We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors

Transcription

1 We are IntechOpen, the world leading publiher of Open Acce book Built by cientit, for cientit 3,5 8,.7 M Open acce book available International author and editor Download Our author are among the 5 Countrie delivered to TOP % mot cited cientit.% Contributor from top 5 univeritie Selection of our book indexed in the Book Citation Index in Web of Science Core Collection (BKCI) Intereted in publihing with u? Contact book.department@intechopen.com Number diplayed above are baed on latet data collected. For more information viit

2 Chapter 3 Homotopy Perturbation Method to Solve Heat Conduction Equation Anwar Ja'afar Mohamed Jawad Additional information i available at the end of the chapter Introduction Fin are extenively ued to enhance the heat tranfer between a olid urface and it convective, radiative, or convective radiative urface. Finned urface are widely ued, for intance, for cooling electric tranformer, the cylinder of air-craft engine, and other heat tranfer equipment. In many application variou heat tranfer mode, uch a convection, nucleate boiling, tranition boiling, and film boiling, the heat tranfer coefficient i no longer uniform. A fin with an inulated end ha been tudied by many invetigator [Sen, S. Trinh(986)]; and [Unal (987)]. Mot of them are immered in the invetigation of ingle boiling mode on an extended urface. Under thee circumtance very recently, [Chang (5)] applied tandard Adomian decompoition method for all poible type of heat tranfer mode to invetigate a traight fin governed by a power-law-type temperature dependent heat tranfer coefficient uing 3 term. [Liu (995)] found that Adomian method could not alway atify all it boundarie condition leading to boundarie error. The governing equation for the temperature ditribution along the urface are nonlinear. In conequence, exact analytic olution of uch nonlinear problem are not available in general and cientit ue ome approximation technique to approximate the olution of nonlinear equation a a erie olution uch a perturbation method; ee [Van Dyke M. (975)], and Nayfeh A.H. (973)], and homotopy perturbation method; ee [He J. H. (999, (), and(3)]. In thi chapter, we applied HPM to olve the linear and nonlinear equation of heat tranfer by conduction in one-dimenional in two lab of different material and thickne L.. The perturbation method Many phyic and engineering problem can be modelled by differential equation. However, it i difficult to obtain cloed-form olution for them, epecially for nonlinear Jawad, licenee InTech. Thi i an open acce chapter ditributed under the term of the Creative Common Attribution Licene ( which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited.

3 6 Heat Tranfer Phenomena and Application one. In mot cae, only approximate olution (either analytical one or numerical one) can be expected. Perturbation method i one of the well-known method for olving nonlinear problem analytically. In general, the perturbation method i valid only for weakly nonlinear problem[nayfeh ()]. For example, conider the following heat tranfer problem governed by the nonlinear ordinary differential equation, ee [Abbabandy (6)]:, ( + εu) u + u=, u() = () where > i a phyical parameter, the prime denote differentiation with repect to the time t. Although the cloed-form olution of u(t) i unknown, it i eay to get the exact reult u, () = / ( + ε ), a mentioned by [Abbabandy (6)]. Regard that ε a a perturbation quantity, one can write u(t) into uch a perturbation erie 3 ε ε ε 3 u( t) = u ( t) + u ( t) + u ( t) + u ( t) +... () Subtituting the above expreion into () and equating the coefficient of the like power of ε, to get the following linear differential equation, u + u =, u () =, (3),, u + u = u u, u () =, (4),,, u + u = ( u u + u u ), u () =, (5),,,, Solving the above equation one by one, one ha Thu, we obtain ut () a a perturbation erie which give at t = the derivative u + u = ( u u + u u + u u ), u () =, (6) () t u t = e () t t = u t e e t t 3 3t u() t = e e + e (7) 3 3 ut ( ) = e t + ε( e t e t ) + ε ( e t e t + e t ) +... (8), u () = + ε ε + ε ε + ε ε + ε ε + ε ε +... (9)

