THE STUDY OF HEAT TRANSFER PHENOMENA BY USING MODIFIED HOMOTOPY PERTURBATION METHOD COUPLED BY LAPLACE TRANSFORM

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1 THE STUDY OF HEAT TRANSFER PHENOMENA BY USING MODIFIED HOMOTOPY PERTURBATION METHOD COUPLED BY LAPLACE TRANSFORM Uriel FILOBELLO-NINO, Hector VAZQUEZ-LEAL *, Agusti Leobardo HERRERA-MAY,,3, Roberto Carlos AMBROSIO-LAZARO 3, Victor Mauel JIMENEZ-FERNANDEZ, Mario Alberto SANDOVAL-HERNANDEZ 4, Oscar ALVAREZ-GASCA, Beatriz Elea PALMA-GRAYEB. * Facultad de Istrumetació Electróica, Uiversidad Veracruzaa, Circuito Gozalo Aguirre Beltrá S/N, Xalapa, Veracruz, 9, México Micro ad Naotechology Research Ceter, Uiversidad Veracruzaa, Calzada Ruiz Cortíes 455, 9494 Boca del Río, Veracruz, México. 3 Maestría e Igeiería Aplicada, Facultad de Igeiería de la Costrucció y el Hábitat, Uiversidad Veracruzaa, Calzada Ruíz Corties 455, 9494, Boca del Río, Veracruz, Mexico. 4 Natioal Istitute of Astrophysics, Optics ad Electroics, Luis Erique Erro #, Sata María Toatzitla, 784, Puebla, México. * Correspodig author; hvazquez@uv.mx. Itroductio I this paper, we preset Modified Homotopy Perturbatio Method Coupled by Laplace Trasform (MHPMLT) to solve oliear problems. As case study MHPMLT is employed i order to obtai a approximate solutio for the oliear differetial equatio that describes the steady state of a heat oe-dimesioal flow. The compariso betwee approximate ad exact solutios shows the practical potetiality of the method. Key words: Homotopy Perturbatio Method; Laplace Trasform; Heat coductio The heat trasfer laws are of paramout importace i the desig ad operatio of equipmet i may idustrial applicatios as well as i pure scieces. Therefore, it is importat to search for accurate aalytical approximate solutios for the equatios describig these pheomea. However, it is well kow that oliear differetial equatios that describe them are difficult to solve [-3]. Laplace Trasform (L.T.) plays a relevat role i mathematics because its methods allow to solve may problems i sciece ad egieerig [4]. It is well kow that Laplace Trasform is a powerful tool useful for solvig liear ordiary differetial equatios with costat coefficiets ad iitial coditios ad to solve some cases of differetial equatios with variable coefficiets ad partial differetial equatios [4]. The cotributio of L.T. to oliear ordiary differetial equatios has required its combiatio with other techiques. Thus, [] reported a combiatio of Homotopy Perturbatio (HPM) ad L.T. methods (LT-HPM) i order to obtai highly accurate approximate solutios for these equatios. At this time, it is clafiried that the couplig of LT ad HPM is kow by aother ame i the literature: A modified Homotopy Perturbatio Method coupled by Laplace Trasform (MHPMLT) [5,6,7]. As a matter of fact, [5] amed this modificatio as He-Laplace

