Cent. Eur. J. Eng. 4(4) 014 341-351 DOI: 10.478/s13531-013-0176-8 Central European Journal of Engineering Analytical accuracy of the one imensional heat transfer in geometry with logarithmic various surfaces Research Article A. Vahabzaeh 1, M. Fakour 1, D. D. Ganji, I. Rahimipetroui 1 1 Department of Mechanical Engineering, Sari Branch, Islamic Aza University, Sari, Iran Department of Mechanical Engineering, Babol University of science an technology, Babol, Iran Receive 7 April 014; accepte 15 July 014 Abstract: In this stuy, heat transfer an temperature istribution equations for logarithmic surface are investigate analytically an numerically. Employing the similarity variables, the governing ifferential equations have been reuce to orinary ones an solve via Homotopy perturbation metho (HPM), Variational iteration metho (VIM), Aomian ecomposition metho (ADM). The influence of the some physical parameters such as rate of effectiveness of temperature on non-imensional temperature profiles is consiere. Also the fourth-orer Runge- Kutta numerical metho (NUM) is use for the valiity of these analytical methos an excellent agreement are observe between the solutions obtaine from HPM, VIM, ADM an numerical results. Keywors: Heat transfer logarithmic various heat generations variational iteration metho (VIM) homotopy perturbation metho (HPM) Aomian ecomposition metho (ADM) Versita sp. z o.o. 1. Introuction In the heart of all the ifferent engineering sciences, everything explains itself in some kins of mathematical relations an most of these problems an phenomena are moele by linear an nonlinear equations. Therefore, many ifferent methos have recently introuce to solve these equations. Analytical methos have mae a comeback in research methoology after taking a backseat to the numerical techniques for the latter half of the preceing century. Most scientific problems an phenomena such as heat transfer occur nonlinearly. Except a limite number of these problems, it is ifficult E-mail: g_avoo@yahoo.com to fin the exact analytical solutions for them. Therefore, approximate analytical solutions are searche an were introuce, among which Aomian Decomposition Metho (ADM) [1 3], Variational Iteration Metho (VIM) [4 7], Homotopy Perturbation Metho (HPM) [8] are the most effective an convenient ones for both weakly an strongly nonlinear equations. Perturbation metho [9] provies the most versatile tools available in nonlinear analysis of engineering problem, but its limitations restrict its application [10, 11]. Perturbation metho [1] is base on assuming a small parameter. The majority of nonlinear problems, especially those having strong nonlinearity, have no small parameters at all. The approximate solutions obtaine by the perturbation methos, in most cases, are vali only for small values of the small parameter. Generally, the perturbation solutions are uniformly vali as long as a 341
Analytical accuracy of the one imensional heat transfer in geometry with logarithmic various surfaces scientific system parameter is small. However, we cannot rely fully on the approximations, because there is no criterion on which the small parameter shoul exist. Thus, it is essential to check the valiity of the approximations numerically an/or experimentally. To overcome these ifficulties, some new methos have been propose such as VIM, HPM, ADM an so on. Among these solution techniques, the Variational iteration metho an the Aomian ecomposition metho are the most transparent methos of solution of fractional ifferential an integral equations, because they provie immeiate an visible symbolic terms of analytic solutions, as well as numerical approximate solutions to both linear an nonlinear ifferential equations without linearization or iscretization. The HPM, propose first by He [13 15], for solving ifferential an integral equations, linear an nonlinear, has been the subject of extensive analytical an numerical stuies. The metho, which is