4 Homotopy Perturbation Method to Solve Heat Conduction Equation 63 Obviouly, the above erie i divergent for ε, a hown in Fig.. Thi typical example illutrate that perturbation approximation are valid only for weakly nonlinear problem in general. In view of the work by [Abbabandy (6)], the HAM extend a erie approximation beyond it initial radiu of convergence. Figure. Comparion of the exact and approximate olution of (). Solid line: exact olution u, () = / ( + ε ) ; Dahed-line: 3th-order perturbation approximation; Hollow ymbol: 5th-order approximation given by the HPM; Filled ymbol: 5th-order approximation given by the HAM when h = /(+ ε ). To overcome the retriction of perturbation technique, ome non-perturbation technique are propoed, uch a the Lyapunov' artificial mall parameter method [Lyapunov A.M. (99)], the -expanion method [Karmihin et al(99)], the homotopy perturbation method [He H., J.,(998)], and the variational iteration method (VIM), [He H., J.,(999)],. Uing thee non-perturbation method, one can indeed obtain approximation even if there are no mall/large phyical parameter. However, the convergence of olution erie i not guaranteed. For example, by mean of the HPM, we obtain the ame and exact approximation of Eq.(), a the perturbation reult in Eq.(9), that i divergent for >, a hown in Fig.. ; For detail, ee [Abbabandy (6)]. Thi example how the importance of the convergence of olution erie for all poible phyical parameter. From phyical point of view, the convergence of olution erie i much more important than whether or not the ued analytic method itelf i independent of mall/large phyical parameter. If one doe not keep thi in mind, ome uele reult might be obtained. For example, let u conider the following linear differential equation [Ganji et al (7)]: u + u = u, x R, t > t x xxt () ux (,) = e x ()

5 64 Heat Tranfer Phenomena and Application It exact olution read x t u ( x, t) = e exact () By mean of the homotopy perturbation method, [Ganji et al (7)] wrote the original equation in the following form: ϕ(, : ) (, : ) (, : ) (, : ) ( ) xt p ϕ [ xt p ϕ p p xt p ϕ + + xt p ] = t t x x t 3 (3) ubject to the initial condition ϕ( x,: p) = e x (4) where p [;] i an embedding parameter. Then, regarding p a a mall parameter, [Ganji et al (7)] expanded ϕ ( xt, : p) in a power erie ( xt, : p) u( xt, ) u ( xt, ). p m ϕ = + (5) which give the olution. For p =, and ubtitute (5) into the original equation (3) and initial condition in (4), then equating the coefficient of the like power of p, one can get governing equation and the initial condition for um( x, t ). In thi way, [Ganji et al (7)] obtained the mth-order approximation and the 5th-order approximation read + m= k= m m uxt (, ) u( xt, ) + u( xt, ) (6) k x e uhpm( x, t) [ t + 66t + 47t + 33t t t+ 7] (7) 7 However, for any given x, the above approximation enlarge monotonouly to the poitive infinity a the time t increae, a hown in Fig.. Unfortunately, the exact olution monotonouly decreae to zero! Let δ t u u = (8) u () exact HAM where δ () t denote the relative error of the HPM approximation (7). A hown in Fig., the relative error δ(t) monotonouly increae very quickly: In fact, it i eay to find that the HPM erie olution (6) i divergent for all x and t except t = which however correpond to the given initial condition in (). In other word, the convergence radiu of the HPM olution erie (7) i zero. It hould be emphaized that, the variational iteration method (VIM) obtained exactly the ame reult a (7) by the 6th exact

6 Homotopy Perturbation Method to Solve Heat Conduction Equation 65 iteration ee; [He H. J., (999)], and [Ganji et al (7)]. Thi example illutrate that both of the HPM and the VIM might give divergent approximation. Thu, it i very important to enure the convergence of olution erie obtained. Figure. Approximation of () given by the homotopy perturbation method. Dahed-line: exact olution(); Solid line: the 5th-order HPM approximation(7); Dah-dotted line: the relative error (t) defined by(8). 3. Outline of Homotopy Perturbation Method (HPM) The homotopy analyi method (HAM) ha been propoed by Liao in hi PhD diertation in [Liao (99)]. Liao introduced the o-called auxiliary parameter in [Liao (997a)] to contruct the following two-parameter family of equation: ( p) L( u u ) = hpn( u) (9) where u i an initial gue. [Liao (997a)] pointed out that the convergence of the olution erie given by the HAM i determined by h, and thu one can alway get a convergent erie olution by mean of chooing a proper value of h. Uing the definition of Taylor erie with repect to the embedding parameter p (which i a power erie of p ), [Liao (997b)] gave general equation for high-order approximation. [He J. H. (999)] followed Dr. Liao early idea of Homotopy Perturbation Method (HPM) when he contructed the one-parameter family of equation: ( plu ) ( ) + pnu ( ) = () where Eq.() repreented pecial cae of Eq.(9) for convergent olution of (HAM) at h =. To illutrate the baic idea of thi method, conider the following general nonlinear differential equation [ee Ghaemi et al ()].

