Numerical solution of the imprecisely defined inverse heat conduction problem

doi:10.1088/1674-1056/24/5/050203

Abstract This paper investigates the numerical solution of the uncertain inverse heat conduction problem. Uncertainties present in the system parameters are modelled through triangular convex normalized fuzzy sets. In the solution process, double parametric forms of fuzzy numbers are used with the variational iteration method (VIM). This problem first computes the uncertain temperature distribution in the domain. Next, when the uncertain temperature measurements in the domain are known, the functions describing the uncertain temperature and heat flux on the boundary are reconstructed. Related example problems are solved using the present procedure. We have also compared the present results with those in [Inf. Sci. (2008) 178 1917] along with homotopy perturbation method (HPM) and [Int. Commun. Heat Mass Transfer (2012) 39 30] in the special cases to demonstrate the validity and applicability.

Keywords: triangular fuzzy number double parametric form of fuzzy numbers uncertain inverse heat conduction variational iteration method (VIM)

Received: 21 August 2014
Revised: 25 November 2014
Accepted manuscript online:

PACS:	02.60.Cb	(Numerical simulation; solution of equations)
	02.60.-x	(Numerical approximation and analysis)
	02.90.+p	(Other topics in mathematical methods in physics)
	07.05.Mh	(Neural networks, fuzzy logic, artificial intelligence)

Corresponding Authors: S. Chakraverty
E-mail: sne_chak@yahoo.com

About author: 02.60.Cb; 02.60.-x; 02.90.+p; 07.05.Mh

Cite this article:

Smita Tapaswini, S. Chakraverty, Diptiranjan Behera Numerical solution of the imprecisely defined inverse heat conduction problem 2015 Chin. Phys. B 24 050203

