ADOMIAN DECOMPOSITION METHOD FOR THREE-DIMENSIONAL DIFFUSION MODEL IN FRACTAL HEAT TRANSFER INVOLVING LOCAL FRACTIONAL DERIVATIVES

Transcription

1 THERMAL SCIENCE, Year 215, Vol. 19, Suppl. 1, pp. S137-S141 S137 ADOMIAN DECOMPOSITION METHOD FOR THREE-DIMENSIONAL DIFFUSION MODEL IN FRACTAL HEAT TRANSFER INVOLVING LOCAL FRACTIONAL DERIVATIVES by Zhi-Ping FAN a, Hassan Kamil JASSIM b,c, Ravinder Krishna RAINA d, and Xiao-Jun YANG e a School of Computer Science and Educational Software, Guangzhou University, Guangzhou, China b Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiryah, Iraq c Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran d Department of Mathematics, College of Technology and Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur, Rajasthan, India e Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, China Original scientific paper DOI: /TSCI15S1S37F The non-differentiable analytical solution of the 3-D diffusion equation in fractal heat transfer is investigated in this article. The Adomian decomposition method is considered in the local fractional operator sense. The obtained result is given to show the sample and efficient features of the presented technique to implement fractal heat transfer problems. Key words: Adomian decomposition method, diffusion equation, fractal heat transfer, local fractional derivative Introduction The theory of local fractional calculus attracts researchers from mathematical physics and engineering applications [1-8]. This interest spans the works of diffusion phenomena with non-differentiability [9-11]. The 3-D diffusion model in fractal heat transfer involving local fractional derivatives (LFD) was presented as [2, 8]: subject to the initial and boundary conditions: 2 Φ( yz,,, τ ) η Φ ( yz,,, τ) = τ (1) Φ ( yz,,,) = f( yz,, ) (2a) Φ (, yz,, τ) =Φ ( ayz,,, τ) = g( yz,, τ) (2b) 1 Corresponding author; dyangiaojun@163.com

2 S138 Fan, Z. P. et al.: Adomian Decomposition Method for Three-Dimensional Diffusion THERMAL SCIENCE, Year 215, Vol. 19, Suppl. 1, pp. S137-S141 Φ (,, z, τ) =Φ ( bz,,, τ) = g ( z,, τ) (2c) 2 Φ (, y,,) t =Φ (, yct,,) = g(, yt,) (2d) where the local fractional Laplace operator is defined as [1, 2, 4-8]: = + + y z (3) η β is a non-differentiable diffusion coefficient, and Φ( yzτ,,, ) is satisfied with the non-differentiable concentration distribution [2, 9]. Recently, the authors [1] suggested the local fractional Adomian decomposition method (LFADM) to consider 1-D diffusion equation on Cantor time-space. Based on it, Yan et al. considered the Laplace equation within the LFD [11]. Baleanu et al. developed non-differential solution to wave equation on Cantor sets within the LFD [12]. The main target of this manuscript to utilize the method to implement the 3-D diffusion model in fractal heat transfer. 3-D diffusion model in fractal heat transfer We first rewrite the problem (1) in the local fractional operator form: ( ) (2 ) (2 ) (2 ) Lτ Φ ( yz,,, τ) = η [ L Φ ( yz,,, τ) + Lyy Φ ( yz,,, τ) + Lzz Φ ( yz,,, τ)] (4) where the local fractional differential operators (see A1 of the Appendi) L L (2 ), (2 ) yy, and are defined by: (2 ) L zz L ( ) τ (.) = (.), τ (2 ) L (2 ) L zz 2 (.) = (.), 2 2 (2 ) L yy Adopting the inverse operator (see A2 of the Appendi) and using the initial condition leads to: 2 ( ), t (.) = (.), 2 y L (.) = (.) (5a,b,c,d) 2 z ( ) ( ) τ τ (,,, τ ) L L Φ yz = ( ) (2 ) (2 ) (2 ) τ yy zz ( ) L τ to both sides of (4) = η L [ Φ Φ ( yz,,, τ) +Φ Φ ( yz,,, τ) + L Φ ( yz,,, τ)] (6) Hence, we get: Φ ( yzτ,,, ) = ( ) (2 ) (2 ) (2 ) t yy zz = η L [ L Φ ( yz,,, t) + L Φ ( yz,,, t) + L Φ ( yz,,, t)] +Φ ( yz,,,) (7) According to the LFADM we decompose the unknown function Φ ( yzτ,,, ) as an infinite series: Substituting (8) into (7) yield: Φ ( yz,,, τ) = Φn ( yz,,, τ) (8) n=

