Fractional Cattaneo heat equation in a semi-infinite medium

doi:10.1088/1674-1056/22/1/014401

Chin. Phys. B ›› 2013, Vol. 22 ›› Issue (1): 14401-014401.doi: 10.1088/1674-1056/22/1/014401

• ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS • 上一篇下一篇

Fractional Cattaneo heat equation in a semi-infinite medium

续焕英^a, 齐海涛^{a b}, 蒋晓芸^c

a School of Mathematics and Statistics, Shandong University (Weihai), Weihai 264209, China;
b State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Aerospace Engineering,College of Engineering, Peking University, Beijing 100871, China;
c School of Mathematics, Shandong University, Jinan 250100, China

收稿日期:2012-04-20 修回日期:2012-06-18 出版日期:2012-12-01 发布日期:2012-12-01
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11102102, 11072134, and 91130017), the Natural Science Foundation of Shandong Province, China (Grant No. ZR2009AQ014), and the Independent Innovation Foundation of Shandong University, China (Grant No. 2010ZRJQ002).

Fractional Cattaneo heat equation in a semi-infinite medium

Xu Huan-Ying (续焕英)^a, Qi Hai-Tao (齐海涛)^{a b}, Jiang Xiao-Yun (蒋晓芸)^c

a School of Mathematics and Statistics, Shandong University (Weihai), Weihai 264209, China;
b State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Aerospace Engineering,College of Engineering, Peking University, Beijing 100871, China;
c School of Mathematics, Shandong University, Jinan 250100, China

Received:2012-04-20 Revised:2012-06-18 Online:2012-12-01 Published:2012-12-01
Contact: Qi Hai-Tao E-mail:htqi@sdu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11102102, 11072134, and 91130017), the Natural Science Foundation of Shandong Province, China (Grant No. ZR2009AQ014), and the Independent Innovation Foundation of Shandong University, China (Grant No. 2010ZRJQ002).

摘要/Abstract

摘要： To better describe the phenomenon of non-Fourier heat conduction, the fractional Cattaneo heat equation is introduced from the generalized Cattaneo model with two fractional derivatives of different orders. The anomalous heat conduction under the Neumann boundary condition in a semi-infinity medium is investigated. Exact solutions are obtained in series form of the H-function by using the Laplace transform method. Finally, numerical examples are presented graphically when different kinds of surface temperature gradient are given. The effects of fractional parameters are also discussed.

关键词: Caputo fractional derivative, non-Fourier heat conduction, Cattaneo equation, H-function

Abstract: To better describe the phenomenon of non-Fourier heat conduction, the fractional Cattaneo heat equation is introduced from the generalized Cattaneo model with two fractional derivatives of different orders. The anomalous heat conduction under the Neumann boundary condition in a semi-infinity medium is investigated. Exact solutions are obtained in series form of the H-function by using the Laplace transform method. Finally, numerical examples are presented graphically when different kinds of surface temperature gradient are given. The effects of fractional parameters are also discussed.

Key words: Caputo fractional derivative, non-Fourier heat conduction, Cattaneo equation, H-function

中图分类号: (Heat conduction)

44.10.+i

44.05.+e (Analytical and numerical techniques) 02.30.Gp (Special functions)

续焕英, 齐海涛, 蒋晓芸. Fractional Cattaneo heat equation in a semi-infinite medium[J]. Chin. Phys. B, 2013, 22(1): 14401-014401.

Xu Huan-Ying (续焕英), Qi Hai-Tao (齐海涛), Jiang Xiao-Yun (蒋晓芸). Fractional Cattaneo heat equation in a semi-infinite medium[J]. Chin. Phys. B, 2013, 22(1): 14401-014401.