2 Method for simplicity. Thus, from here o, we agree to call the proposed method MHPMLT. This work proposes MHPMLT method i the search for approximate solutios for the oliear ordiary differetial equatio with Dirichlet boudary coditios defied o a fiite iterval [8] which describes the steady state oe-dimesioal heat coductio i a slab with temperature-depedet thermal coductivity [9]. The case of equatios with boudary coditios o ifiite itervals has bee reported for some authors [,] ad correspod ofte to problems defied o semi-ifiite rages. However, the methods for solvig these problems are differet from those that will be preseted i this study [8]. Noliear problems frequetly arise i sciece ad egieerig, whereby, it is very importat to search o differetial equatios that describe them. I recet years, there have bee proposed several methods focused to fid approximate solutios to oliear differetial equatios; such as those based o: variatioal approaches [], tah method [3], exp-fuctio [4], Adomia s decompositio method (ADM) [5], parameter expasio [6], homotopy aalysis method (HAM) [,3], perturbatio method [7], ad homotopy perturbatio method (HPM) [,5-,8-3], sice the mai solutio process of this article is HPM, ext we briefly metio some of the last developmets of this method; such as the couplig of HPM ad Frobeius method [], multiple scales HPM method [], Parametrized HPM [], Noliearities Distributio Homotopy Perturbatio Method used to fid the solutio of Troesch Problem [3]; amog may others. The paper is orgaized as follows. I Sectio, we itroduce the basic idea of stadard HPM method. For Sectio 3 we itroduce the ecessary aspects of MHPMLT. Additioally i Sectio 4 the basic equatios for the heat trasfer problem metioed are derived. Sectio 5 presets the applicatio of the proposed method, i the search for a approximate solutio for the oliear ordiary differetial equatio, which describes the steady state oe-dimesioal coductio of heat i a slab with temperature-depedet thermal coductivity. Besides, a discussio o the results is preseted i Sectio 6. Fially, a brief coclusio is give i Sectio 7.. Stadard HPM The stadard homotopy perturbatio method (HPM) was proposed by Ji Hua He, it was itroduced like a powerful tool to approach various kids of oliear problems. The Homotopy Perturbatio Method (HPM) is cosidered as a combiatio of the classical perturbatio techique ad the homotopy (whose origi is i the topology), but ot restricted to small parameters as occur with traditioal perturbatio methods. For example, HPM method requires either small parameter or liearizatio, but oly few iteratios to obtai highly accurate solutios [9,4]. To figure out how HPM works, cosider a geeral oliear differetial equatio i the form A( u) f ( r) =, r, () with the followig boudary coditios B( u, u / ) =, r, () where A is a geeral differetial operator, B is a boudary operator, () f r a kow aalytical fuctio ad is the domai boudary for. A ca be divided ito two operators L ad N, where L is liear ad N oliear; so that () ca be rewritte as

3 3 L( u) + N( u) f ( r) =. (3) Geerally, a homotopy ca be costructed as [9,4] H( U, p) = ( p)[ L( U) L( u)] + p[ L( U) + N( U) f ( r)] =, p [,], r. or (4) H( U, p) = L( U) L( u ) + p[ L( u ) + N( U) f ( r)] =, p [,], (5) where p is a homotopy parameter, whose values are withi rage of ad, ad u is the first approximatio for the solutio of (3) that satisfies the boudary coditios. Assumig that solutio for (4) or (5) ca be writte as a power series of p as U = v + v p + v p + (6)... Substitutig (6) ito (5) ad equatig idetical powers of p terms there ca be foud values for the sequece,,, Whe p yields to the approximate solutio for () i the form U = v + v + v + v... 3 (7) 3. Basic Idea of Modified Homotopy Perturbatio Method Coupled by Laplace Trasform The objective of this sectio is showig how MHPMLT ca be employed to fid aalytical approximate solutios for ordiary differetial equatios (ODES) as (3). For this purpose, MHPMLT follows the same steps of stadard HPM util (5), ext we apply Laplace trasform o both sides of homotopy equatio (5), to obtai L( U) L( u ) + p L( u ) + N( U) f ( r) = (8) usig the differetial property of L.T, we have [4] ( ) s U s U () s U()... U () = or L( u ) pl( u ) + p N( U ) + f ( r) ( ) ( U ) = () ().. () s U + s U + + U + s L( u) pl( u) p N( U ) f ( r) + + s applyig iverse Laplace trasform to both sides of (), we obtai (9) ()

