THERMAL SCIENCE, Year 017, Vol. 1, No. 4, pp. 1833-1838 1833 EXACT TRAVELING WAVE SOLUTIONS FOR A NEW NON-LINEAR HEAT TRANSFER EQUATION by Feng GAO a,b, Xiao-Jun YANG a,b,* c, an Yu-Feng ZHANG a School of Mechanics an Civil Engineering, China University of Mining an Technology, Xuzhou, China** b State Key Laboratory for Geo-Mechanics an Deep Unergroun Engineering, China University of Mining an Technology, Xuzhou, China*** c College of Mathematics, China University of Mining an Technology, Xuzhou, China Introuction Original scientific paper https://oi.org/10.98/tsci16051076g In this paper, we propose a new non-linear partial ifferential equation to escribe the heat transfer problems at the extreme excess temperatures. Its exact traveling wave solutions are obtaine by using Cornejo-Perez an Rosu metho. Key wors: traveling wave solution, non-linear heat transfer equation, Cornejo-Perez an Rosu metho, exact solution, heat transfer The approximate, analytical, numerical, an exact solutions for the non-linear heat transfer problems have been investigate by a great many of engineers an scientists for many years [1-]. Many technologies were propose to fin the solutions for them, such as the homotopy perturbation metho [3], heat-balance integral metho [4], variational iteration metho [5], an integral transforms [6], an so on. Recently, Cornejo-Perez an Rosu [7] propose a new metho to fin the exact solutions to non-linear orinary ifferential equations (ODE) an to apply to erive the exact traveling wave solutions for the non-linear partial ifferential equations (PDE) [8]. The targets of the previous paper are to aress a new non-linear heat transfer equation an to fin its exact traveling wave solutions. Mathematical moel propose Let us consier the non-linear heat transfer equation [1, 4]: α = ΛΘ [ ( ξ, τ)] where α [Wm 1 K 1 ] is the heat iffusion coefficient an Λ[Θ(ξ, τ)] the non-linear heat source. When Λ[Θ(ξ, τ)] is expane for the constants C i etermine by the temperature fiel: * Corresponing author, e-mail: yangxiaojun@163.com ** Now in: State Key Laboratory for Geo-Mechanics an Deep Unergroun Engineering, School of Mechanics an Civil Engineering, China University of Mining an Technology, Xuzhou, China *** Now in: School of Mechanics an Civil Engineering, China University of Mining an Technology, Xuzhou, China (1)
1834 THERMAL SCIENCE, Year 017, Vol. 1, No. 4, pp. 1833-1838 Equation (1) can be re-written: n i ΛΘξτ [ (, )] = C Θ( ξτ, ) () i=0 i n i α = C (, ) iθ ξτ (3) i=0 As a special case of eq. (), we consier the raiative loss of heat at the extreme excess temperature fiel shown in fig. 1, enote by Λ(ξ, τ): 4 = + (4) Λξτ (, ) βθξτ (, ) κθ ( ξτ, ) Figure 1. The extreme excess temperature fiel where β an κ are constants, an Θ(ξ, τ) is the excess temperature fiel. The governing equation at the extreme excess temperature can be written as: 4 = α βθξτ (, ) κθ ( ξτ, ), ( ξτ, ) [0, ) [0, ) (5) At the low excess temperature, eq. () reas [9]: = α βθξτ (, ), ( ξτ, ) [0, ) [0, ) where β is a constant an Θ(ξ, τ) the low excess temperature fiel. At the high excess temperature, eq. () can be re-written [10]: 4 = α κθ ( ξτ, ), ( ξτ, ) [0, ) [0, ) (6) (7) where κ is relate to Stefan's constant an Θ(ξ, τ) the excess temperature fiel. Equation (1) aopting in the previous paper is calle the heat transfer equation at the extreme excess temperature an the parameters β an κ are etermine by the ifferent temperature fiels. Analysis of metho In this section, we introuce the traveling wave transformation metho base on theory of Cornejo-Perez an Rosu group [7, 8]. In orer to illustrate the methoology, we consier the PDE with respect to ξ an τ given by: (, ) k ℵ,,, Θξτ (, ), Θ ( ξτ, ) 0 = where k is a positive integer. (8)