a coupling of the traitional perturbation metho an homotopy in topology, eforms continuously to a simple problem which is easily solve. This metho has a significant avantage in that it provies an analytical approximate solution to a wie range of nonlinear problems in applie sciences. In this paper, the Homotopy perturbation metho, Variational iteration metho, an the Aomian ecomposition methos, as simple, accurate an computationally efficient analytical tools, are use to solve the nonlinear heat transfer equation. The accuracy of these methos is emonstrate by comparing its results with those generate by numerical metho.. Basic iea of Homotopy- Perturbation Metho, Variational Iteration Metho an Aomian Decomposition Metho.1. Homotopy-perturbation metho Consier the following equation: with the bounary conition of: A(u) f(r) = 0, r Ω (1) ( B u, u ) = 0, r Γ () n where L is a linear operator, while N is nonlinear operator, B is a bounary operator, Γ is the bounary of the omain Ω an f(r) is a known analytic function, where A(u) is efine as follows: A(u) = L(u) + N(u) (3) Homotopy-perturbation structure is shown as: or H(ν, p) = L(ν) L(u 0 ) + pl(u 0 ) + p[n(ν) f(r)] = 0 (4) H(ν, p) = (1 p)[l(ν) L(u 0 )] + p[a(ν) f(r)] = 0 (5) where ν(r, p)ω [0, 1] R (6) where r Ω an p [0, 1] is an impeing parameter, u 0 is an initial approximation which satisfies the bounary conitions. consiering Eqs. (4) an (5), we have H(ν, 0) = L(ν) L(u 0 ) = 0, H(ν, 1) = A(ν) f(r) = 0 (7) The changing process of p from zero to unity is just that of ν(r, p) from u 0 to u(r). In topology, this calle eformation, L(ν) L(u 0 ) an L(ν) + N(ν) f(r) are homotopic. The basic assumption is that the solution of Eqs. (4) or (5) can be expresse as a power series in p: ν = ν 0 + p ν 1 + p ν +... (8) an the best approximation is: u = lim p 1 ν = ν 0 + ν 1 + ν +... (9) The above convergence is iscusse in [16 18]... Variational iteration metho To clarify the basic ieas of VIM, we consier the following ifferential equation: L(u) + N(u) = g(t) (10) where L is a linear operator, N is a nonlinear operator an g(t) is a heterogeneous term. Accoring to VIM, we can write own a correction functional as follows: u n+1 = u n (t) + t 0 λ (Lu n (τ) + Fũ n (τ) g(τ)) τ (11) where λ is a general Lagrangian multiplier [19, 0] which can be ientifie optimally via the variational theory. The subscript n inicates the n-th approximation an ũ n consiere as a restricte variation [19, 0]. 34
A. Vahabzaeh, M. Fakour, D. D. Ganji, I. Rahimipetroui.3. Aomian Decomposition Metho To clarify the basic ieas of ADM, we consier the following ifferential equation: L(u) + R(u) + N(u) = g(t) (1) where R is the highest orer erivative which is assume to be easily invertible, L the linear ifferential operator of less orer than R, N presents the nonlinear terms an g(t) is a heterogeneous term. Applying the invers operator L 1 to the both sies of Eq. (1), an using the given conitions we obtain: u = f(x) L 1 (Ru) L 1 (Nu) (13) By equating terms in Eq. (17), the first few Aomian s polynomials A 0, A 1, A, A 3 an A 4 are given A 0 = F(u 0 ) (18) A 1 = u 1 F (u 0 ) (19) A = u F (u 0 ) + 1! u 1F (u 0 ) (0) A 3 = u 3 F (u 0 ) + u 1 u F (u 0 ) + 1 3! u3 1F (u 0 ). (1) Now that the A k are known, Eq. (15) can be substitute in Eq. (14) to specify the terms in the expansion for the solution of Eq. (16). where the function f(x) represents the terms arising from integration the source term g(x), using given conitions. For nonlinear ifferential equations, the nonlinear operator N(u) = F(u) is represente by an infinite series of the so-calle Aomian polynomials. F(u) = A m (14) m=0 The polynomials A m are generate for all kin of nonlinearity so that epens only on A 0, epens on u 0, A 1 an u 1 an so on. The Aomian metho efines the solution u(x) by the series: u = u m (15) m=0 In the case of F(u), the infinite series is a Taylor expansion about u 0. In other wors F(u) = F(u 0 ) + F (u 0 )(u u 0 ) + F (u 0 ) (u