4 4 U s U () + s U() U () + ( ) s = L( u) pl( u) p N( U ) f ( r) + + s Assumig that solutios of (3) ca be expressed as a power series of p = p v () U = () the substitutig () ito (), we get ( ) s U () s U ().. U () s p =, (3) = + L( u) pl( u) + p N( p ) + f ( r) = comparig coefficiets of p, with the same power leads to ( ) p : = ( s U () s U().. U ()) L( u) ) s, p : = ( N( ) L( u ) f ( r) + ) s, p : = N(, ) s, (4) 3 p : 3 = N(,, ) s, j p : j = N(,,,..., j) s, Assumig that the iitial approximatio has the form: U() = u =, U () =,.., U () = the the approximate solutio may be obtaied as follows ; u U = lim = p (5) 4. Goverig equatios. The goal of this work is the searchig for a approximate solutio for the oliear problem, which describes the steady state oe-dimesioal coductio of heat i a slab with thermal coductivity liearly depedet o the temperature (see Figure ). The trasferred eergy caused by the temperature differece betwee two adjacet parts of a body is called heat coductio [5]. Let u( x, y, z, t ) be the temperature of the above metioed slab x, y, z at timet, ad K, ad the thermal coductivity, specific heat, ad desity of at a poit ( ) the solid respectively; the it is verified that the temperature obeys the followig partial differetial equatio [6].

5 5 u u u u K + K + K = (6) x x y y z z t which is kow as the heat coductio equatio. I the cases where K is a costat, the above equatio reduces to the followig kow liear partial differetial equatio. K u u u u + + = x y z t (7) Figure shows a oe-dimesioal coductio of heat through a isulated slab. A substace with a high thermal coductivity is a good heat coductor; o the cotrary, oe with a small thermal coductivity is a poor coductor of heat, or equivaletly a good thermal isulator. The K value depeds o the temperature, icreasig slightly whe it icreases, but ca be cosidered almost costat throughout a substace if the temperature differece betwee the parts is ot too large [5]. I the case of heat flow uder steady coditios, the temperature does ot deped o time t, such that u/ t=. Thus (6) becomes i u u u K + K + K = x x y y z z while (7) becomes i the Laplace equatio for u (8) u = (9) Solutios to the liear heat coductio equatios for costat thermal coductivity (7) ad (9) are studied i detailed, for istace i [5,6]. Noetheless ulike the above, i geeral K is depedet o temperature, ad i this case (6) ad (8), are oliear. This article cosiders the case of steady coditios for the oe dimesioal coductio of heat i a slab of thickess L, assumig a temperature depedet thermal coductivity K [9]. Supposig that the temperatures of the two opposite faces of the slab are uiformly maitaied at T adt, wheret T ; the, the goverig equatio is obtaied as a oe dimesioal case of (8), so that d du K = () dx dx subject to the followig boudary coditios

6 6 u() = T, u( L) T =. () For the sake of simplicity, we assume that the thermal coductivity varies liearly with temperature, thus [9] (see discussio). ( ) K = K + u T () where K is the thermal coductivity at temperaturet ad the costat is defied below. I order to employ MHPMLT to obtai a hady accurate aalytical approximate solutio for the heat problem above described, we rewrite () as follows. After performig the idicated derivative i (), we get d u dk du K dx + = (3) dx dx Next, it is suggested the itroductio of the followig dimesioless quatities [9]. u T x K K y =, z =, = ( T T) =, (4) T T L K ote that the last equality of (4) defies. Employig the chai rule, it is possible to deduce that d d d =, dx L dz dx O the other had, the first equatio of (4) ca be writte as d = (5) L dz u = T + y( T T ) (6) Thus, substitutig (6) ito (), yields i K = K + y (7) After substitutig (5)-(7) ito (3), it is obtaied d y + y d y + dy = dz dz dz where boudary coditios () adopt the simpler form (8) y () =, y () =. (9) 5. Case Study The objective of this sectio is employ MHPMLT, to fid a aalytical approximate solutio for the oliear problem give by (8) ad (9). We will see that it is possible to fid a hady solutio by applyig MHPMLT method. Idetifyig terms:

7 7 L( y) = y( z) (3) N y y z y z y z ( ) = ( ( ) ( ) + ( )), (3) where prime deotes differetiatio respect to z. I accordace with (4), we propose or (3) ( p)( y y) + p y + yy + y = Applyig Laplace trasform we get, y = y + p y yy y (33) ( ( )) s Y () s A s = y + p y yy y (34) where we have defied Y ( s) = ( y( z)), A= y() with iitial coditio y () =. Solvig for Y() s ad applyig Laplace iverse trasform ( ( )) A y() z = + + y + p y yy y s s s Next, we assume a series solutio for yz, ( ) i the form (35) where y( z) = p ( z) (36) = ( z) = + Az (37) is chose as the first approximatio for the solutio of (8) that satisfies the coditios y () =, y () = A. Substitutig (36) ad (37) ito (35), we get A p = + + ( y + p ( y ( + p + p +..)( + p + p +..))) = s s s (38) ( + p + p +..). s O comparig the coefficiets of like powers of p we have p : () z = + A s s, (39) p : ( z) = { + } s, (4)

8 8 p ( z) = { } : + + s, (4) K After solvig the above Laplace trasforms for ( z), ( z), ( z),.. we obtai p : ( z) = + Az, (4) p : ( ) Az z =, (43) 3 3 p : ( ) A z A z z = +. (44) K ad so o. By substitutig (4)-(44) ito (5) ad evaluatig the limit p, results i a hady secod order approximatio. A 3 3 y( z) = + Az + ( + ) z + A z. (45) I order to calculate the value of A, we require that (45) satisfies the boudary coditio y () =. Cosiderig the case studies =.5, =, ad =.5, we obtai respectively the solutios y( z) = z 3 (46) z z, y z z z 3 ( ) = , (47) y( z) = z 3 (48) z z. It should be metioed that the problem ()-() has the exact solutio (see discussio). ( ) ( ) T + T + H u =, (49) where T T T H = + ( T ) T + ( T T ) x. (5) L 6. Discussio This work employed MHPMLT i the search for a hady accurate aalytical approximate solutio for the oliear secod order ordiary differetial equatio with fiite boudary coditios, which describes the steady state oe-dimesioal heat coductio i a slab with thermal coductivity, liearly depedet o the temperature. As it is well kow, the temperature field of a body approaches asymptotically to steady state coditios; therefore the kowledge of the statioary solutio is relevat

9 9 because it determies the fial temperature distributio alog the slab. The case of thermal coductivity liearly depedet o the temperature arises, for istace, i the case of a pure metal such as copper. For this metal, at a temperature ragig from to 5 K, the thermal coductivity is modelled maily by electros ad icreases liearly with temperature (see ()) [7]. For other temperature rages, this depedecy is o loger liear, ad should be the subject of future ivestigatios, usig for istace MHPMLT, to model the temperature distributio i these cases [7]. From (4) is iferred that MHPMLT is expressed i terms of the iitial coditios for a give differetial equatio, therefore, our procedure was aimed to express the approximate solutios i terms of A [8], so that y () ca be determied, just requirig that the approximate solutio satisfies the boudary coditio y () =. This coditio defies a algebraic equatio for A, whose solutio cocludes the procedure by obtaiig a aalytical approximate solutio for the proposed problem. Figures -4 show the compariso betwee umerical solutios ad approximate solutios (46)- (48) for =.5, =, ad =.5 respectively. It ca be oticed that curves are i good agreemet, whereby it is iferred the potetiality of MHPMLT i the search for approximate solutios of oliear problems with fiite boudary coditios [8]. Nevertheless, i more precise terms, it is possible to verify the accuracy of our results by calculatig the square residual error (S.R.E) of approximate solutios (46)-(48). S.R.E is defied as ( ()) b a R u r dr, where a ad b are the ed poits, ad ur () is a approximate solutio to the equatio to be solved (3), i our case (8) [9] ad the residual R( u( r )) results of substitutig ur () ito differetial equatio to be solved. The resultig values were respectively of.4,.747 ad.6745 which cofirms the accuracy of the proposed solutios. The square residual error (S.R.E.) is i geeral terms a positive umber, represetative of the total error committed, by usig the approximate solutio ur ()[9]. Figure. Compariso umerical solutio of the oliear problem give by (8) ad (9)) for ad MHPMLT approximatio (46).