[1]	Hetmaniok E, Nowak I, Slota D and Witula R 2012 Int. Commun. Heat Mass Transfer 39 30
[2]	Slota D 2011 Numer. Heat Transfer Part A 59 755
[3]	Chen H T and Wu X Y 2007 Numer. Heat Transfer Part B 51 159
[4]	Cialkowski M J, Frackowiak A and Grysa K 2007 Int. J. Heat Mass Transfer 50 217
[5]	Deng S and Hwang Y 2007 Int. J. Heat Mass Transfer 50 2089
[6]	Frackowiak A, Botkin N D, Cialkowski M and Hoffmann K H 2010 Int. J. Heat Mass Transfer 53 2123
[7]	Grvsa K and Lesniewska R 2010 Inverse Problems Sci. Eng. 18 3
[8]	Grzymkowski R and Slota D 2006 Comput. Math. Appl. 51 3
[9]	Monde M, Arima H, Liu W, Mitsutake Y and Hammad J 2003 Heat Transfer-Asian Research 32 618
[10]	Wroblewska A, Frackowiak A and Cialkowski M J 2008 Arch. Thermodyn. 1 1
[11]	Woodfielda P L, Mondeb M and Mitsutakeb Y 2006 Int. J. Heat Mass Transfer 49 187
[12]	Fernandes A P, Sousa P F B, Borges V L and Guimaraes G 2010 Appl. Math. Model 34 4040
[13]	Kameli H and Kowsar F 2012 Int. Commun. Heat Mass Transfer 39 1410
[14]	Johansson B T Lesnic D and Reeve T 2012 Int. Commun. Heat Mass Transfer 39 887
[15]	Jiang M and Wang M 2013 Chin. Phys. Lett. 30 010201
[16]	Xia L 2011 Chin. Phys. Lett. 28 120202
[17]	Chen J and Zhang Y 2010 Chin. Phys. Lett. 27 026502
[18]	WangQ H, Li Z H, Lai J C and He A Z 2007 Chin. Phys. Lett. 24 107
[19]	Cui K and Yang W 2005 Chin. Phys. Lett. 22 2738
[20]	Ding L, Liu P G, He J G, Zakaria A and LoVetri J 2014 Acta Phys. Sin. 63 044102 (in Chinese)
[21]	MaY M, Lü B, Liu Y, Qia L and Pei 2013 Acta Phys. Sin. 62 204202 (in Chinese)
[22]	Cao X Q, Song J Q, Zhang W M, Zhao Y L and Liu B N 2013 Acta Phys. Sin. 62 170504 (in Chinese)
[23]	Zhang P and Zhang 2013 Acta Phys. Sin. 62 164201 (in Chinese)
[24]	Gu T, Rao H and Zhang Q 2013 Acta Phys. Sin. 62 169501 (in Chinese)
[25]	Wang Q and Wang M 2014 Chin. Phys. B 23 028703
[26]	Cui C, Liu X and Duan S 2013 Chin. Phys. 22 104501
[27]	Xu W L, Zhang J C, Li X, Xu B Q and Tao G S 2013 Chin. Phys. B 22 010203
[28]	Wang F and Zhang R 2012 Chin. Phys. B 21 050204
[29]	Li L andZhang Q 2011 Chin. Phys. B 20 100201
[30]	Huang C H and Tsai C C 1998 Heat Mass Transfer 34 47
[31]	Fan C L, Sun F R and Yang L 2009 Inverse Prob. Sci. Eng. 17 885
[32]	Ozisik M N and Oriande H R B 2000 Inverse Heat Transfer: Fundamentals and Applications (New York: Taylor & Francis)
[33]	Chang S L and Zadeh L A 1972 IEEE Trans. Syst. Man. Cyber. 2 30
[34]	Dubois D and Prade H 1982 Fuzzy Sets Syst. 8 225
[35]	Kaleva O 1990 Fuzzy Sets Syst. 35 389
[36]	Seikkala S 1987 Fuzzy Sets Syst. 24 319
[37]	Bede B 2008 Inf. Sci. 178 1917
[38]	Ma M, Friedman M and Kandel A 1999 Fuzzy Sets Sys. 105 13
[39]	Mikaeilvand N and Khakrangin S 2012 Neural Comput. Applic. 21 S307
[40]	Akin Ö, Khaniyev T, Oruç Ö and Türksen I B 2013 Expert Syst. Appl. 40 953
[41]	Khastan A, Nieto J J and Rodriguez-Lopez R 2011 Fuzzy Sets Syst. 17 20
[42]	Chalco-Cano Y and Roman-Flores H 2008 Chaos Soliton. Frac. 38 112119
[43]	Behera D and Chakraverty S 2014 Annals Fuzzy Math. Inf. 7 401
[44]	Tapaswini S and Chakraverty S 2014 Fire Safety J. 66 8
[45]	Tapaswini S and Chakraverty S 2014 The Scientific World J. 2014 1
[46]	He J H 1999 Int. J. Nonlinear Mech. 34 699
[47]	He J H 2000 Appl. Math. Comput. 34 115
[48]	Abbaoui K and Cherruault Y 1995 Comput. Math. Appl. 29 103
[49]	Maidi A and Corriou J P 2013 Appl. Math. Comput. 219 8632
[50]	Wazwaz A M 2009 Appl. Math. Comput. 212 120
[51]	Huang Y J and Liu H K 2013 Appl. Math. Model. 378 118
[52]	Mosayebidorcheh S and Mosayebidorche T 2012 Int. J. Heat Mass Transfer 55 6589
[53]	Miansari M, Ganji D D and Miansari M 2008 Phys. Lett. A 372 779
[54]	Ganji D D, Afrouzi G A and Talarposhti R A 2007 Phys. Lett. A 368 450
[55]	Khaleghi H, Ganji D D and Sadighi A 2007 Num. Heat Transfer Part A 52 25
[56]	Hanss M 2005 Applied Fuzzy Arithmetic: An Introduction with Engineering Applications (Berlin: Springer-Verlag)
[57]	Jaulin L, Kieffer M, Didrit O and Walter E 2001 Applied Interval Analysis (France: Springer)
[58]	Ross T J 2004 Fuzzy Logic with Engineering Applications (New York: John Wiley & Sons)
[59]	Zimmermann H J 2001 Fuzzy Set Theory and its Application (London: Kluwer Academic Publishers)
[60]	Chakraverty S 2014 Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems (USA: IGI Global Publication)