u 0) + F (u 0 ) (u u 0) 3 3! +...! (16) By rewriting Eq. (15) as u u 0 = u 1 + u + u 3 +..., substituting it into Eq. (16) an then equating two expressions for F(u) foun in Eq. (16) an Eq. (14) efines formulas for the Aomian polynomials: F(u) = A 0 + A 1 + A + A 3 +... = F(u 0 ) + F (u 0 )(u 1 + u +...) + F (u 0 ) (u 1 + u +...) +...! (17) 3. Applications 3.1. Heat transfer problem escription with a various logarithmic surfaces The one-imensional heat transfer in a logarithmic various surface A(x) an logarithmic various heat generation G(x) was stuie (Figure 1). It is also assume that the conuction coefficient, k, can be variable as a function of temperature. The energy equation an the bounary conitions for this geometry are as follows: where ( k(t )A(x) T ) + G(x) = 0 () x x {x = 0 T = T 0, x = L T = T L (3) {A(x) = A 0 e ax, G(x) = G 0 e ax (4) Assuming k as a linear function of temperature, we have: k(t ) = k 0 (1 + βt ) (5) Here, β shows the rate of effectiveness of temperature variation on thermal conuctivity coefficient an k 0 is the thermal conuctivity of the fin at the ambient. After simplification, we have: α x θ(x) + α βt 0 θ(x) x θ(x) + β T 0 x θ(x) + x θ(x) + β T 0 θ(x) x θ(x) + c e αx = 0 (6) 343
Analytical accuracy of the one imensional heat transfer in geometry with logarithmic various surfaces where θ = T T 0, c = G k 0 AβT 0 (7) With making the bounary conitions imensionless we have: x = 0 1, x = L θ = T L T 0 = z (8) An α, z, L are constants to be etermine through the initial conitions. 3.1.1. VIM applie to the problem In orer to solve Eq. (6) with bounary conitions (8) using VIM, we construct a correction functional, as follows: x { θ n+1 =θ n + λ α 0 τ θ n(τ) ( + βt 0 n(τ)) τ θ + τ θ n(τ) } +βt 0 θ n (τ) τ θ n(τ) + ce αx τ ( + αβt 0 θ n (τ) ) τ θ n(τ) (9) Its stationary conitions can be obtaine as follows: λ (x) + αλ (x) = 0, 1 λ (x) = 0, λ(x) = 0 (30) The Lagrangian multiplier can therefore be ientifie as: λ = eαx e ατ αe αx (31) As a result, we obtain the following iteration formula: x ) { θ n+1 = θ n + ( eαx e ατ α 0 αe ατ τ θ n(τ) + αβt 0 θ n (τ) x θ n(τ) + βt 0 τ θ n(τ) } + τ θ n(τ) + βt 0 θ n (τ) τ θ n(τ) + ce ατ τ (3) Now we start with an arbitrary initial approximation that satisfies the initial conition: θ 0 (x) = e αl z (z 1)e αx + (33) 1 + e αl 1 + e αl Using the above Variational formula (9), we have: x { ( θ 1 = θ 0 + λ α 0 τ θ 0(τ) ( ( + βt 0 0(τ)) τ θ + ( +βt 0 θ 0 (τ) τ θ 0(τ) ) τ θ 0(τ) ) } + ce αx τ + αβt 0 θ 0 (τ) τ θ 0(τ) ) (34) Substituting Eq. (33) in to Eq. (34) an after simplification, in the same manner, the rest of components were obtaine by using the Maple package. Accoring to the VIM, we can conclue (α = 40, T 0 = 10, c = an β = 0): θ(x) = 0.5 + 0.775e 4x + (0.0315( 4e 8x + 4e 4x + 16xe 4x ))e 8x (35) an so on. In the same manner the rest of the components of the iteration formula can be obtaine. 3.1.. HPM applie to the problem A homotopy perturbation metho can be constructe as follows: ( ) H(θ, p) = (1 p) x θ(x) + α x θ(x) ( + p α x θ(x) + αβt 0 θ(x) x θ(x) +βt 0 x θ(x) + x θ(x) ) +βt 0 θ(x) x θ(x) + ce αx (36) One can now try to obtain a solution of Eq. (36) in the form of: ν(θ) = ν 0 (x) + p p 1 (x) +... (37) where v i (x), i = 0, 1,,... are functions yet to be etermine. Accoring to Eq. (36) the initial approximation to satisfy initial conition is: ν 0 (x) = θ(x) = e αl z (z 1)e αx + (38) 1 + e αl 1 + e αl Substituting Eqs. (37) an (38) into Eq. (36) yiels: x θ 1(c)+ c e + βt 0(z 1) α (e αx ) +α αx ( 1 + e αl ) x θ 1(x) = 0 (39) The solution of Eq. (39) may be written as follow (α = 4, T 0 = 10, c =, z = 0.1, L = 5 an β = 0): θ 1 (x) = 0.93868 10 + 0.58511e 4x 1 + 1.58553e 4x 8 x 1.58553e 4x 4 x 1.0570e 4x 8 + 0.58511e 4x 4 0.58511e 4x 1 x + 0.58511e 4x x (40) In the same manner, the rest of components were obtaine by using the Maple package. Accoring to the HPM, we 344