10 Figure 3. Compariso umerical solutio of the oliear problem give by (8) ad (9)) for ad MHPMLT approximatio (47). Figure 4. Compariso umerical solutio of the oliear problem give by (8) ad (9) for ad MHPMLT approximatio (48). The parameter, turs out to be of paramout importace for our study. Thus, would be small for the case of small differece of temperature of the two opposite faces of the slab (see (4)) ad for the same reaso, for two give arbitrary poits o the slab, this limit correspods to the simpler case i which, thermal coductivity is almost costat (see ()). From (8), it is deduced that for steady coditios, temperature varies liearly with z i this limit. A more iterestig case would occur if the temperature gradiet alog the slab was ot ecessarily small ad correspoded to bigger values of. From (8), it is clear that the oliear term becomes importat too. A relevat fact from MHPMLT follows from equatios like (8), which ca be writte i the form L( z) + N( z) =, where Lz () is liear ad N( z) oliear [8]. It is well kow that classical methods of approximatio as perturbatio method PM [8] provides i geeral,

11 better results for small perturbatio parameters. From this poit of view, ca be visualized as a parameter of smalless that measures how greater is the cotributio of liear term Lz () tha the oe of N( z ). I geeral, it is easier to fid aalytical approximate solutios for small values of tha for big values of the same. Figures -4 ad the values of square residual error obtaied show a oticeable fact, that (47) ad (48) provide a good approximatio as solutios of (8), despite of the fact that perturbatio parameters = ad =.5 are ideed large. Thus, i priciple, the proposed methodology is ot restricted to small parameters [8] ad it is able to explai the pheomeo uder study for a wide rage of values of the aforemetioed parameter i a simple way. O the other had, it is very importat to emphasize that, it is possible to improve the accuracy of our results (see Figure. 4), cosiderig higher order approximate solutios. I [9] optimal homotopy perturbatio method (OHPM) was employed i order to provide a approximate solutio for (8). Although the solutio reported is hady ad has good accuracy, this method is more complicated for applicatios tha MHPMLT. OHPM requires costruct a homotopy, which icludes the presece of certai fuctios, provided of some parameters which are determied i order to cotrol the covergece of solutios. This procedure is usually loger ad difficult tha MHPMLT which may times requires oly of calculatig elemetary Laplace trasforms i a systematic way. At differece of other methods (for istace PM) which iclude the boudary coditios from the begiig of the problem at the lowest order approximatio, MHPMLT estimates oe of the iitial coditios ukow at first, requirig that the whole proposed solutio satisfies oe of the boudary coditios (the other boudary coditio is satisfied from the begiig of the procedure) thus, it is esured that the approximate solutio fits correctly o both boudaries of the iterval (the above is provided, by calculatig the value of A ) [8]. Although this case study admitted the exact solutio (49), it is ecessary to make the followig observatios. We ote that (49) ad (5) provide a solutio too log ad somehow cumbersome for practical applicatios. Eve though (46)-(48) are oly approximate solutios for the propose problem, they are hady, accurate ad for the same reaso ideal for practical applicatios. Other theoretical ad practical reasos related with the above, i favor of usig the proposed method is that, the secod order approximatio (45) provided by MHPMLT is ot oly hady but it is expressed i terms of the perturbatio parameter. Thus, from previously explaied, at least for the case of moderated temperature gradiets it is possible to employ (45) istead of the rather cumbersome ad complicated exact solutio (49) with small loss of precisio. (45) is expressed i terms of physical parameter which allows i priciple estimate the cotributios of the differet power terms. For example if is small so that we keep just the first two terms of (45), the the temperature varies very approximately i a liear way with z. I cosequece, this observatio is eve more importat, regardig that the proposed method is ot restricted to small parameters by which icreases the fruitfuless of the proposed method because it is possible to employ (45) istead of (49) for a wider iterval of. Fially, maybe the mai reaso for proposig MHPMLT, is that the majority of oliear differetial equatios that describe heat problems do ot admit a exact solutio.