[1]	Bidirectional visible light absorber based on nanodisk arrays Qi Wang(王琦), Fei-Fan Zhu(朱非凡), Rui Li(李瑞), Shi-Jie Zhang(张世杰), and Da-Wei Zhang(张大伟). Chin. Phys. B, 2023, 32(3): 030205.
[2]	Adaptive multi-step piecewise interpolation reproducing kernel method for solving the nonlinear time-fractional partial differential equation arising from financial economics Ming-Jing Du(杜明婧), Bao-Jun Sun(孙宝军), and Ge Kai(凯歌). Chin. Phys. B, 2023, 32(3): 030202.
[3]	Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems Beibei Zhu(朱贝贝), Lun Ji(纪伦), Aiqing Zhu(祝爱卿), and Yifa Tang(唐贻发). Chin. Phys. B, 2023, 32(2): 020204.
[4]	Comparison of differential evolution, particle swarm optimization, quantum-behaved particle swarm optimization, and quantum evolutionary algorithm for preparation of quantum states Xin Cheng(程鑫), Xiu-Juan Lu(鲁秀娟), Ya-Nan Liu(刘亚楠), and Sen Kuang(匡森). Chin. Phys. B, 2023, 32(2): 020202.
[5]	Characteristics of piecewise linear symmetric tri-stable stochastic resonance system and its application under different noises Gang Zhang(张刚), Yu-Jie Zeng(曾玉洁), and Zhong-Jun Jiang(蒋忠均). Chin. Phys. B, 2022, 31(8): 080502.
[6]	Substitutions of vertex configuration of Ammann-Beenker tiling in framework of Ammann lines Jia-Rong Ye(叶家容), Wei-Shen Huang(黄伟深), and Xiu-Jun Fu(傅秀军). Chin. Phys. B, 2022, 31(8): 086101.
[7]	Research and application of stochastic resonance in quad-stable potential system Li-Fang He(贺利芳), Qiu-Ling Liu(刘秋玲), and Tian-Qi Zhang(张天骐). Chin. Phys. B, 2022, 31(7): 070503.
[8]	Most probable transition paths in eutrophicated lake ecosystem under Gaussian white noise and periodic force Jinlian Jiang(姜金连), Wei Xu(徐伟), Ping Han(韩平), and Lizhi Niu(牛立志). Chin. Phys. B, 2022, 31(6): 060203.
[9]	Coupled flow and heat transfer of power-law nanofluids on non-isothermal rough rotary disk subjected to magnetic field Yun-Xian Pei(裴云仙), Xue-Lan Zhang(张雪岚), Lian-Cun Zheng(郑连存), and Xin-Zi Wang(王鑫子). Chin. Phys. B, 2022, 31(6): 064402.
[10]	Data-driven parity-time-symmetric vector rogue wave solutions of multi-component nonlinear Schrödinger equation Li-Jun Chang(常莉君), Yi-Fan Mo(莫一凡), Li-Ming Ling(凌黎明), and De-Lu Zeng(曾德炉). Chin. Phys. B, 2022, 31(6): 060201.
[11]	Correlation and trust mechanism-based rumor propagation model in complex social networks Xian-Li Sun(孙先莉), You-Guo Wang(王友国), and Lin-Qing Cang(仓林青). Chin. Phys. B, 2022, 31(5): 050202.
[12]	Anti-function solution of uniaxial anisotropic Stoner-Wohlfarth model Kun Zheng(郑坤), Yu Miao(缪宇), Tong Li(李通), Shuang-Long Yang(杨双龙), Li Xi(席力), Yang Yang(杨洋), Dun Zhao(赵敦), and De-Sheng Xue(薛德胜). Chin. Phys. B, 2022, 31(4): 040202.
[13]	Ultra-broadband absorber based on cascaded nanodisk arrays Qi Wang(王琦), Rui Li(李瑞), Xu-Feng Gao(高旭峰), Shi-Jie Zhang(张世杰), Rui-Jin Hong(洪瑞金), Bang-Lian Xu(徐邦联), and Da-Wei Zhang(张大伟). Chin. Phys. B, 2022, 31(4): 040203.
[14]	A new algorithm for reconstructing the three-dimensional flow field of the oceanic mesoscale eddy Chao Yan(颜超), Jing Feng(冯径), Ping-Lv Yang(杨平吕), and Si-Xun Huang(黄思训). Chin. Phys. B, 2021, 30(12): 120204.
[15]	Novel energy dissipative method on the adaptive spatial discretization for the Allen-Cahn equation Jing-Wei Sun(孙竟巍), Xu Qian(钱旭), Hong Zhang(张弘), and Song-He Song(宋松和). Chin. Phys. B, 2021, 30(7): 070201.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Numerical solution of the imprecisely defined inverse heat conduction problem

Cite this article:

Online attention