopy perturbation solutions for the two problems namely (i) unsteady convective–radiative equation and (ii) non-linear convective–radiative conduction equation are only valid for weak non-linearity. It is worth mentioning that homotopy perturbation method (HPM) does not provide exact and convergent solutions in most of the situations. Moreover, HPM have no criteria for establishing the convergence of the series solution. In the present study, it is shown and commented that the results in the two examples are valid only for small values of the parameters. The homotopy analysis method (HAM) is used in obtaining more meaningful and valid solutions. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction The main purpose of the present attempt is two fold. Firstly to show that HPM does not provide a valid solution for the large values of the parameters. Secondly to provide an explicit, analytic and valid solution by HAM. The HAM has been proposed by Liao in his PhD dissertation [2] in 1992. In [2] Liao followed the traditional concept of homotopy to construct the following one-parameter family of equations ð1−pÞLðuÞ þ pN ðuÞ ¼ 0;

ð1Þ

in which L is an auxiliary linear operator, N is a non-linear operator related to the original non-linear problem N (u)= 0 and p is an embedding parameter. However, in [3], Liao expressed Eq. (1) in a different form: ð1−pÞLðuÞ ¼ −pN ðuÞ:

ð2Þ

Moreover, Liao found that in some cases the series solution provided by the traditional concepts of homotopy is divergent. This disadvantage has been removed by introducing an auxiliary parameter [4,5] in 1997. The new concept is used to construct the following two-parameter family of equation ð1−pÞLðu−u0 Þ ¼ JpN ðuÞ;

ð3Þ

where u0 is an initial guess. It is straightforward to note that Eq. (1) is a special case of Eq. (3) when J = −1. Liao pointed out [2,5,10,12–14] that the convergence of the solution series given by the HAM is determined by J, and thus one can always get a convergent series solution by means of choosing a proper value of J. So, the auxiliary parameter J

provides us a simple way to ensure the convergence of HAM series solutions. Using the definition of Taylor series with respect to the embedding parameter p (which is a power series of p), Liao gave a general equations for high-order approximations. Moreover in recent papers by Hayat et al. [6] and Sajid et al. [7,8] it is explicitly proved that HPM is a special case of HAM when J = −1. Since the results of HPM are just valid for the weak non-linearity so it has the same drawbacks as in the traditional perturbation methods. This is surprising that various workers are still claiming that HPM does not require any small parameter. The results presented in [1] are corrected here by using HAM [9]. The readers are referred to the articles [10–30] for the HAM solutions in various problems. 2. The first example (unsteady non-linear convective–radiative equation) The problem considered in [1] is dθ dθ þ θ þ 1 θ þ 2 θ4 ¼ 0; dτ dτ

ð4Þ

θð0Þ ¼ 1:

ð5Þ

2.1. HAM solution 2.1.1. Zeroth-order deformation problem The temperature θ(τ) can be expressed by the following set of base functions fexpð− kτ Þjk N 0g

ð6Þ

in the form of the following series ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (M. Sajid). 0735-1933/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2008.08.010

θð X Þ ¼

∞ X k¼0

am;k e−kτ ;

ð7Þ

60

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where θm ðτ Þ ¼

1 ∂m b θ ðτ; pÞ m! ∂pm

j

;

ð16Þ

p¼0

and the convergence of the series (15) depends upon the values of J. The value of J is chosen in such a way that the series (15) is convergent at p = 1. By using Eq. (15) one obtains θðτÞ ¼ θ0 ðτÞ þ

∞ X

θm ðτÞ:

ð17Þ

m¼1

2.1.2. mth-order deformation problems Here we first differentiate Eq. (12) m times with respect to p then divide by m! and setting p = 0 we obtain

Fig. 1. J-curves for different values of 1.

