A NEW TECHNOLOGY FOR SOLVING DIFFUSION AND HEAT EQUATIONS

Transcription

1 THERMAL SCIENCE: Year 7, Vol., No. A, pp A NEW TECHNOLOGY FOR SOLVING DIFFUSION AND HEAT EQUATIONS by Xiao-Jun YANG a and Feng GAO a,b * a School of Mechanics and Civil Engineering, China Universiy of Mining and Technology, Xuzhou, China b Sae Key Laboraory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China Universiy of Mining and Technology, Xuzhou, China Original scienific paper DOI:.98/TSCI6446Y In his paper, a new echnology combing he variaional ieraive mehod and an inegral ransform similar o Sumudu ransform is proposed for he firs ime for soluions of diffusion and hea equaions. The mehod is accurae and efficien in developmen of approimae soluions for he parial differenial equaions. Key words: hea ransfer, diffusion equaion, hea equaion, inegral ransform, approimae soluion Inroducion The classical hea conducion has been discussed in he lieraure more han in wo cenuries wih ineger [-3] of non-local fracional [4-6], and local fracional [7-9] operaors. Analyical soluions are classical approach in soluion ransien hea conducion problems [-3] and recenly efficien approimae echniques o non-linear (wih non-linear diffusion coefficiens were developed [-]. The variaional ieraion mehod (VIM proposed by He [3] was used widely o solve linear and non-linear hea-ransfer problems [4] boh direc and inverse []. This idea of his echnology was eended o a soluion mehodology compromising is basic idea wih inegral ransform mehods, such Laplace [5] and Sumudu [6]. In he cone, of he inegral ransforms applied, we menion a new inegral ransform resembling he Sumudu ransform (see more deails in [7] mainly addressed o soluion of hea conducion problems. The presen paper addresses a new echnology combining he VIM and an inegral ransform similar o he Sumudu ransform o solve diffusion and hea equaions. Mahemaical models Following he Fourier (Fick law, he rae hea energy per uni area, i. e. he hea flu q( yz,,, is proporional o he gradien ( yz,,, [] wih he hermal conduciviy (TC κ as a ranspor coefficien: q( yz,,, =κ yz,,, (a ( * Corresponding auhor; jsppw@sohu.com

2 34 THERMAL SCIENCE: Year 7, Vol., No. A, pp or equivalenly: ( yz,,, ( yz,,, ( yz,,, q( yz,,, κ i j = + + k y z (b In he case of a volumeric hea source g(, y, z, he 3-D version of he model ( is []: ( yz,,, κ ( yz,,, ρc = g( yz,,, (a or equivalenly: ( yz,,, ( yz,,, ( yz,,, ( yz,,, κ + + c g ρ = ( yz,,, y z (b where ρ is he mass densiy (MD, and c is he specific hea capaciy (SHC. Applying he energy balance and coninuiy equaions he 3-D diffusion equaion of hea reads [3]: ( yz,,, α ( yz,,, = (3a or equivalenly: ( yz,,, ( yz,,, ( yz,,, ( yz,,, α + + = y z (3b where α is he hermal diffusiviy (TD. The -D version of eq. (a is: (, (, α = h (,, (, (, L (, (4a and he diffusion equaion in -D conducion hea ransfer from eq. (3a: (, (, g(, α =, (, (, L (,, h(, = ρc (4b subjeced o iniial and boundary condiions: (, θ( (, = ϒ ( ( L, = (4c Analysis of he mehod compromising VIM and inegral ransform For he seek of clariy in he eplanaions, we consider he following differenial equaion in he operaor form: = ϒ, ( (4d (4e

3 THERMAL SCIENCE: Year 7, Vol., No. A, pp where Λ = / and Ξ= α /. Applying he VIM [3], he funcional reads: n n n n Λ +Ξ = h (5 + (, = (, + λ( τ Λ (, τ Ξ (, τ h(, τ dτ (6 Furher, applying he inegral ransform o eq. (6 we ge: n+ (, = n(, + Y λ( τ Λ n(, τ Ξn(, τ h(, τ dτ = { } (, Y{ ( } Y (, (, h(, = n + λ Λn Ξn Considering he variaion of eq. (7 wih respec o ( n,, we have: { } ( ( { ( } ( ( h( δn+, = δn, + δ Y λ Y n, n,, Λ Ξ = From he inermediae resul of eq. (8 we receive ha: { } ( { ( } ( ( h( = + λ ( δ{ Λ n (, } = δn+, = + Y λ δ Y Λn, Ξn,, = Thus, finally one obains: = + λ ( δ n(, n(, = = + λ ( = = λ ( (7 (8 (9 = ( Therefore, we developed an ieraion algorihm applying he inegral operaor from eqs.(6 and (, namely: ( ( { ( } ( h( n+, = n, + Y Λ n, + Ξn,, ( Consequenly, eq. ( allows obaining he inegral ransform soluion in he form: (, = lim n (, ( n which reduces o: { n } (, Y lim (, = (3 n Eamples of approimae soluions for diffusion and hea problems Eample Le us consider he -D diffusion eq. (4b wih he iniial-boundary value condiions:

4 36 THERMAL SCIENCE: Year 7, Vol., No. A, pp ( ( L,, (, ep (, = = ep ( α, = ep L ep α ( ( (4a,b,c From eq. (, we can consruc he following ieraive algorihm wih he inegral operaor: (, ( ( ( n n, n+, = n, Y + α (5a subjeced o he iniial value condiion: Thus, we have: ( { ( } (, = Y, = ep (5b ( ( (, = ep + α (5c (, ep( ( α α 3 3 ( ( ( = + + (5d 3, = ep + α + α + α (5e and so on. Therefore, he inegral ransform soluion for he diffusion eq. (4b wih he iniial-boundary value condiions in eqs. (4a,b,c reads: { n } (, Y lim (, = = { ( ( α α α } 3 3 Y ep... = ep n = = ( ep( α The 3-D graphs corresponding o he cases α =, α =, α = 3, and α = 4, are displayed in fig. -4, respecively. Eample As he second eample, we consider he hea eq. (4a wih he iniial-boundary value condiions: (, = ep (, (, ( L, = ep ( α, = ep( L ep( α (7a,b,c where h(, =. The ieraion algorihm wih he inegral operaor reads: (, (, n n n+ (, = n(, Y + α subjeced o he iniial value condiion: ( { ( } ( (6 (8a, = Y, = ep (8b

5 THERMAL SCIENCE: Year 7, Vol., No. A, pp α = α = 6 5 (, (, Figure. The approimae soluion of he diffusion equaion for he TD α = (, Figure 3. The approimae soluion of he diffusion equaion for he TD α = From eqs. (8a,b, we have he following (up o n = as illusraive eample: ( ( (, = ep + α (8c ( ( (, ep α α = + + (8d ( ( ( 3, ep α α α 3 3 = (8e Therefore, he inegral ransform soluion for he hea eq. (4a wih he iniial-boundary condiions (7a,b,c is: { n } (, Y lim (, = = n { ( ( α α α } Y ep... α = = = ( ep( α = ep The corresponding 3-D plos for he cases α =, α =, α = 3, and α = 4, are presened in figs. 5-8, respecively..5 Figure. The approimae soluion of he diffusion equaion for he TD α = (, α = 4 Figure 4. The approimae soluion of he diffusion equaion for he TD α = 4 (9

6 38 THERMAL SCIENCE: Year 7, Vol., No. A, pp α = α = 6 5 (, (, Figure 5. The approimae soluion of he diffusion equaion for he TD α =.5.5 Figure 6. The approimae soluion of he diffusion equaion for he TD α =.5 (, Figure 7. The approimae soluion of he diffusion equaion for he TD α = 3.5 α = 3 (, α = Figure 8. The approimae soluion of he diffusion equaion for he TD α = 4 Conclusion The addressed an approimae soluion mehod compromising he VIM and an inegral ransform similar o Sumudu ransform. The soluion echnology was eemplified by soluions of -D ransien hea conducion. The proposed mehod is accurae and efficien and allows sraigh forwardly develop approimae soluions for he hea-ransfer equaions by conducion..5 Nomenclaure c he SHC, [Jkg K ] ime, [s] space co-ordinae, [m] Greek symbols α he TD, [m s ] κ he TC, [Wm K ] ρ he MD, [kgm 3 ] ϕ(, emperaure, [K] ( = Y τ ( inegral ransform, [ ]