THERMAL SCIENCE, Year 017, Vol. 1, No. 4, pp. 1833-1838 1835 The TTM can be written: where ϑ is a constant. Due to the chain rules: θ = ξ ϑτ (9) Θξτ (, ) Θξτ (, ) Θξτ (, ) Θθ ( ) = ϑ an = θ θ Equation (8) becomes the ODE with respect to θ given by: Θθ ( ) Θθ ( ) k ℵ,, ( ), ( ) 0 Θθ Θ θ = ξ ξ Following the Cornejo-Perez an Rosu group [7, 8], eq. (11), can be written: 1( ) ( ) 0 ν Θ θ θ ν Θ Θ = which reuces to the following equation of the factorize form which leas to: Θθ ( ) ν1( Θ) Θθ ( ) Θ + ν1( Θ) + ν( Θ) ν1( Θ) ν( Θ) ( Θ) 0 θ Θ + Ξ = θ (10a,b) (11) (1) (13) ν 1( Θ ) ( ) Θ + ν1( Θ) + ν( Θ) = Λ( Θ), ν1( Θ) ν( Θ) = Π Θ (14a,b) Θ Θ where Θ(θ) = Θ, ν 1 [Θ(ϑ)] = ν 1 (Θ), ν [Θ(ϑ)] = ν (Θ), Λ[Θ(ϑ)] = Λ(Θ), an Π[Θ(ϑ)] = Π(Θ). Thus, eq. (1) can be re-written as the secon orer ODE with respect to θ given by: Θθ ( ) Θθ ( ) Λ ( Θ) + Π ( Θ) = 0 (15) θ θ Thus, we obtain the following ODE from eq. (15) that: Θθ ( ) Θθ ( ) ν1( ΘΘθ ) ( ) = 0, ν( ΘΘθ ) ( ) = 0 (16a,b) θ θ Furthermore, we irectly write the solutions of eq. (15) from eqs. (16a) an (16b). Conveniently, we call this technology as the CPRM [8]. Traveling wave solutions for the non-linear heat transfer problem which leas to: In this section, the CPRM is use to solve the non-linear heat transfer equation. From eqs. (5), (10a), an (10b) we have the non-linear ODE: Θθ ( ) Θθ ( ) 4 ϑ α βθ ( θ ) κθ ( θ ) θ = θ (17)
1836 THERMAL SCIENCE, Year 017, Vol. 1, No. 4, pp. 1833-1838 Θθ ( ) ϑ Θθ ( ) β κ 4 + Θθ ( ) Θ ( θ) = 0 (18) θ α θ α α Following eq. (1), we give: 1( ) ( ) 0 σ Θ θ θ σ Θ Θ = (19) which yiels the following equation in the factorize form: where Θθ ( ) σ1( Θ) Θθ ( ) Θ + σ1( Θ) + σ( Θ) σ1( Θ) σ( Θ) ( Θ) 0 θ Θ + Ξ = θ σ1( Θ) ϑ κ β 3 Θ + σ1( Θ) + σ( Θ) =, σ1( Θ) σ( Θ) = Θ ( θ) Θ α α + κ Thus, we have from eq. (1b) that: (0) (1a,b) 3 3 1 κ β κ β σ 1( Θ) = i Θ ( θ), σ ( Θ) = Η1,1 i Θ ( θ) ± (a,b) Η1,1 α α α α where H 1,1 is an unknown coefficient. From eqs. (a) an (b), eq. (1a) can be written as: which leas to: 3 3 3 3i κ 1 ( ) i κ ( ) β H 1,1 i κ ( ) β Θ θ + Θ θ + Θ θ ± = ϑ (3) H1,1 α H1,1 α α α α α From eq. (4a), we obtain: 5 1 ϑ + H = 0, ± H = (4a,b) H αβ 1,1 1,1 1,1 H1,1 which lea from eq. (19) to: 5 H 1,1 = ± (5) 3 3 κ β 5 κ β σ 1( Θ) =± i Θ ( θ), σ( Θ) =± i Θ ( θ) ± (6a,b) 5 α α α α It follows from eqs. (6a) an (6b) that:
THERMAL SCIENCE, Year 017, Vol. 1, No. 4, pp. 1833-1838 1837 3 Θθ ( ) ( ) ( ) 0 5 i κ β ± Θ θ Θθ = (7) θ α α 3 Θθ ( ) 5 ( ) ( ) 0 i κ β ± Θ θ ± Θθ = θ α α By integrating eqs. (7) an (8), we obtain the following exact solutions, which are given by: (8) where 5 Θθ ( ) κ tanh βθ 1 or Θθ ( ) κ tanh βθ = 1 4β + 10a = + 4β 8a Thus, we obtain the exact traveling wave solutions, which are given by: κ β ( ξ ϑt ) Θξt (, ) = tanh 1 4β + 10a κ 5 β ( ξ ϑt ) Θξt (, ) = tanh 1 4β + 8a (9a,b) (30a) (30b) 5 ϑ = 5 αβ (30c) with β > 0. The travelling wave solutions of eq. (4) for the ifferent parameters are illustrate in figs. (a) an (b). Figure. (a) The travelling wave solution (30a) for β = 1, α = 1, an κ = 1, (b) the travelling wave solution (30b) for β = 1, α = 1, an κ = 1 Conclusion In the present work, a non-linear heat transfer equation was propose for the first time. With the help of the CPRM, its exact traveling wave solutions were graphically illus-
1838 THERMAL SCIENCE, Year 017, Vol. 1, No. 4, pp. 1833-1838 trate. The CPRM to obtain the exact solutions were as an alternative technology propose to fin a class of the non-linear PDE in mathematical physics. Acknowlegments This work is supporte by the State Key Research Development Program of the People s Republic of China (Grant No. 016YFC0600705), the Natural Science Founation of China (Grant No. 5133004), Priority Acaemic Program Development of Jiangsu Higher Eucation Institutions (PAPD014), Sichuan Sci-Technology Support Program (01FZ014), National Natural Science Funament of China (Grant No. 1137131361), Innovation Team of Jiangsu Province Hoste by China University of Mining an Technology (014), an Key Discipline Construction by China University of Mining an Technology (Grant No. XZD 0160). Nomenclature α heat iffusion coefficient, [Wm 1 K 1 ] β constant, [1s 1 ] Θ(ξ, τ) exess temperature, [K] κ constant, [K 3 s 1 ] ξ space co-orinate, [m] τ time co-orinate, [s] References [1] Polyanin, A. D., Zaitsev, V. F., Hanbook of Nonlinear Partial Differential Equations, CRC Press, Boca Raton, Fla., USA, 004 [] Yang, X.-J., A New Integral Transform Operator for Solving the Heat-Diffusion Problem, Applie Mathematics Letters, 64 (017), Feb., pp. 193-197 [3] Ganji, D. D., The Application of He's Homotopy Perturbation Metho to Nonlinear Equations Arising in Heat Transfer, Physics letters A, 355 (006), 4, pp. 337-341 [4] Hristov, J., Integral Solutions to Transient Nonlinear Heat (Mass) Diffusion with a Power-Law Diffusivity: A Semi-Infinite Meium with Fixe Bounary Conitions, Heat Mass Transfer, 5 (016), 3, pp. 635-655 [5] He, J.-H., Maximal Thermo-Geometric Parameter in a Nonlinear Heat Conuction Equation, Bulletin of the Malaysian Mathematical Sciences Society, 39 (016),, pp. 605-608 [6] Yang, X.-J., A New Integral Transform with an Application in Heat-Transfer Problem, Thermal Science, 0 (016), Suppl. 3, pp. S677-S681 [7] Cornejo-Perez, O., Rosu, H. C., Nonlinear Secon Orer ODE: Factorizations an Particular Solutions, Progress of Theoretical Physics, 114 (005), 3, pp. 533-538 [8] Griffiths, G., Schiesser, W. E., Traveling Wave Analysis of Partial Differential Equations: Numerical an Analytical Methos with MATLAB an Maple, Acaemic Press, New York, USA, 010 [9] Carslaw, H. S., Introuction to the Mathematical Theory of the Conuction of Heat in Solis, Macmillan, Lonon, 010 [10] Carslaw, H. S., Jaeger, J. C., Conuction of Heat in Solis, Oxfor University Press, Oxfor, UK, 1959 Paper submitte: May 1, 016 Paper revise: June 30, 016 Paper accepte: August 3, 016 017 Society of Thermal Engineers of Serbia. Publishe by the Vinča Institute of Nuclear Sciences, Belgrae, Serbia. This is an open access article istribute uner the CC BY-NC-ND 4.0 terms an conitions.