L½θm ðτ Þ − χm θm−1 ðτ Þ ¼ JRm ðτ Þ;

ð18Þ

θm ð0Þ ¼ 0;

ð19Þ

2 3 m−1 m−1 X k X l X X dθm−1 þ θm−1 þ 1 Rm ðτÞ ¼ 4 θm−1−k θk′ þ 2 θm−1−k θk−l θl−j θj ;5; ð20Þ dτ k¼0 k¼0 l¼0 j¼0

 χm ¼

0; 1;

mV1; : mN1:

ð21Þ

The general solutions of Eqs. (19)–(21) can be written as −τ θm ðτ Þ ¼ θ★ m ðτ Þ þ C1 e ;

where θ★ m (τ) is the particular solution and the constants are determined by the boundary conditions (19). In the next subsection, the linear non-homogeneous Eqs. (18)–(20) are solved using Mathematica in the order m = 1,2,3,…

Fig. 2. J-curves for different values of 2.

where am,k are the coefficients. Invoking the so-called Rule of solution expressions for θ(τ) and Eq. (5) the initial guess θ0(τ) and linear operators L are −τ

θ0 ðτÞ ¼ e ;

ð8Þ

Lð f Þ ¼ f ′ þ f ;

ð9Þ

where L½C1 e− τ  ¼ 0;

ð10Þ

and C1 is a constant. The non-linear operator is:

2.2. Analysis of the results We note that the explicit, analytic expression in Eq. (17) is the series solution of the problem. One can find the convergence region and rate of approximation by choosing the proper values of the auxiliary parameter J. To see this the J-curves are plotted for different values of the parameters 1 and 2 in Figs. 1 and 2. Fig. 1 depicts that the interval for admissible values of J shrinks towards zero by increasing 1, however there is no change in the value of J when we vary 2 keeping 1 fixed and is shown in Fig. 2. Since the results of HPM can be obtained as a special case of HAM when J =−1 but Fig. 1 shows that J =−1 is valid only for 1 ≥0.5. Therefore

h i ∂b  4 θ ðτ; pÞ b ∂b θ ðτ; pÞ N b θ ðτ; pÞ ¼ θ ðτ; pÞ θ ðτ; pÞ : ð11Þ þ θ ðτ; pÞ þ 1 b þ 2 b ∂τ ∂τ

The problem at the zeroth-order is h i h i θ ðτ; pÞ ; ð1−pÞL b θ ðτ; pÞ−θ0 ðτ Þ ¼ pJN b

ð12Þ

b θ ð0; pÞ ¼ 1;

ð13Þ

where J is the auxiliary non-zero parameter. For p = 0 and p = 1, we have b θ ðτ; 0Þ ¼ θ0 ðτÞ;

b θ ðτ; 1Þ ¼ θðτÞ:

ð14Þ

The initial guess θ0(τ) tends to θ(τ) as p varies from 0 to 1. Due to Taylor's series expansion b θ ðτ; pÞ ¼ θ0 ðτÞ þ

∞ X m¼1

θm ðτÞpm ;

ð22Þ

ð15Þ Fig. 3. Temperature θ for 1 = 2.0, 2 = 0 for exact, HPM and HAM solutions.

M. Sajid, T. Hayat / International Communications in Heat and Mass Transfer 36 (2009) 59–62

61

in the form of the following series θð X Þ ¼ b0;0 þ

∞ X

bm;k X k ;

ð26Þ

k¼0

in which bm,k are the coefficients. Invoking the so-called Rule of solution expressions for θ(X) and Eq. (24) the initial guess θ0(X) and linear operators L are θ0 ð X Þ ¼ 1−X;

ð27Þ

L1 ð f Þ ¼ f ″;

ð28Þ

where L1 ½C2 X þ C3  ¼ 0; Fig. 4. Temperature θ for 1 = 2.0, 2 = 0.5 for HPM and HAM solutions.

ð29Þ

and C2, C3 are the constants and the non-linear operator is: h i ∂2 b θ ð X; pÞ 2 b N1 b −N θ ð X; pÞ θ ð X; pÞ ¼ ∂X 22 3 !2  4 2b b θ ð X; p Þ ∂ θ ð X; p Þ ∂ 5− 2 θbð X; pÞ : þ θbð X; pÞ þ 1 4 2 ∂X ∂X The problem at the zeroth-order is

ð30Þ

h i h i ð1−pÞL1 θbð X; pÞ − θ0 ð X Þ ¼ pJN 1 b θ ð X; pÞ ;

ð31Þ

′ b θ ð0; pÞ ¼ 0;

ð32Þ

b θ ð1; pÞ ¼ 1:

For p = 0 and p = 1, we have b θ ð X; 1Þ ¼ θð X Þ:

b θ ð X; 0Þ ¼ θ0 ð X Þ;

ð33Þ

The initial guess θ0(X) tends to θ(X) as p varies from 0 to 1. Due to Taylor's series expansion Fig. 5. J-curves for different values of 1.

b θ ð X; pÞ ¼ θ0 ð X Þ þ where

ln θ þ 1 ðθ−1Þ þ τ ¼ 0:

θm ð X Þ ¼

θ′ð0Þ ¼ 0;

θð1Þ ¼ 1:

ð23Þ

1 ∂m b θ ð X; pÞ m! ∂pm

j

ð35Þ

; p¼0

and the convergence of the series (34) depends upon the values of J. The value of J is chosen in such a way that the series (34) is convergent at p = 1. Then by using Eq. (33) one obtains θð X Þ ¼ θ0 ð X Þ þ

∞ X

θm ð X Þ:

ð36Þ

m¼1

L1 ½θm ð X Þ−χm θm−1 ð X Þ ¼ JR1m ð X Þ;

ð37Þ

θm ð0Þ ¼ θm ð1Þ ¼ 0;

ð38Þ

where ð24Þ

3.1. HAM solution

R1m ð X Þ ¼

2 m−1  X  d θm−1 − N2 θm−1 þ 1 θ′m−1−k θ′k þ θm−1−k θ″k 2 dX k¼0

−2

3.1.1. Zeroth-order deformation problem The temperature θ(X) can be expressed by the set of base functions n o X k jk z 0

ð34Þ

3.1.2. mth-order deformation problems Here we first differentiate Eq. (31) m times with respect to p then divide by m! and setting p = 0 we obtain

3. The second example (non-linear convective–radiative conduction equation) The problem considered in reference [1] is: "  2 # 2 2 d θ d θ dθ 2 −2 θ4 ¼ 0; − N θ þ 1 θ 2 þ dX dX 2 dX

θm ð X Þpm ;

m¼1

one cannot get convergent results using HPM for values 1 N0.5. In this case one can also obtain an exact solution when 2 =0 of the form

The graphical results for the temperature θ for the exact, HPM and HAM solutions are presented in Fig. 3 for 1 = 0.2 when 2 = 0. In this case it is clear that exact and HAM solutions are overlapping, however the HPM solution is different from that of exact solution and this difference increases when we increase 1 (i.e. for strong non-linearity) and is shown in Fig. 3. In Fig. 4 the comparison between HPM and HAM solutions is presented when 2 is non-zero.

∞ X

ð25Þ

m−1 X k X l X k¼0 l¼0

ð39Þ

θm−1−k θk−l θl−j θj :

j¼0

The general solutions of Eqs. (19)–(21) can be written as θm ð X Þ ¼ θ★ m ð X Þ þ C1 X þ C2 ;

ð40Þ

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adjust the convergence regions where and whenever necessary. It is also pointed out that claim of HPM solutions as convergent ones for all non-linear problems is erroneous. References

Fig. 6. J-curves for different values of 2.

Fig. 7. Comparison between HPM and HAM solutions when 1 = 0.2 and 2 = 0.2.

where θ★ m (X) is the particular solution and the constants are determined by the boundary conditions (38). In the next subsection, the linear non-homogeneous Eqs. (19)–(21) are solved using Mathematica in the order m = 1,2,3,… 3.2. Analysis of the results The procedure is the same as considered in previous example. The J-curves are plotted for four different values of the parameter 1 and 2 in Figs. 5 and 6. One can easily observe that the range of admissible values of J decreases by increasing both 1 and 2. The comparison between HPM and HAM solutions for 1 = 0.2 and 2 = 0.2 is displayed in Fig. 7. 4. Concluding remarks In this paper the solutions of [1] are reproduced using homotopy analysis method. It is shown for example 1 of [1] that HPM results are divergent for strong non-linearity. The comparison between the exact, HPM and HAM solutions for a special case of example 1 is also given. The same conclusion has been drawn for the results of example 2 of [1]. Hence it is concluded that HPM results are divergent for strong non-linearity. However HAM provides us a simple way to control and

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