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An inversion problem of estimating parameters for a one-dimensional time fractional thermal wave equation with fractional heat flux conditions and Caputo fractional derivatives is investigated. To begin with, the analytical solution of the direct problem is obtained. Then, based on the parameter sensitivity analysis, the least-squares method is used to estimate both the fractional order α and the relaxation time τ simultaneously. Finally, two different heat flux distributions are given as different boundary conditions to perform the simulation experiments, respectively. By analyzing the degree of fitting curves, results show that the least-squares method performs well in parameter estimation for this fractional thermal wave equation. This study provides an effective method of estimating the parameters of fractional thermal wave equations.
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Keywords:
- fractional thermal wave equation /
- fractional heat flux /
- parameter estimation /
- least-squares method
[1] Xin B G, Chen T, Liu Y Q 2011 Acta Phys. Sin. 60 048901 (in Chinese) [辛宝贵, 陈通, 刘艳芹 2011 60 048901]
[2] Wei T, Luo M K, Hua Y 2013 Acta Phys. Sin. 62 210503 (in Chinese) [蔚涛, 罗懋康, 华云 2013 62 210503]
[3] Rajneesh K, Vandana G 2013 Chin. Phys. B 22 074601
[4] Metzler R, Klafter J 2004 J. Phys. A: Math. Gen. 37 R161
[5] Liu K C, Wang Y N, Chen Y S 2012 Int. J. Therm. Sci. 58 29
[6] Mainardi F 2010 Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (London: Imperial College Press) pp57-77
[7] Dong J P 2011 J. Phys. A: Math. Theor. 44 215204
[8] Yu Y J, Wang Z H 2013 Chin. Phys. Lett. 30 110201
[9] Yang Y Q, Chen Y 2009 Chin. Phys. Lett. 26 100501
[10] Li K T, Zhang W R, Fang H Q, Xie W J, Liu J, Zheng M, Wang X H, Wang W, Tan W C, Cheng H P 2012 Biophys. J. 102 1011
[11] Ahmadikia H, Fazlali R, Moradi A 2012 Int. Commun. Heat Mass Transfer 39 121
[12] Povstenko Y Z 2011 J. Therm. Stresses 34 97
[13] Jiang X Y, Qi H T 2012 J. Phys. A: Math. Theor. 45 485101
[14] Qi H T, Xu H Y, Guo X W 2013 Comput. Math. Appl. 66 824
[15] Murio D A 2008 Comput. Math. Appl. 56 1138
[16] Ghazizadeh H R, Maerefat M, Azimi A 2010 J. Comput. Phys. 229 7042
[17] Ge H X, Cheng R J 2014 Chin. Phys. B 23 040203
[18] Xia L L 2011 Chin. Phys. Lett. 28 120202
[19] Yang G W, Cui K 2005 Chin. Phys. Lett. 22 2738
[20] Chen Z J, Zhang S Y 2010 Chin. Phys. Lett. 27 026502
[21] Guo Y X, Yu Y, Huang H J 2001 Chin. Phys. B 10 1
[22] Mei F X, Chen X W 2000 Chin. Phys. B 9 721
[23] Zhang H B 2002 Chin. Phys. B 11 1
[24] Jiang L Y, Yu Y, Guo Y X 2001 Chin. Phys. B 10 181
[25] Chen Z Y, Huang N N, Li Z G 1994 Chin. Phys. B 3 1
[26] Zeng Y C, Fu Z J, Chen Z 2008 Acta Phys. Sin. 57 46 (in Chinese) [曾以成, 付志坚, 陈争 2008 57 46]
[27] Tong H Q, Gao F 2006 Acta Phys. Sin. 55 577 (in Chinese) [童恒庆, 高飞 2006 55 577]
[28] You Y, Dai D, Ma X K 2002 Acta Phys. Sin. 51 2459 (in Chinese) [尤勇, 戴栋, 马西奎 2002 51 2459]
[29] Peng H P, Li L X, Yang Y X 2007 Acta Phys. Sin. 56 51 (in Chinese) [彭海朋, 李丽香, 杨义先 2007 56 51]
[30] Tian W Y, Li C, Deng W H, Wu Y J 2012 Math. Comput. Simulat. 85 45
[31] Murio D A 2008 Comput. Math. Appl. 56 2371
[32] Zhang Y 2012 Ph. D. Dissertation (Shanghai: Fudan University) (in Chinese) [张瀛 2012 博士学位论文 (上海: 复旦大学)]
[33] Compte A, Metzler R 1997 J. Phys. A: Math. Gen. 30 7277
[34] Podlubny I 1999 Fractional Differential Equations (London: Academic Press) pp78-106