参考文献 38

[1]	Joseph D D and Prezioso L 1989 Rev. Mod. Phys. 61 41
[2]	Özicsik M N 1993 Heat Conduction (2nd edn.) (New York: John Wiley & Sons)
[3]	Wang L Q, Zhou X S and Wei X H 2008 Heat Conduction (Berlin: Springer)
[4]	Podlubny I 1999 Fractional Differential Equations (New York: Academic Press)
[5]	Hilfer R 2000 Applications of Fractional Calculus in Physics (Singapore: World Scientific)
[6]	Kilbas A A, Srivastava H M and Trujillo J J 2006 Theory and Applications of Fractional Differential Equations (Amsterdam: Elsevier)
[7]	Magin R L 2006 Fractional Calculus in Bioengineering (Connecticut: Begell House Publishers)
[8]	Chen W, Sun H G and Li X C 2010 Fractional Derivative Modelling of Mechanical and Engineering Problems (Beijing: Science Press) (in Chinese)
[9]	Herrmann R 2011 Fractional Calculus: An Introduction for Physicists (Singapore: World Scientific)
[10]	Compte A and Metzler R 1997 J. Phys. A: Math. Gen. 30 7277
[11]	Mainardi F 1996 Appl. Math. Lett. 9 23
[12]	Mainardi F 1996 Chaos Soliton. Fract. 7 1461
[13]	Atanackovic T M, Pilipovic S and Zorica D 2007 J. Phys. A: Math. Theor. 40 5319
[14]	Povstenko Y Z 2011 J. Therm. Stresses 34 97
[15]	Qi H T and Jiang X Y 2011 Physica A 390 1876
[16]	Atanacković T, Konjik S, Oparnica L and Zorica D 2012 Continuum Mech. Thermodyn. 24 293
[17]	Magin R L 2010 Comput. Math. Appl. 59 1586
[18]	Emmanuel S and Berkowitz B 2007 Transp. Porous Med. 67 413
[19]	Valdes-Parada F J, Ochoa-Tapia J A and Alvarez-Ramirez J 2006 Physica A 369 318
[20]	Lewandowska K D and Kosztolowicz T 2008 Acta Phys. Pol. B 39 1211
[21]	Kosztolowicz T and Lewandowska K D 2009 J. Phys. A: Math. Theor. 42 055004
[22]	Luchko Y and Punzi A 2011 Int. J. Geomath. 1 257
[23]	Ghazizadeh H R, Maerefat M and Azimi A 2010 J. Comput. Phys. 229 7042
[24]	Ge H X, Liu Y Q and Cheng R J 2012 Chin. Phys. B 21 010206
[25]	Ghazizadeh H R, Azimi A and Maerefat M 2012 Int. J. Heat Mass Transfer 55 2095
[26]	Jiang R Q 1997 Transient Shock Effect in Thermal Conduction, Quality Diffusion and Moment Transfer (Beijing: Science Press) (in Chinese)
[27]	Yilbas B S 2012 Laser Heating Applications (Amsterdam: Elsevier)
[28]	Wang X and Xu X 2001 Appl. Phys. A 73 107
[29]	Jiang F M, Liu D Y and Zhou J H 2003 Microscale Thermophys. Eng. 6 331
[30]	Xu H Y, Zhang Y C, Song Y Q and Chen D Y 2004 Chin. Phys. 13 1758
[31]	Tao Y J, Huai X L and Li Z G 2006 Chin. Phys. Lett. 23 2487
[32]	Lam T T 2010 Int. J. Therm. Sci. 49 1639
[33]	Xu M Y and Tan W C 2001 Sci. China Ser. A 44 1387
[34]	Xu M Y and Tan W C 2003 Sci. China Ser. G 46 145
[35]	Langlands T A M, Henry B L and Wearne S L 2009 J. Math. Biol. 59 761
[36]	Debnath L and Bhatta D 2007 Integral Transforms and Their Applications (2nd edn.) (Boca Raton: Chapman & Hall/CRC)
[37]	Mathai A M, Saxena R K and Haubold H J 2010 The H-Function: Theory and Applications (Berlin: Springer)
[38]	Xu M Y and Tan W C 2006 Sci. China Ser. G 49 257

Fractional Cattaneo heat equation in a semi-infinite medium

Fractional Cattaneo heat equation in a semi-infinite medium

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