7 66 Heat Tranfer Phenomena and Application Au ( ) f( r) =, r Ω () With boundary condition u Bu (, ) =, r Γ n () where A i a general differential operator, B i a boundary operator, f(r) i a known analytic function, and i the boundary of the domain. The operator A can be generally divided into linear and nonlinear part, ay L and N. Therefore () can be written a Lu () + Nu () f() r = (3) [He (999)] contructed a homotopy vrp (, ): Ω [,] Rwhich atifie: H(v, p) = ( - p)[l(v) - L(v )] + p[a(v) - f(r)] = (4) where r Ω, p [,] that i called homotopy parameter, and v i an initial approximation of (9). Hence, it i obviou that: and H(v, ) = L(v) - L(v ) = (5) H(v, ) = [ A( v) - f(r) ] = (6) In topology, L(v) - L(v ) i called deformation, and [ A( v) - f(r) ] i called homotopic. The embedding parameter p monotonically increae from zero to unit a the trivial problem H(v, ) = in (5) i continuouly deform the original problem in (6), H(v, ) =. The embedding parameter p [,] can be conidered a an expanding parameter. [Nayfeh A.H. (985)] Apply the perturbation technique due to the fact that p, can be conidered a a mall parameter, the olution of () or (3) can be aumed a a erie in p, a follow: when p, the approximate olution, i.e., 3 v= v + pv+ p v + p v (7) u= lim v= v + v + v + v +... (8) p 3 The erie (8) i convergent for mot cae, and the rate of convergence depend on A(v), [He, L. (999)].

8 Homotopy Perturbation Method to Solve Heat Conduction Equation Application of Homotopy Perturbation Method HPM An analytic method for trongly nonlinear problem, namely the homotopy analyi method (HAM) wa propoed by Liao in 99, ix year earlier than the homotopy perturbation method by [He H., J.,(998)], and the variational iteration method by [He H., J.,(999)]. Different from perturbation technique, the HAM i valid if a nonlinear problem contain mall/large phyical parameter. More importantly, unlike all other analytic technique, the HAM provide u with a imple way to adjut and control the convergence radiu of olution erie. Thu, one can alway get accurate approximation by mean of the HAM. In the next ection, HPM i applied to olve the linear and nonlinear equation of heat tranfer by conduction in one-dimenional in a lab of thickne (L). [Anwar ()] olved the linear and non-linear heat tranfer equation by mean of HPM. 4.. Non-Linear Heat tranfer equation Conider the heat tranfer equation by conduction in one-dimenional in a lab of thickne L. The governing equation decribing the temperature ditribution i: d dt ( k ), x [, L] dx dx = (9) Where the two face are maintained at uniform temperature T and T with T > Tthe lab make of a material with temperature dependent thermal conductivity k = k( T) ; ee [Rajabi A.(7)]. The thermal conductivity k i aumed to vary linearly with temperature, that i: T T k = k + T = T T L = T [ ε ] (), ( ) T T (3) where ε i a contant and k i the thermal conductivity at temperature T. Introducing the dimenionle quantitie T T θ = T T, k k ε = k x X = X [,] L where k i the thermal conductivity at temperature T, then (9) reduce to dθ ( ) d θ + ε dx =, dx ( + εθ ) X [,] θ() =, θ() = (3) The problem i formulated by uing (9) a: ( pl ) [ θ( Xp, ) θ ( X)] + pn[ θ( Xp, )] = (3)

9 68 Heat Tranfer Phenomena and Application Where the Linear operator: L( θ),, = θ (33) and, θ XX = (34) from Eq.(3), the initial gue i: θ ( X) = X (35) and the linear operator: and the nonlinear operator of θ( Xp, ) i: LCX [ + C ] = (36) N[ θ( X, p)] = θ( X, p) + ε[ θ( X, p). θ( X, p) + ( θ( X, p) ) ] = XX XX X (37) and: θ(, p) =, θ(, p) = (38) where p [,] i an embedding parameter. For p = and, we have θ( X,) = θ ( X) θ( X,) = θ( X) (39) θ ( X) tend to θ ( X ) a p varie from to. Due to Taylor erie expanion: θ( Xp, ) θ( Xp, ) = θo( X) + (4)! p = and the convergence of erie (4) i convergent at p = obtain = p=. Then by uing (35) and (36) one θ ( X) = θ ( X) + θ ( X) (4) For the -th- order problem, if we firt differentiate Eq.(3) time with repect to p then divide by! and etting p = we obtain:,,,,,, n n n n n= L [ θ ( X ) uθ ( X )] + [ θ ( X ) + ε ( θ. θ + θ θ )] = (4)