A. Vahabzaeh, M. Fakour, D. D. Ganji, I. Rahimipetroui can conclue (α = 4, T 0 = 10, c =, z = 0.1, L = 5 an β = 0): θ(x) = lim p 1 ν(x) = ν 0 (x) + ν 1 (x) + ν (x) +... (41) Therefore, θ(x) = 0.64604e 4x 1 + 0.0738796.153488e 4x 1 x 3.303e 4x 8 x.153488e 4x 4 x + 1.909e 4x 8 0.53837e 4x 16 + 0.53837e 4x 16 x + 0.53837e 4x x (4) an so on. In the same manner the rest of the components of the iteration formula can be obtaine. 3.1.3. ADM applie to the problem In orer to apply ADM to nonlinear equation in fluis problem, we rewrite Eq. (6) in the following operator form: L xx x θ(x) = α x θ(x) αβt 0 θ(x) x θ(x) βt 0 x θ(x) βt 0 θ(x) x θ(x) where the notation: ce αx (43) L xx = x (44) is the linear operator. By using the inverse operator, we can write Eq. (43) in the following form: [ θ(x) = Lxx 1 α x θ(x) αβt 0 θ(x) x θ(x) ] βt 0 x θ(x) βt 0 θ(x) x θ(x) ce αx where the inverse operator is efine by: where L 1 xx = x x 0 θ(x) N 1 (θ) = αβt 0 θ(x) x ( θ(x) N (θ) = βt 0 x θ(x) N 3 (θ) = βt 0 θ(x) x 0 (45) x x (46) ) (47) The nonlinear operators N 1 (θ), N (θ), N 3 (θ) are efine by the following infinite series: N i (θ) = A i,n, i = 1,, 3 (48) n=0 where A i,n is calle Aomian polynomials an efine by [3]: [ [ n ]] A i,n = 1 n n! λ N n i λ i f i, n = 0, 1,,... (49) i=0 Hence we obtain the components series solution by the following recursive relation: [ θ n+1 (x) = Lxx 1 α x θ n(x) αβt 0 θ n (x) x θ n(x) ( ] βt 0 n(x)) x θ βt 0 θ n (x) x θ n(x) ce αx (50) where n 0. Aomian s polynomials formula, Eq. (49), is easy to set computer co to get as many polynomials as we nee in the calculation. We can give the first few Aomian s polynomials of the A i,n as: e αz α z (z 1)e αx βt 0 + (z 1)e αx 1 + e αl 1 + e A 1,0 = αl 1 + e αl (51) A,0 = βt 0(z 1) α (e αx ) (5) ( 1 + e αl ) ( e αz α z (z 1)e αx βt 0 + 1 + e αl 1 + e αl ) (z 1)e αx A 3,0 = 1 + e αl (53) an so on, the rest of the polynomials can be constructe in a similar manner. Using the recursive relation, Eq. (50) an Aomian s polynomials formula, Eq (49), with the initial conitions, Eq. (8), gives (α = 4, T 0 = 10, c =, z = 0.1, L = 5 an β = 0): θ(x) = 0.01 + 0.09e 4x θ 1 (x) = 0.5xe 4x + 0.9e 4x + 0.01 (54) In the same manner, the rest of components were obtaine by using the Maple package. Accoring to the ADM, we can conclue (α = 4, T 0 = 10, c =, z = 0.1, L = 5 an β = 0): θ(x) = 0.5xe 4x + 1.8e 4x + 0.0 (55) an so on. In the same manner the rest of the components of the iteration formula can be obtaine. 345
Analytical accuracy of the one imensional heat transfer in geometry with logarithmic various surfaces Table 1. Comparison between the Numerical results an HPM, VIM an ADM solution for θ(x) when β = 0, α =, T 0 = 10, c =, z = 0.1, L = 5. x VIM HPM ADM NUM Error of VIM Error of HPM Error of ADM 0.0 1.0000000000 1.00000000000 1.00000000000 1.0000000000 0.0000000000 0.0000000000 0.0000000000 0.5 0.556355696 0.5563556970 0.5563556990 0.556355355 0.0000000340 0.000000034 0.0000000345 1.0 0.156418876 0.1564188760 0.1564188780 0.156418895 0.0000000019 0.0000000019 0.00000000 1.5 0.104089934 0.10408993410 0.10408993430 0.1040899333 0.0000000090 0.0000000008 0.0000000011.0 0.100637370 0.1006373700 0.100637370 0.100637376 0.0000000060 0.0000000006 0.0000000009.5 0.100097608 0.1000976090 0.10009760310 0.100097609 0.0000000001 0.0000000000 0.0000000003 3.0 0.1000147391 0.10001473910 0.10001473930 0.1000147374 0.0000000017 0.0000000017 0.000000000 3.5 0.100001965 0.1000019660 0.1000019690 0.100001955 0.0000000010 0.0000000011 0.0000000014 4.0 0.1000003193 0.10000031940 0.10000031970 0.1000003189 0.0000000004 0.0000000005 0.0000000008 4.5 0.1000000410 0.10000004100 0.10000004130 0.1000000408 0.000000000 0.000000000 0.0000000005 5.0 0.1000000000 0.10000000000 0.10000000000 0.1000000000 0.00000000000 0.0000000000 0.0000000000 Figure 1. Geometry of the problem. 