12 7. Coclusios I this work MHPMLT was employed to provide a hady aalytical approximate solutio for the secod order oliear differetial equatio which describes the steady state oedimesioal heat coductio problem i a slab of thermal coductivity liearly depedet o the temperature defied with Dirichlet boudary coditios o a fiite iterval. MHPMLT method expressed the problem of fidig a approximate solutio for the above metioed differetial equatio, i terms of solvig a algebraic equatio for some ukow iitial coditio [8]. The square residual error of the approximate solutios shows that it is a method with high potetial i the search for solutios of boudary value oliear problems, eve for the case of large perturbatio parameters. Fially, we ote a importat additioal advatage from the proposed method; this does ot require of solvig several recurrece differetial equatios; ideed it requires oly of calculatig elemetary Laplace trasforms i a systematic way. Therefore, LTHPM is a tool with high potetial for practical applicatios i sciece ad egieerig [8]. Ackowledgmet The authors would like to thak Rogelio-Alejadro Callejas-Molia, ad Roberto Ruiz-Gomez for their cotributio to this project. Refereces [] Amiikha, H., Hemmatezhad, M., A ovel Effective Approach for Solvig Noliear Heat Trasfer Equatios, Heat Trasfer- Asia Research, 4 (), 6, pp [] Rashidi, M.M., et al., A study of o-ewtoia flow ad heat trasfer over a o-isothermal wedge usig the homotopy aalysis method, Chemical EgieerigCommuicatios, 99 (), pp [3] Rashidi, M., et al., Aalytic approximate solutios for steady flow over a rotatig disk i porous medium with heat trasfer by homotopy aalysis method, Computers & Fluids, 54 (), pp. -9. [4] Spiegel, M.R., Teoría y Problemas de Trasformadas de Laplace, primera edició. Serie de compedios Schaum, Mc-Graw Hill, México, 988. [5] Mishra, H.K., ad Nagar, A.K., He-Laplace method for liear ad oliear partial differetial equatios. Joural of Applied Mathematics (), pp. -6. [6] Liu, Z. J., et al., Hybridizatio of homotopy perturbatio method ad Laplace trasformatio for the partial differetial equatios. Thermal Sciece, (7),4, pp [7] He, J.H., Asymptotic methods for solitary solutios ad compactos, Abstract ad applied aalysis., (). DOI:.55//96793 [8] Filobello-Nio U., et al., Laplace trasform-homotopy perturbatio method as a powerful tool to solve oliear problems with boudary coditios defied o fiite itervals, Computatioal ad Applied Mathematics, 34 (5),, pp. -6. DOI=.7/s z. [9] Marica, V. ad Herisau, N., Noliear Dyamical Systems i Egieerig, first editio. (Spriger-Verlag, Berli Heidelberg ). [] Amiikhah H., Aalytical Approximatio to the Solutio of Noliear Blasius Viscous Flow Equatio by LTNHPM, Iteratioal Scholarly Research Network ISRN Mathematical Aalysis, (), pp. - Article ID , doi:.54// [] Kha, M., et al., A ew study betwee homotopy aalysis method ad homotopy perturbatio trasform method o a semi ifiite domai, Mathematical ad Computer Modellig, 55 (), pp

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