The Solutions of Nonlinear Heat Conduction Equation via Fibonacci&Lucas Approximation Method. Zehra Pınar a. Turgut Öziş b.

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1 The Solutios of Noliear Heat Coductio Equatio via Fiboacci&Lucas Approximatio Method Zehra Pıar a Turgut Öziş b a Namık Kemal Uiversity, Faculty of Sciece & Letters, Departmet of Mathematics,, Tekirdağ, Turkey b Ege Uiversity, Sciece Faculty, Departmet of Mathematics, Borova-İzmir, Turkey Abstract: To obtai ew types of exact travellig wave solutios to oliear partial differetial equatios, a umber of approximate methods are kow i the literature. I this study, we exted the class of auxiliary equatios of Fiboacci&Lucas type equatios. The proposed Fiboacci&Lucas approximatio method produces may ew solutios. Cosequetly, we itroduce ew exact travellig wave solutios of some physical systems i terms of these ew solutios of the Fiboacci&Lucas type equatio. I additio to usig differet asatz, we use determie differet balacig priciple to obtai optimal solutios. Key Words: Noliear heat coductio equatio, Fiboacci&Lucas equatio, travellig wave solutios, oliear partial differetial equatio, auxiliary equatio b Correspodig author: Zehra Pıar (zpiar@ku.edu.tr, )

2 1. Itroductio The ispectio of oliear wave pheomea physical systems is of great iterest from both mathematical ad physical poits of view. I most cases, the theoretical modelig based o oliear partial differetial equatios (NLPDEs) ca accurately describe the wave dyamics of may physical systems. The critical importace is to fid closed form solutios for NLPDEs of physical sigificace. This could be a very complicated task ad, i fact, is ot always possible sice i various realistic problems i physical systems. So, searchig for some exact physically sigificat solutios is a importat topic because of wide applicatios of NLPDEs i biology, chemistry, physics, fluid dyamics ad other areas of egieerig[,3]. Sice may of the most useful techiques i aalysis are formal or heuristic the tred i recet years has also bee to justify ad provide the ew procedures or methods rigorously[18]. Hece, over the past decades, a umber of approximate methods for fidig travellig wave solutios to oliear evolutio equatios have bee proposed/or developed ad furthermore modified [6-0]. The solutios to various evolutio equatios have bee foud by oe or other of these methods. The techique of these methods cosist of the solutios of the oliear evolutio equatios such that the target solutios of the oliear evolutio equatios ca be expressed as a polyomial i a elemetary fuctio which satisfies a particular ordiary differetial equatio. Recetly, to determie the solutios of oliear evolutio equatios, may exact solutios of various auxiliary equatios have bee utilized [-7]. I this paper, we will examie the cosequeces of the choice of the Fiboacci&Lucas type equatio for determiig the solutios of the oliear evolutio equatio i cosideratio ad more we search for additioal forms of ew exact solutios of oliear differetial equatios which satisfyig Fiboacci&Lucas type equatio(s). To obtai wave solutios of oliear partial differetial equatios via Fiboacci&Lucas trasformatio optimal idex value is proposed.. Fiboacci & Lucas Polyomials I this sectio, we determie Fiboacci&Lucas polyomials. Ordiary differetial equatios satisfied by two families of Fiboacci ad Lucas polyomials are derived usig idetities which relate them to the geeralized polyomials, ad opolyomial solutios are deduced from correspodig solutios of the partial differetial equatios.

3 The sigle variable polyomials F (1, z) ad L (1, z), 0, with the properties F ( x, y) x F (1, ), L ( x, y) x L (1, ), where y/ x, are referred to as the Fiboacci ad Lucas polyomials, respectively, by Doma ad Williams [1]. Galvez ad Devesa [4] have show that they satisfy the ordiary differetial equatios d F df (1 4 ) [( 1) ( 5) ] ( 1)( ) F 0, (1) d d d L dl (1 4 ) [( 1) ( 3) ] ( 1) L 0. () d d Usig the earlier results, it ca be show that a secod liearly idepedet solutio of the Eq.(1) is L (1, ) / 1 4, ad a secod liearly idepedet solutio of the Eq.() is 1 4 (1, ). F The polyomials F(,1) ad L(,1), also referred to as Fiboacci ad Lucas polyomials by Hoggatt ad Bickell [5], are related to the geeralized polyomials by F x y y F L x y y L ( 1)/ / (, ) (,1), (, ) (,1), where x/ y, that they satisfy the ordiary differetial equatios d F df (4 ) 3 ( 1) F 0, (3) d d d L dl (4 ) L 0, (4) d d which also have solutios L (,1) / 4 ad 4 F(,1), respectively. As see above, Eqs(1)-(4) are depeds o idex value, so the solutios of Eqs(1)-(4) are differet solutios for each idex values. But, oe of the importat questios is Which idex value produce a available basis for the give problem? ad the aswer is give i the followig sectio. 3. Methodology