10 Homotopy Perturbation Method to Solve Heat Conduction Equation 69 Where: θ () = θ () = (43) u, =, > (44) The general olution of (4) can be written a: where θ * ( X) i the particular olution. * θ ( X ) = θ ( X ) + θ ( X ) (45) The linear non-homogeneou (4) i olved for the order =,, 3,..., for =, (4) become: Then d θ dθ dθ d θ ε θ θ θ dx + ( + ) = dx dx dx () =, () = (46) d θ + ε = (47) dx the olution of (47) give : ε θ = ( ) X X (48) For =, Eq.(4) become: Solution of (49) give : θ θ dθ d θ θ ε θ θ θ θ d ( d d ) = dx dx dx dx dx, () =, () = (49) Then, olution of (3) i: ε 3 θ = [ X X X] (5) ε 3 ε θ ( X) = ( X) + ( X X ) + [ X X X] (5) Reult of θ obtained for different value of ε are preented in Table and Fig. 3. Clearly, for mall value for ε then (5) i a good approximation to the olution. That mean for ε =, then k = k, for ε =, then k = k. However, a ε increae, (5) produce

11 7 Heat Tranfer Phenomena and Application inaccurate divergent reult. The reult obtained via HPM are compared to thoe via General Approximation Method GAM obtained by Khan R. A.(9). For thi problem, it i found that HPM produce agreed reult compared to GAM. X ε =.5 ε =.8 ε =. ε =.5 ε = Table. Solution θ via X for different value of ε. Special computer program wa ued a pecial cae, the temperature ditribution along a road of length (L = m) when T = C and T = 5 C, are preented in Table and Fig.4. X ε =.5 ε =.8 ε =. ε =.5 ε = Table. Solution T via X for different value of ε 4.. Linear Heat tranfer equation In thi ection we conider the linear one-dimenional equation of heat tranfer by conduction (diffuion equation) [Anderon (984)]: for initial condition T T α = x, t > t x (5)

12 Homotopy Perturbation Method to Solve Heat Conduction Equation thita ep x Figure 3. Graphical reult θ via X obtained by HPM for different value of ε. 9 8 T X ep Figure 4. Graphical reult of Temperature via X obtained by HPM for different value of ε. and boundary condition Tx (,) = gx ( ) = in( π. x) (53) T(, t) = T(, t) = (54) α i thermal conductivity that i aumed contant with temperature. To olve the parabolic partial differential equation (5) uing HPM, we conider a correction functional equation a: T u T T ( p)[ ] + p[ α ] = t t t x (55) Then:

13 7 Heat Tranfer Phenomena and Application T u u T p αp = t t t x (56) T T αp = t x (57) α p ( T + pt + p T + p T +...) ( T pt p T p T..) = t x (58) For zeroth order of p: T = t (59) Then T ( x, t) = in( π. x) For firt order of p: T T α = t x (6) T 4 in(. x ) t + πα π = (6) T ( x, t ) = in( π. x ) 4 π α in( π. x ) t (6) For econd order of p: T t T α = x (63) 4 T ( xt, ) = in( π. x) 4 π α in( π. xt ) + 8 π α in( π. xt ) (64) Uing equation (56) for other order of p, we can obtain the following reult: Txt (, ) = in( π. x)[ (4 π α. t) + (4 π α. t)...] (65) It i obviou that Txt (, ) converge to the exact olution a increaing order of p: Txt (, ) = in( π. x).exp( 4 π α. t) (66) Fig.5 and Fig.6 repreent the HPM olution T(x, t) for α =.5, and α =. repectively for x, t.4.

14 Homotopy Perturbation Method to Solve Heat Conduction Equation 73 X t= t =. t =. t =.3 t = E E E E E- Table 3. Solution T(x, t) for x, t.4 at α = T X 5 t 4 6 Figure 5. Solution T(x, t) for x, t.4 and α =.5.

15 74 Heat Tranfer Phenomena and Application X t= t =. t =. t =.3 t = Table 4. Solution T(x, t) for x, t.4 at α =...5 T t 5 5 X 5 Figure 6. Solution T(x, t) for x, t.4 and α =.