7 THERMAL SCIENCE: Year 7, Vol., No. A, pp Acknowledgemen This work is suppored by he Sae Key Research Developmen Program of he People s Republic of China (Gran No. 6YFC675 and he Prioriy Academic Program Developmen of Jiangsu Higher Educaion Insiuions (PAPD4. References [] Carslaw, H. S., Jaeger, J. C., Conducion of Hea in Solids, Clarendon Press, Oford, UK, 959 [] Ozisik, M. N., Hea Conducion, John Wiley and Sons, New York, USA, 993 [3] Incropera, F. P., Dewi, D. P., Inroducion o Hea Transfer, John Wiley and Sons, New York, USA 99 [4] Povsenko, Y. Z. Fracional Hea Conducion Equaion and Associaed Thermal Sress, Journal of Thermal Sresses, 8 (4,, pp. 83- [5] Khalil, H., Khan, R. A., A New Mehod Based on Legendre Polynomials for Soluions of he Fracional Two-Dimensional Hea Conducion Equaion, Compuers & Mahemaics wih Applicaions, 67 (4,, pp [6] Tarasov, V. E., Hea Transfer in Fracal Maerials, Inernaional Journal of Hea and Mass Transfer, 93 (6,, pp [7] Yang, X. J., e al., Local Fracional Homoopy Perurbaion Mehod for Solving Fracal Parial Differenial Equaions Arising in Mahemaical Physics, Romanian Repors in Physics, 67 (5, 3, pp [8] Zhang, Y., e al., Local Fracional Variaional Ieraion Algorihm II for Non-Homogeneous Model Associaed wih he Non-Differeniable Hea Flow, Advances in Mechanical Engineering, 7 (5,, pp. -7 [9] Yang, X.-J., e al., Canor-Type Cylindrical-Coordinae Mehod for Differenial Equaions wih Local Fracional Derivaives, Physics Leers A, 377 (3, 8, pp [] Hrisov, J., An Approimae Analyical (Inegral-Balance Soluion o a Nonlinear Hea Diffusion Equaion, Thermal Science, 9 (5,, pp [] Galakionov, V. A., On New Eac Blow-Up Soluions for Nonlinear Hea Conducion Equaions wih Source and Applicaions, Differenial and Inegral Equaions, 3 (99, 5, pp [] Geng, F., Lin, Y., Applicaion of he Variaional Ieraion Mehod o Inverse Hea Source Problems, Compuers & Mahemaics wih Applicaions, 58 (9,, pp. 98- [3] He, J., Variaional Ieraion Mehod for Delay Differenial Equaions, Communicaions in Nonlinear Science and Numerical Simulaion, (997, 4, pp [4] Ganji, D. D., e al., Applicaion of Variaional Ieraion Mehod and Homoopy Perurbaion Mehod for Nonlinear Hea Diffusion and Hea Transfer Equaions, Physics Leers A, 368 (7, 6, pp [5] Hesameddini, E., Laifizadeh, H., Reconsrucion of Variaional Ieraion Algorihms Using he Laplace Transform, Inernaional Journal of Nonlinear Sciences and Numerical Simulaion, (9, -, pp [6] Goswami, P., Alqahani, R. T., Soluions of Fracional Differenial Equaions by Sumudu Transform and Variaional Ieraion Mehod, Journal of Nonlinear Science and Applicaions, 9 (6, 3, pp [7] Yang, X.-J., A New Inegral Transform Mehod for Solving Seady Hea-Transfer Problem, Thermal Science, (6, Suppl. 3, pp. S639-S64 Appendi The inegral ransform of he funcion ϕ( is defined by [7]: ( ( ( = Y τ = τ e d τ, τ> (A The epression (A suggess ha he inegral eiss for some, where θ ( τ,τ and Y is he inegral ransform. The inverse inegral ransform is given by [7]: τ { ( ( } { ( } ( Y Y = Y = (A

8 4 THERMAL SCIENCE: Year 7, Vol., No. A, pp The properies of he inegral ransform are briefly oulined [7]: (R Suppose ha ( =Y[ ( τ ] and ( =Y[ ( τ ]. Then, we have: ( + ( ( + ( Y a τ b τ = a b (A3 where a and b are consans. (R Suppose ha ( = Y[ τ ( ] and he derivaive of ϕ(τ is ϕ ( (τ. Then, we have: ( ( ( ( Y τ = (R3 Suppose ha ( Y[ τ ( ] = and he inegral of ϕ(τ is τ ( τ dτ. Then, we have: τ Y ( τ d τ = ( (A5 (R4 Le μ be a consan. Then we have: (R5 (R6 Y ep ( µ Y= [ ] = µ [ τ] Y = (A4 (A6 (A7 (A8 Paper submied: April, 6 Paper revised: May, 6 Paper acceped: June, 6 7 Sociey of Thermal Engineers of Serbia Published by he Vinča Insiue of Nuclear Sciences, Belgrade, Serbia. This is an open access aricle disribued under he CC BY-NC-ND 4. erms and condiions