4 The fudametal ature of the auxiliary equatio techique is give by may authors i the literature [6-15] ad it is applied to this ew approximatio. Let us have a oliear partial differetial equatio ( ) (5) ad let by meas of a appropriate trasformatio which is depeded o Fiboacci&Lucas type equatio(s), this equatio is reduced to oliear ordiary differetial equatio ( ). (6) For large class of the equatios of the type (6) have exact solutios which ca be costructed via fiite series ( ) ( ) (7) Here, ( ) are parameters to be further determied, is a iteger fixed by a balacig priciple ad elemetary fuctio ( ) is the solutio of some ordiary differetial equatio referred to as the auxiliary equatio[15,16,17,19, 1]. It is worth to poit out that we happe to kow the geeral solutio(s), ( ), of the auxiliary equatio beforehad or we kow at least exact aalytical particular solutios of the auxiliary equatio. The outlie of the method: A) Defie the solutio of Eq.(6) by the asatz i form of fiite series i Eq.(7) where ( ) are parameters to be further determied, is a iteger fixed by a balacig priciple ad elemetary fuctio ( ) is the solutios of Eqs(1-4) be cosidered which are depeded o idex values which helps us to obtai wave solutio for the give problem. But, for each idex values, we obtai differet solutios of auxiliary equatios(1)-(4), so we eed to determie idex values to cotrol the availability of the give problem. To determie idex values of Fiboacci&Lucas fuctio from the balacig priciple we proposed a ovel balacig give below m u is highest degree m (8) 1,0,1 mod( m) B) Substitute Eq.(7) ito ordiary differetial equatio Eq.(6) to determie the parameters ( ) with the aid of symbolic computatio.

5 C) Isert predetermied parameters ad elemetary fuctio ( ) of the auxiliary equatio ito Eq.(7) to obtai travellig wave solutios of the oliear evolutio equatio i cosideratio. It is very apparet that determiig the elemetary fuctio ( ) via auxiliary equatio is crucial ad plays very importat role fidig ew travellig wave solutios of oliear evolutio equatios. This fact, ideed, compel researchers to search for a ovel auxiliary equatios with exact solutios. I this study, we use Fiboacci& Lucas differetial equatios are give above. 4. Travellig Wave Solutios of Noliear Heat Coductio Equatio I Terms of Fiboacci&Lucas Equatio Regardig As A Asatz. I this sectio, we cosider the followig oliear heat coductio equatio u u pu qu t xx 0 (9) where pq, are real costats. To use Fiboacci&Lucas approximatio method, we cosider the determied variables, istead of the wave variable. Although the wave trasformatio is ot used to obtai the wave solutio, it is obtaied by Fiboacci&Lucas trasformatios which are determied by., The, variables carries Eq. ( ) ito the ordiary differetial equatio. From the balacig priciple, 0 1 is obtaied. Therefore, the asatz yields U( ) g g z( ) (10) where z( ) may be determied by the solutio of Eqs.(1)-(4). Case 1: We cosider y/ x trasformatio ad Eq(1) is cosidered as a auxiliary equatio, The variables carries Eq. ( ) ito the ordiary differetial equatio U( ) U( ) U ( ) U( ) 3 8t 8 U( ) t 1 U( ) t pu( ) qu( ) 0 (11) x x x x ad here the asatz is assumed as followig U g0g1f ( ) ( ) (1) Hece, substitutig Eqs.(1) ad(1) ito Eq.(11) ad lettig each coefficiet of F ( ) to be zero, we obtai algebraic equatio system ad solvig the system by the aid of Maple 16, we ca determie the coefficiets:

6 i) C C, C C, g ( 3 ) 3 qx (1 ) qx 6qx 1qx 48qx 4qx g1 1 qx (1 ) ( ) ii) C C, C g g1 g1 8 ( 3 ) 0 3 qx (1 ) 7 px (1 ) 5( ) Substitutig the above coefficiets ito asatz (1) with the solutio of Lucas type equatio, we obtai oe of ew solutio of oliear heat coductio equatio ( 3 ) qx 6qx 1qx 48qx 4qx u( x, t) 3 qx (1 ) 1 qx (1 ) ( ) t C1hypergeom,,[ 3 9],1 x ( 83 ) t t C 1 hypergeom,,[ 7 3 ],1 x x u( x, t) t C1hypergeom,,[ 3 9],1 x g 1 7 ( 83 ) px (1 ) t hypergeom, 3 1,[ 7 3 ],1 t 5( 1 1g1 36g1 8 8 ) x x 8 ( 3 ) 3 qx (1 ) To obtai suitable solutio, 1 is obtaied from Eq(8).For the special values of parameters, the solutios are show i Figure 1.

7 (a) (b) Figure 1. (a) is for the first solutio for C, q 1, C1 0.5, 1, (b) is for the secod sloutio where p 1, C1 1, g1 1, 1, If idex values are chose arbitrary, ot usig balacig formula Eq(8), the solutios of Eqs(1)-(4) do t costruct a basis for the give equatio. I that case, the behavior of solutios chages respect to idex values. Therefore, to obtai travellig wave solutios for the give equatio, we eed to use the balacig formula Eq(8). For this example, if idex values are take as,3,4,..., which are ot satisfy the Eq(8), the for these idex values we do t obtai solutios or these idex values distort wave behavior of the obtaied solutios. It is see i the Figure. (a) (b)

8 Figure. (a) is for the idex value, (b) is for the 4 idex value Case : We cosider x/ y trasformatio ad Eq(4) is cosidered as a auxiliary equatio. The variables carries Eq. ( ) ito the ordiary differetial equatio ( ) ( ) ( ) 3 U( ) pu ( ) qu( ) 0 (13) 3/ U x U U t t t ad here the asatz is assumed as followig from the balacig priciple U g0g1l ( ) ( ) (14) Hece, substitutig Eqs.(4) ad(14) ito Eq.(13) ad lettig each coefficiet of L ( ) to be zero, we obtai algebraic equatio system ad solvig the system by the aid of Maple 16, we ca determie the coefficiets: C C, C C,, p p, g 0, g g ad 4 arcta 4 C1 C1, C,, g 4 arcta 4 p, g 0, g g 4 arcta t(4 ) Substitutig the above coefficiets ito asatz (14) with the solutio of Lucas type equatio, we obtai ew solutios of oliear heat coductio equatio. 1 x x 1 x x U( x, t) x g 1 C1 si 8pt px arcta C cos 8pt px arcta t 4t x t 4t x x x x x 4 arcta cos arcta t x 4t x 4t x U ( x, t) x g1 C1si arcta 4t x g1 t To obtai optimal solutio, 1is obtaied from Eq(8).For the special values of parameters, the solutios are show i Figure 3.

9 (a) (b) Figure 3: (a) is the figure of the first solutio, (b) is for the secod solutio for special values C1 1, p 1, g1 1, 1. Case 3: We cosider y/ x trasformatio ad Eq() is cosidered as a auxiliary equatio, The variables carries Eq. ( ) ito the ordiary differetial equatio U( ) U( ) U ( ) U( ) 3 8t 8 U( ) t 1 U( ) t pu( ) qu( ) 0 (11) x x x x ad here the asatz is assumed as followig U g0g1l Hece, substitutig Eqs.() ad(15) ito Eq.(11) ad lettig each coefficiet of L ( ) to be zero, we obtai algebraic equatio system ad solvig the system by the aid of Maple 16, we ca determie the coefficiets: i) C 0, C g ( ) ( ) (15) ( 1) 3 qx (1 ) qx ( ) ii) C 0, C pqx 7 pq x g 81pq x g 88pqx 81pq x g 144 pqx 7 pq x g 18 (1 ) ( ) qx g1 qx qx g1qx g1qx qx qx g1qx qx 8 ( 1) g0 3qx (1 )