16 Homotopy Perturbation Method to Solve Heat Conduction Equation Dicuion For example., clearly, for ε, (5) i a good approximation to the olution. That mean for ε =, then k = k, and for ε =, then k = k. However, a ε increae, (5) produce inaccurate divergent reult. For example, (66) i a good approximation to the olution a α value decreaed. That mean a α increae, (66) produce inaccurate divergent reult. 6. Concluion Homotopy Perturbation Method HPM i applied to olve the linear and nonlinear partial differential equation. Two numerical imulation are preented to illutrate and confirm the theoretical reult. The two problem are about heat tranfer by conduction in two lab. Reult obtained by the homotopy perturbation method are preented in table and figure. Reult are compared with thoe tudied by the generalized approximation method by [Sajida et al (8)]. Homotopy Perturbation Method i conidered a effective method in olving partial differential equation. Author detail Anwar Ja'afar Mohamed Jawad Al-Rafidain Univerity College, Baghdad, Iraq 7. Reference Abbabandy S. (6), The application of homotopy analyi method to nonlinear equation ariing in heat tranfer. Phy. Lett. A 36, 9-3. Anderon D. A. (984), J. C. Tannehill, R. H. Pletcher, Computational Fluid Mechanic and Heat Tranfer, McGraw-Hill company. Anwar J. M. Jawad,()"Application of Homotopy Perturbation Method in Heat Conduction Equation", Al-ba'ath Univerity journal, Syria vol. 3. Chang, M.H., (5). A decompoition olution for fin with temperature dependent urface heat flux, Int.J. Heat Ma Tranfer 48: Ganji D. D., H. Tari, M.B. Jooybari (7), Variational iteration method and homotopy perturbation method for nonlinear evolution equation. Comput. Math. Appl. 54, 8-7. Ghaemi, M., Tavaoli Kajani M.(), Application of He homotopy perturbation method to olve a diffuion-convection problem, Mathematical Science Vol. 4, No He H., J.,(998), Approximate analytical olution for eepage flow with fractional derivative in porou media. Comput. Meth. Appl. Mech. Eng. 67, He H. J., (999),Variational iteration method: a kind of nonlinear analytical technique: ome example. Int. J. Non-Linear Mech. 34,

17 76 Heat Tranfer Phenomena and Application He J. H. (3), Homotopy perturbation method a new nonlinear analytical technique, Appl. Math. Comput. 35, He J. H. (), A coupling method of a homotopy technique and a perturbation technique for non-linear problem, Int. J. Nonlinear Mech., 35, (37). He J. H.(999), Homotopy perturbation technique, J. Compt. Method Appl. Mech. Eng., 78, Karmihin A. V., A.I. Zhukov, V.G. Koloov(99), Method of Dynamic Calculation and Teting for Thin-Walled Structure. Mahinotroyenie, Mocow. Khan R. A.(9), The generalized approximation method and nonlinear Heat tranfer equation, Electronic Journal of Qualitative Theory of Differential Equation, No., - 5; Lyapunov A. M. (99), General Problem on Stability of Motion. Taylor & Franci,London, (Englih tranlation). Liao S. J. (99), On the propoed homotopy analyi technique for nonlinear problem and it application, Ph.D. diertation, Shanghai Jio Tong Univerity, (99). Liao S. J. (997a), Numerically olving nonlinear problem by the homotopy analyi method, Compt. Mech., Liao S. J. (997b), A kind of approximate olution technique which doe not depend upon mall parameter (II): an application in fluid mechanic, Int. J. Non-linear Mech. 3, Liu, G.L., (995). Weighted reidual decompoition method in nonlinear applied mathematic, in: Proceeding of the 6th congre of modern mathematical and mechanic, Suzhou, China: Nayfeh A.H. (973), Perturbation Method, Wiley, New York. Nayfeh A.H. (985), Problem in Perturbation, John Wiley, New York. Nayfeh A.H. (), Perturbation Method. Wiley, New York. Rajabi A.(7), D. D. Gunji and H. Taherian, Application of homotopy perturbation method in nonlinear heat conduction and convection equation, Phyic Letter A, 36, Sajida M., T. Hayatb (8), Comparion of HAM and HPM method in nonlinear heat conduction and convection equation, Nonlinear Analyi, Real World Application, Sen, A.K. S. Trinh, (986). An exact olution for the rate of heat tranfer from a rectangular fin governed by a power law type temperature dependence, Int. J. Heat Ma Tranfer, 8: Songxin Liang, David J. Jeferey, (9),Comparion of homotopy analyi method and homotopy perturbation method through an evolution equation, Communication in Nonlinear Science and Numerical Simulation, Volume 4, Iue, p Unal, H.C., (987). A imple method of dimenioning traight fin for nucleate pool boiling, Int. J. Heat Ma Tranfer, 9: Van Dyke M. (975), Perturbation Method in Fluid Mechanic, Annotated Edition, Parabolic Pre, Stand ford, CA.