Time fractional dual-phase-lag heat conduction equation

doi:10.1088/1674-1056/24/3/034401

Abstract We build a fractional dual-phase-lag model and the corresponding bioheat transfer equation, which we use to interpret the experiment results for processed meat that have been explained by applying the hyperbolic conduction. Analytical solutions expressed by H-functions are obtained by using the Laplace and Fourier transforms method. The inverse fractional dual-phase-lag heat conduction problem for the simultaneous estimation of two relaxation times and orders of fractionality is solved by applying the nonlinear least-square method. The estimated model parameters are given. Finally, the measured and the calculated temperatures versus time are compared and discussed. Some numerical examples are also given and discussed.

Keywords: fractional dual-phase-lag model thermal wave non-Fourier heat conduction analytical solution

Received: 24 July 2014
Revised: 02 October 2014
Accepted manuscript online:

PACS:	44.10.+i	(Heat conduction)
	44.05.+e	(Analytical and numerical techniques)
	45.10.Hj	(Perturbation and fractional calculus methods)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11102102, 11472161, and 91130017), the Natural Science Foundation of Shandong Province, China (Grant No. ZR2014AQ015), and the Independent Innovation Foundation of Shandong University, China (Grant No. 2013ZRYQ002).

Corresponding Authors: Jiang Xiao-Yun
E-mail: wqjxyf@sdu.edu.cn

Cite this article:

Xu Huan-Ying (续焕英), Jiang Xiao-Yun (蒋晓芸) Time fractional dual-phase-lag heat conduction equation 2015 Chin. Phys. B 24 034401

[1]	Wang L Q, Zhou X S and Wei X H 2008 Heat Conduction (Berlin: Springer)
[2]	Tzou D Y 1995 ASME J. Heat Transfer 117 8
[3]	Tzou D Y 1997 Macro- to Micro-scale Heat Transfer: the Lagging Behavior (New York: Taylor & Francis)
[4]	Shen B and Zhang P 2008 Int. J. Heat Mass Transfer 51 1713
[5]	Xu F, Lu T J Seffen K A and Ng E Y K 2009 Appl. Mech. Rev. 62 050801
[6]	Tzou D Y and Chiu K S 2001 Int. J. Heat Mass Transfer 44 1725
[7]	Liu K C 2005 J. Phys. D: Appl. Phys. 38 3722
[8]	Mitra K, Kumar S, Vedavarz A and Moallemi M K 1995 ASME J. Heat Transfer 117 568
[9]	Antaki P J 2005 ASME J. Heat Transfer 127 189
[10]	Podlubny I 1999 Fractional Differential Equations (New York: Academic Press)
[11]	Hilfer R 2000 Applications of Fractional Calculus in Physics (Singapore: World Scientific)
[12]	Magin R L 2006 Fractional Calculus in Bioengineering (Connecticut: Begell House Publishers)
[13]	Chen X, Kang J H, Fu C J and Tan W C 2013 PLOS ONE 8 e57093
[14]	Diao L J, Zhang X F and Chen D Y 2014 Acta Phys. Sin. 63 038401 (in Chinese)
[15]	Magin R L 2010 Comput. Math. Appl. 59 1586
[16]	Suki B, Barabási A L and Lutchen K R 1994 J. Appl. Physiol. 76 2749
[17]	Chen Q, Suki B and An K N 2004 J. Biomech Eng. 126 666
[18]	Kiss M Z, Varghese T and Hall T J 2004 Phys. Med. Biol. 49 4207
[19]	Kohandel M, Sivaloganathan S, Tenti G and Darvish K 2005 Phys. Med. Biol. 50 2799
[20]	Craiem D and Armentano R 2007 Biorheology 44 251263
[21]	Chen M, Jia L B and Yin X Z 2011 Chin. Phys. Lett. 28 088703
[22]	Vosika Z B, Lazovic G M, Misevic G N and Simic-Krstic J B 2013 PLoS ONE 8 e59483
[23]	Compte A and Metzler R 1997 J. Phys. A: Math. Gen. 30 7277
[24]	Povstenko Y Z 2005 J. Therm. Stresses 28 83
[25]	Jiang X Y and Xu M Y 2010 Physica A 389 3368
[26]	Povstenko Y Z 2011 J. Therm. Stresses 34 97
[27]	Ezzat M A 2011 Int. J. Therm Stress 50 449
[28]	Jiang X Y and Qi H T 2012 J. Phys. A: Math. Theor. 45 485101
[29]	Jiang H, Liu F, Turner I and Burrage K 2012 J. Math. Anal. Appl. 389 1117
[30]	Wang Q F, Dai B D and Li Z F 2013 Chin. Phys. B 22 080203
[31]	Li Q H, Chen S S and Zeng J H 2013 Chin. Phys. B 22 120204
[32]	Ge H X and Cheng R J 2014 Chin. Phys. B 23 040203
[33]	Wang Y, Ouyang J and Yang B X 2010 Acta Phys. Sin. 59 6757 (in Chinese)
[34]	Qi H T and Jiang X Y 2011 Physica A 390 1876
[35]	Xu H Y, Qi H T and Jiang X Y 2013 Chin. Phys. B 22 014401
[36]	Xu H Y, Qi H T and Guo X W 2013 Comput. Math. Appl. 66 824
[37]	Qi H T and Guo X W 2014 Int. J. Heat Mass Transfer 76 535
[38]	Atanacković T M, Pilipović S, Stanković B and Zorica D 2014 Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes (London: ISTE-Wiley) p. 257
[39]	Fan W P and Jiang X Y 2014 Acta Phys. Sin. 63 140202 (in Chinese)
[40]	Ghazizadeh H R, Azimi A and Maerefat M 2012 Int. J. Heat Mass Transfer 55 2095
[41]	Sun H G, Sheng H, Chen Y Q, Chen W and Yu Z B 2013 Chin. Phys. Lett. 30 046601
[42]	Ezzat M A, El-Karamanyb A S and Ezzat S M 2012 Nucl. Eng. Des. 252 267
[43]	Ezzat M A, El-Karamany A S and Fayik M A 2012 Arch. Appl. Mech. 82 557
[44]	Ezzat M A, El-Bary A A and Fayik M A 2013 Mech. Adv. Mater. Struc. 20 593
[45]	Xu M Y and Tan W C 2006 Sci. China Ser. G 49 257
[46]	Klafter J and Sokolov I M 2005 Phys. World 18 29
[47]	Li H B and Li Z 2010 Chin. Phys. B 19 054401
[48]	Ghazizadeh H R, Maerefat M and Azimi A 2010 J. Comput. Phys. 229 7042
[49]	Pennes H H 1948 J. Appl. Physiol. l 93
[50]	Mathai A M, Saxena R K and Haubold H J 2010 The H-Function: Theory and Applications (Berlin: Springer)

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