10 iii) p ( ) qx(1 ) ( 1 8 ) qx 6qx qx 48qx 4qx g1 1 qx (1 )( ) 8 ( 1) g0 3 qx (1 ) Substitutig the above coefficiets ito asatz (15) with the solutio of Lucas type equatio, we obtai oe of ew solutio of oliear heat coductio equatio t hypergeom,,[ 3 3 ],1 ( 43 ) x t g ( 1) 18 u( x, t) 1 3 qx (1 ) 9 x qx ( ) t( 1) pqx 7 pq x g1 81pq x g1 88 pqx 81pq x g1 144 pqx 7 pq x g1 t u( x, t) t 1 18 qx g1(1 ) (4qx qx 84g1qx 144g1qx 1qx 30 6qx 60g1qx 3qx ) x 3qx 1 x t hypergeom,,[ 3 3 ],1 x ( 43 ) u( x, t) t( 1) 56 3qx 6qx qx 48qx 4qx t 1 qx (1 )( ) x 4 3qx t C1hypergeom,,[5 3 ],1 x ( 43 ) t t C 1 hypergeom,,[ 3 3 ],1 x x To obtai optimal solutio, 1 is obtaied from Eq(8).For the special values of parameters, the solutios are show i Figure 4.

11 (a) (b) (c) Figure 4. (a) is for the first solutio for C 1, q 1, g1 1, 1, (b) is for the secod sloutio where p 1, q 1, g1 1, 1, (c) is for the third solutio C 1, C1 1, q 1, 1 Case 4: We cosider x/ y trasformatio ad Eq(3) is cosidered as a auxiliary equatio, the variables carries Eq. ( ) ito the ordiary differetial equatio ( ) ( ) ( ) 3 U( ) pu ( ) qu( ) 0 (13) 3/ U x U U t t t ad here the asatz is assumed as followig

12 U g g F -1 ( ) 0 1 ( ) (16) Hece, substitutig Eqs.(3) ad(16) ito Eq.(13) ad lettig each coefficiet of F ( ) to be zero, we obtai algebraic equatio system ad solvig the system by the aid of Maple 16, we ca determie the coefficiets: g C 3 g C g C 16 g C 4 4 3g C C g Substitutig the above coefficiets ito asatz (16) with the solutio of Lucas type equatio, we obtai oe of ew solutio of oliear heat coductio equatio. u( x, t) g g C 3 g C g C 16 g C 4 4 3g C g C 4 4 To obtai optimal solutio, 1is obtaied from Eq(8).For the special values of parameters, the solutios are show i Figure 5. Figure 5. Graph of solutio of Eq.(9) forc, g1 si( xt), 1 Coclusio

13 As it is see, the key idea of obtaiig ew travellig wave solutios for the oliear equatios is usig the exact solutios of differet types equatios as a asatz. By meas of Fiboacci&Lucas type equatio, the wave trasformatio is ot used to obtai the wave solutios, the Fiboacci&Lucas trasformatio is used to obtai the wave solutios. But, the wave solutios are ot obtaied by the stadard Fiboacci&Lucas trasformatios. For this reaso, the optimal idex value is proposed i Eq(8). Usig the solutios of Fiboacci&Lucas type equatio, we have successfully obtaied a umber of ew exact periodic solutios of the oliear heat coductio equatio by employig the solutios of the Fiboacci&Lucas type equatio regardig as a auxiliary equatio i proposed method. I this letter, we have obtaied ew solutios of the oliear equatio i had usig the Fiboacci&Lucas type equatio (Eqs.(1-4)) for distict cases. However, it is well kow that the optimal idex value produces ew travellig wave solutios for may oliear problems. The preseted method could lead to fidig ew exact travellig wave solutios for other oliear problems. 5. Refereces [1] B. G. S. Doma & J. K. Williams. "Fiboacci ad Lucas Polyomials.'1 Math. Proc. Cambridge Philos. Soc. 9.0 (1981): [] L.Debath, Noliear partial differetial equatios for scietists ad Egieers ( d ed.) Birkhauser, Bosto (005) [3] G.B.Whitham, A geeral approach to liear ad oliear waves usig a Lagragia, J. Fluid Mech.,(1965) [4] F. J. Galvez & J. S. Dehasa. "Novel Properties of Fiboacci ad Lucas Polyomials." MathProc. Cambridge Philos. Soc. 91 (1985): [5] V. E. Hoggatt, Jr., & M. Bickell "Roots of Fiboacci Polyomials." The Fiboacci Quarterly 11.3 (1973):71-74 [6] C. Yog, L. Biao, Z. Hog-Quig, Geeralized Riccati equatio expasio method ad its applicatio to Bogoyayleskii s geeralized breakig solito equatio, Chiese Physics. 1 (003) [7] G. Cai, Q. Wag, J. Huag, A modified F-expasio method for solvig breakig solito equatio, Iteratioal Joural of Noliear sciece (006) 1-18 [8] X. Zeg, X. Yog, A ew mappig method ad its applicatios to oliear partial differetial equatios, Phy. Lett. A. 37 (008) [9] X. Yog, X. Zeg, Z. Zhag, Y. Che, Symbolic computatio of Jacobi elliptic fuctio solutios to oliear differetial-differece equatios, Comput.Math. Appl. doi: /j.camwa [10] W. X. Ma, T. Huag, Y. Zhag, A multiple Exp-fuctio method for oliear differetial equatios ad its applicatios, Phys. Scr. 8(010) (8pp) [11] T. Ozis, I. Asla, Symbolic computatio ad costructio of New exact travelig wawe solutios to Fitzhugh-Nagumo ad Klei Gordo equatio, Z. Naturforsch. 64a(009) 15-0 [1] T. Ozis, I. Asla, Symbolic computatios ad exact ad explicit solutios of some oliear evolutio equatios i mathematical physics, Commu. Theor. Phys., 51(009) [13] Siredaoreji, Auxiliary equatio method ad ew solutios of Klei Gordo equatios,