4. Results an iscussion The objective of the present stuy is to apply Homotopy perturbation metho, Variational iteration metho, an the Aomian ecomposition methos to obtain an explicit analytic solution of heat transfer equation of logarithmic surface profiles (Figure 1). For showing the efficiency of analytical applie metho a special case is consiere an results are compare with numerical metho as shown in Figure. The numerical solution is performe using the algebra package Maple 16.0, to solve the present case. The package uses a fourth orer Runge-Kutta proceure for solving nonlinear bounary value (B-V) problem [1]. Furthermore, Valiity of HPM, VIM an ADM are shown Figure. Comparison between results obtaine via numerical solution an HPM, VIM an ADM at β = 0, α = 0.4, T 0 = 10, c =, z = 0.1, L = 5. in Table 1. In this tables, the %Error is efine as: %Error = θ(x)num θ(x) Analytical (56) From the graphical representation, the results are prove to be precise an accurate in solving a wie range of mathematical an engineering problems especially Flui mechanic cases. This accuracy gives high confience to us about valiity of this problem an reveals an excellent agreement of engineering accuracy. 346
A. Vahabzaeh, M. Fakour, D. D. Ganji, I. Rahimipetroui Figure 3. Effect of α on θ when β = 0, T 0 = 10, c =, z = 0.1, L = 5. Moreover, Figures 3 to 6 are prepare in orer to see the effects of α an β on the temperature profiles. Figure 3 an Figure 4 are epicte for showing the effect of α on temperature profile respectively. As seen in these figures by increasing α, temperature profiles increases for the values of α in the range 0 < α < 1, whereas by increasing α number, temperature profiles ecrease for the values of α beyon 1.0. In aition, the imensionless temperature istributions along the fin surface with β varying from -50 to 50 are epicte in Figure 5 an Figure 6, respectively. In the case of β > 0, the temperature istributions increases as x increases, reaches a peak value in the neighborhoo of x = 0. an then ecays to zero aroun x = 5.0. For the case of β < 0, 347
Analytical accuracy of the one imensional heat transfer in geometry with logarithmic various surfaces Figure 4. Effect of α on θ when β = 0, T 0 = 10, c =, z = 0.1, L = 5. which is illustrate in Figure 6, the temperature becomes negative an ecreases as x increases. The trough in raial velocity is attaine in the neighborhoo of x = 0.. 5. Conclusion In this paper, the basic iea of the Variational iteration metho, Homotopy-perturbation metho an Aomian ecomposition methos are introuce an then have been successfully applie to governing ifferential equation of geometry with a logarithmic various surface. The 348
A. Vahabzaeh, M. Fakour, D. D. Ganji, I. Rahimipetroui Figure 5. Effect of β on θ when α = 4, T 0 = 10, c =, z = 0.1, L = 5. results obtaine here were compare with the numerical solutions. The results show that these methos enable to convert a ifficult problem into a simple problem which can easily be solve. Important objective of our research is the examination of the convergence of HPM, VIM an ADM. The comparisons of the results obtaine here provie more realistic solutions, reinforcing the conclusions about the efficiency of these methos. Therefore the HPM, VIM, ADM are powerful mathematical tools an can be applie to a large class of linear an nonlinear problems arising in heat transfer equations. 349
Analytical accuracy of the one imensional heat transfer in geometry with logarithmic various surfaces Figure 6. Effect of β on θ when α = 4, T 0 = 10, c =, z = 0.1, L = 5. References [1] Sheikholeslami M., Ganji D. D., Ashoryneja H. R., Investigation of squeezing unsteay nanoflui flow using ADM, Power Technology., 39, 013, 59-65 [] Ganji D. D., Jannatabai M., Mohseni E., Application of He s Variational iteration metho to nonlinear Jaulent-Mioek equations an comparing it with ADM, Journal of Computational an Applie Mathematics., 07, 007, 35-45, 007 [3] Saighi A., Ganji D. D., Analytic treatment of linear an nonlinear Schröinger equations: A stuy with homotopy-perturbation an Aomian ecomposition 350
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