3 THERMAL SCIENCE, Year 215, Vol. 19, Suppl. 1, pp. S137-S141 S139 ( ) (2 ) (2 ) (2 ) n (, y, z,) L L n L yy n L Φ =Φ + η τ Φ + Φ + zz Φn n= n= n= n= (9) The components Φn ( yz,,, τ ), n can be completely determined by using the cursive relationship: Taking Φ ( yz,,, τ ) =Φ ( yz,,,) (1a) 2 ( ) (2 ) (2 ) (2 ) n+ τ η τ n yy n zz n Φ 1 (, y, z, ) = L [ L Φ + L Φ + L Φ ], n (1b) Φ ( yz,,,) = sin ( )cos ( y )cos ( z ) (11a) Φ (, yz,, τ) =Φ (π, yz,, τ) = (11b) Φ (,, z, τ) = Φ (,π, z, τ) = 3 E [ (2 τ) ]sin ( )cos ( z ) (11c) we have: Φ ( y,,, τ) = Φ ( y,,π, τ) = 3 E[ (2 τ) ]sin ( )cos ( y ) (11d) η =.2 (11e) Φ ( yz,,, τ ) = sin ( )cos ( y )cos ( z ) (12a) ( ) (2 ) (2 ) (2 ) n+ 1 τ τ n yy n zz n Φ (, y, z, ) =.2 L [ L Φ + L Φ + L Φ ], n (12b) Consequently, we obtain: Φ ( yz,,, τ ) = sin ( )cos ( y )cos ( z ) (13) ( ) (2 ) (2 ) (2 ) 1 y τ Lτ L T Lyy T Lzz T Φ (,, ) =.2 [ + + ] = 3(.2 τ ) = sin ( )cos ( y )cos ( z ) (14) Γ (1 + ) ( ) (2 ) (2 ) (2 ) 2 y τ Lτ L T1 Lyy T1 Lzz T1 Φ (,, ) =.2 [ + + ] = 2 3(.2 τ ) = sin ( )cos ( y )cos ( z ) (15) Γ (1 + 2 ) ( ) (2 ) (2 ) (2 ) 3 y τ Lτ L T2 Lyy T2 Lzz T2 Φ (,, ) =.2 [ + + ] 3 3(.2 τ ) = sin ( )cos ( y )cos ( z ) (16) Γ (1 + 3 ) and so on. The solution in a non-differentiable series form:

4 S14 Fan, Z. P. et al.: Adomian Decomposition Method for Three-Dimensional Diffusion THERMAL SCIENCE, Year 215, Vol. 19, Suppl. 1, pp. S137-S141 i (.2 τ ) Φ ( yz,,, τ ) = 3sin ( )cos ( y )cos ( z ) ( 1) (17) Γ (1 + i ) is readily obtained. Therefore, the eact solution can be written as: Φ ( yz,,, τ) = 3 E [ (.2 τ) ] sin ( )cos ( y )cos ( z ) (18) Figure 1 shows the eact solution of the 3-D diffusion model in fractal heat transfer when = ln 2 / ln 3, z =, and τ =. i= i Conclusions In this work, the LFADM has been successfully employed to solve the 3-D diffusion model in fractal heat transfer involving LFD. The obtained solution is a nondifferentiable function, which is defined on Cantor function and it discontinuously depend on the LFD. Nomenclature, y, z space co-ordinates, [m] Φ(, y, z, τ) the concentration distribution, [ ] Figure 1. The eact solution of the 3-D diffusion model in fractal heat transfer when = ln2/ln3, and τ = Greek symbols time fractal dimensional order, [ ] τ time, [s] References [1] Yang, X. J., Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong, 211 [2] Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 212 [3] Yang, X. J., et al., Cantor-Type Cylindrical-Coordinate Method for Differential Equations with Local Fractional Derivatives, Physics Letters A, 377 (213), 28-3, pp [4] Yang, X. J., et al., Mathematical Aspects of the Heisenberg Uncertainty Principle within Local Fractional Fourier Analysis, Boundary Value Problems, 213 (213), 1, pp [5] Christianto, V., Rahul, B., A Derivation of Proca Equations on Cantor Sets: a Local Fractional Approach, Bulletin of Mathematical Sciences & Applications, 3 (214), 4, pp [6] Liu, H. Y., et al., Fractional Calculus for Nanoscale Flow and Heat Transfer, International Journal of Numerical Methods for Heat & Fluid Flow, 24 (214), 6, pp [7] Yang, X.-J., et al., Modeling Fractal Waves on Shallow Water Surfaces via Local Fractional Kortewegde Vries Equation, Abstract and Applied Analysis, 214 (214), ID [8] Zhang, Y., et al., On a Local Fractional Wave Equation under Fied Entropy Arising in Fractal Hydrodynamics, Entropy, 16 (214), 12, pp [9] Hao, Y. J., et al., Helmholtz and Diffusion Equations Associated with Local Fractional Derivative Operators Involving the Cantorian and Cantor-Type Cylindrical Coordinates, Advances in Mathematical Physics, 213 (213), ID [1] Yang, X. J., et al., Approimation Solutions for Diffusion Equation on Cantor Time-Space, Proceeding of the Romanian Academy A, 14 (213), 2, pp

5 THERMAL SCIENCE, Year 215, Vol. 19, Suppl. 1, pp. S137-S141 S141 [11] Yan, S. P., et al., Local Fractional Adomian Decomposition and Function Decomposition Methods for Solving Laplace Equation within Local Fractional Operators, Advances in Mathematical Physics, 214 (214), ID [12] Baleanu, D., et al., Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators, Abstract and Applied Analysis, 214 (214), ID Appendi A The local fractional derivative (local fractional differential operator) of Ψ ( ) of order at = is defined as [1, 2, 1-12]: ( ) Ψ Ψ d [ ( ) ( )]) Ψ ( ) ( ) lim = =Ψ = d ( ) where [ Ψ( ) Ψ( )] Γ ( + 1)[ Ψ( ) Ψ ( )]. Its inverse operator (local fractional integral operator) ψ ( ) of order in the interval [ ξζ, ] is given as [1, 2, 1-12]: ζ N 1 ( ) 1 1 ζ ( ) = ( )(d ) = lim ( )( ) (1 ) (1 ) τ j j Γ + Γ + j = (A1) I f ψτ τ f τ τ (A2) where the partitions of the interval [ ξζ, ] are denoted as ( τ j, τ j + 1), with τ j = τ j+ 1 τ j, τ = a, τ N = b, and τ = ma{ τ, τ1,...}, j =,..., N 1. Paper submitted: November 11, 214 Paper revised: February 2, 215 Paper accepted: February 28, 215