14 Chaos, Solitos ad Fractals 31 (007) [14] B. Jag, New exact travellig wave solutios of oliear Klei Gordo equatios, Chaos, Solitos ad Fractals 41 (009) [15] X. Lv, S. Lai, Y.H. Wu, A auxiliary equatio techique ad exact solutios for a oliear Klei Gordo equatio, Chaos, Solitos ad Fractals 41 (009) 8 90 [16] E. Yomba, A geeralized auxiliary equatio method ad its applicatio to oliear Klei Gordo ad geeralized oliear Camassa Holm equatios, Physics Letters A 37 (008) [17] J. Nickel, Elliptic solutios to a geeralized BBM equatio, Physics Letters A 364 (007) 1 6 [18] E.T. Whittaker, G.N. Watso, A Course of Moder Aalysis, Cambridge Uiv. Press, Cambridge, 197. [19] E. Yomba, The exteded Fa s sub-equatio method ad its applicatio to KdV-MKdV, BKK ad variat Boussiesq equatios, Phys. Lett. A, 336(005) [0] Z. Y. Ya, A improved algebra method ad its applicatios i oliear wave equatios,chaos Solitos & Fractals 1(004) [1] M.A. Abdou, A geeralized auxiliary equatio method ad its applicatios, Noliear Dy 5 (008) [] Z. Piar, T. Ozis, A Observatio o the Periodic Solutios to Noliear Physical models by meas of the auxiliary equatio with a sixth-degree oliear term, Commu Noliear Sci Numer Simulat, 18(013) [3] N.K. Vitaov, Z. I. Dimitrova, K. N. Vitaov, O the class of oliear PDEs that ca be treated by the modified method of simplest equatio. Applicatio to geeralized Degasperis Processi equatio ad b-equatio, Commu Noliear Sci Numer Simulat 16 (011) [4] RI. Ivaov, Water waves ad itegrability, Phil Tras R Soc A, 365(007) [5] L. Debath, Noliear water waves, New York, Academic Press; 1994 [6] RS. Johso, The classical problem of water waves: a reservoir of itegrable ad earlyitegrable equatios, J Noli Math Phys 10 (003) 7 9 [7] N. K. Vitaov, Modified method of simplest equatio: Powerful tool for obtaiig exact ad approximate travelig-wave solutios of oliear PDEs, Commu Noliear Sci Numer Simulat 16 (011)

15 List of Figures Figure 1. (a) is for the first solutio for C, q 1, C1 0.5, 1, (b) is for the secod sloutio where p 1, C1 1, g1 1, 1, Figure. (a) is for the idex value, (b) is for the 4 idex value Figure 3: (a) is the figure of the first solutio, (b) is for the secod solutio for special values C 1, p 1, g 1, Figure 4. (a) is for the first solutio for C 1, q 1, g1 1, 1, (b) is for the secod sloutio where p 1, q 1, g1 1, 1, (c) is for the third solutio C 1, C1 1, q 1, 1 Figure 5. Graph of solutio of Eq.(9) forc, g1 si( xt), 1