[35] Debnathand L, Bhatta D 2007 Integral Transforms and Their Applications (2nd Ed.) (London: CRC Press) pp407-425
[36] Frankel J I 1996 Computers Math. Applic 32 117
[37] Chen Y N, Xu Z, Zhao S L, Sun Q J, Yin F F, Dong Y H 2010 Acta Phys. Sin. 59 8113 (in Chinese) [陈跃宁, 徐征, 赵谡玲, 孙钦军, 尹飞飞, 董宇航 2010 59 8113]
[38] Lee H L, Lai T H, Chen W L, Yang Y C 2013 Appl. Math. Model. 37 2630
[39] Ghazizadeh H R, Azimi A, Maerefat M 2012 Int. J. Heat Mass Transfer 55 2095
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[1] Xin B G, Chen T, Liu Y Q 2011 Acta Phys. Sin. 60 048901 (in Chinese) [辛宝贵, 陈通, 刘艳芹 2011 60 048901]
[2] Wei T, Luo M K, Hua Y 2013 Acta Phys. Sin. 62 210503 (in Chinese) [蔚涛, 罗懋康, 华云 2013 62 210503]
[3] Rajneesh K, Vandana G 2013 Chin. Phys. B 22 074601
[4] Metzler R, Klafter J 2004 J. Phys. A: Math. Gen. 37 R161
[5] Liu K C, Wang Y N, Chen Y S 2012 Int. J. Therm. Sci. 58 29
[6] Mainardi F 2010 Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (London: Imperial College Press) pp57-77
[7] Dong J P 2011 J. Phys. A: Math. Theor. 44 215204
[8] Yu Y J, Wang Z H 2013 Chin. Phys. Lett. 30 110201
[9] Yang Y Q, Chen Y 2009 Chin. Phys. Lett. 26 100501
[10] Li K T, Zhang W R, Fang H Q, Xie W J, Liu J, Zheng M, Wang X H, Wang W, Tan W C, Cheng H P 2012 Biophys. J. 102 1011
[11] Ahmadikia H, Fazlali R, Moradi A 2012 Int. Commun. Heat Mass Transfer 39 121
[12] Povstenko Y Z 2011 J. Therm. Stresses 34 97
[13] Jiang X Y, Qi H T 2012 J. Phys. A: Math. Theor. 45 485101
[14] Qi H T, Xu H Y, Guo X W 2013 Comput. Math. Appl. 66 824
[15] Murio D A 2008 Comput. Math. Appl. 56 1138
[16] Ghazizadeh H R, Maerefat M, Azimi A 2010 J. Comput. Phys. 229 7042
[17] Ge H X, Cheng R J 2014 Chin. Phys. B 23 040203
[18] Xia L L 2011 Chin. Phys. Lett. 28 120202
[19] Yang G W, Cui K 2005 Chin. Phys. Lett. 22 2738
[20] Chen Z J, Zhang S Y 2010 Chin. Phys. Lett. 27 026502
[21] Guo Y X, Yu Y, Huang H J 2001 Chin. Phys. B 10 1
[22] Mei F X, Chen X W 2000 Chin. Phys. B 9 721
[23] Zhang H B 2002 Chin. Phys. B 11 1
[24] Jiang L Y, Yu Y, Guo Y X 2001 Chin. Phys. B 10 181
[25] Chen Z Y, Huang N N, Li Z G 1994 Chin. Phys. B 3 1
[26] Zeng Y C, Fu Z J, Chen Z 2008 Acta Phys. Sin. 57 46 (in Chinese) [曾以成, 付志坚, 陈争 2008 57 46]
[27] Tong H Q, Gao F 2006 Acta Phys. Sin. 55 577 (in Chinese) [童恒庆, 高飞 2006 55 577]
[28] You Y, Dai D, Ma X K 2002 Acta Phys. Sin. 51 2459 (in Chinese) [尤勇, 戴栋, 马西奎 2002 51 2459]
[29] Peng H P, Li L X, Yang Y X 2007 Acta Phys. Sin. 56 51 (in Chinese) [彭海朋, 李丽香, 杨义先 2007 56 51]
[30] Tian W Y, Li C, Deng W H, Wu Y J 2012 Math. Comput. Simulat. 85 45
[31] Murio D A 2008 Comput. Math. Appl. 56 2371
[32] Zhang Y 2012 Ph. D. Dissertation (Shanghai: Fudan University) (in Chinese) [张瀛 2012 博士学位论文 (上海: 复旦大学)]
[33] Compte A, Metzler R 1997 J. Phys. A: Math. Gen. 30 7277
[34] Podlubny I 1999 Fractional Differential Equations (London: Academic Press) pp78-106
[35] Debnathand L, Bhatta D 2007 Integral Transforms and Their Applications (2nd Ed.) (London: CRC Press) pp407-425
[36] Frankel J I 1996 Computers Math. Applic 32 117
[37] Chen Y N, Xu Z, Zhao S L, Sun Q J, Yin F F, Dong Y H 2010 Acta Phys. Sin. 59 8113 (in Chinese) [陈跃宁, 徐征, 赵谡玲, 孙钦军, 尹飞飞, 董宇航 2010 59 8113]
[38] Lee H L, Lai T H, Chen W L, Yang Y C 2013 Appl. Math. Model. 37 2630
[39] Ghazizadeh H R, Azimi A, Maerefat M 2012 Int. J. Heat Mass Transfer 55 2095
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