Abstract

In this paper we study the problem of determining two thermal parameters of a cylindrical metal sample. This is an inverse problem in heat conduction where boundary conditions are determined on the basis of temperature measurements taken at the selected internal points in the sample. A hybrid method is used to find the parameters based on the experimental data of the temperature of a metalic sample. Both the direct and inverse problems are described and numerical results are given.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 34
Issue: 1
Year: 2021

DOI: 10.12732/ijam.v34i1.6

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References

1. [1] H.E. Kunze, D. La Torre, F. Mendivil, M.R. Galan and R. Zaki, Fractalbased methods and inverse problems for differential equations: current state-of-the-Art, Mathematical Problems in Engineering, Article ID 737694 (2014), 11 pages.
2. [2] Li Peng and al., Reconstruction of shredded paper documents by feature matching, Mathematical Problems in Engineering, Article ID 514748 (2014), 9 pages.
3. [3] G. Evensen, Data Assimilation, The Ensemble Kalman Filte, Springer (2006).
4. [4] M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics, London (1998).
5. [5] A. Kharab, F. Howari and R.B. Guenther, Calculating three thermal coefficients from one data set, Int. J. of Appl. Math., 32, No 1 (2019), 91–100; doi: 10.12732/ijam.v32i1.9.
6. [6] A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications, Springer Science & Business Media, Mathematics and its Appl. (2012).
7. [7] D. Colton, R. Ewing and W. Rundell (Eds.), Inverse Problems in Partial Differential Equations, SIAM, Philadelphia (1990).
8. [8] G. Berryman, Analysis of approximate inverses in tomography. I. Resolution analysis, Optimization and Engineering, 1, No 1 (2000), 87–115.
9. [9] G. De Carvalho and A.J. Silva Neto, An inverse analysis for polymers thermal properties estimation, Proc. 3rd International Conference on Inverse Problems in Engineering: Theory and Practice, Port Ludlow, WA (1999).
10. [10] H.W. Engl, A.K. Louis and W. Rundell (Editors), Inverse Problems in Geophysical Applications, SIAM, Philadelphia (1996).
11. [11] H.W. Engl and J.R. McLaughlin (Eds.), Inverse Problems and Optimal Design in Industry, B.G. Teubner, Stuttgart (1994).
12. [12] H.W. Engl and W. Rundell (Eds.), Inverse Problems in Diffusion Processes, SIAM, Philadelphia (1995).
13. [13] G. Gripenberg and S.O. Londen, Volterra Iintegral and Functional Equations, Cambridge University Press, Cambridge-New York (1990).
14. [14] F. Greensite, A new treatment of the inverse problem of multivariate analysis, Inverse Problems, 18 (2002).
15. [15] R.M. Gulrajani, P. Savard and F.A. Roberge, The inverse problem in electrocardiography: Solutions in terms of equivalent sources. CRC Crit. Rev. Biomed. Eng., 16 (1988), 171–214.
16. [16] R.M. Gulrajani, F.A. Roberge and P. Savard, The Inverse problem of electrocardiography, In: P.W. Macfarlane and T.D. Lawrie Veitch (Eds.): Comprehensive Electrocardiology. Pergamon Press, Oxford, (1989), 237– 288.
17. [17] B.M. Horacek, The forward and inverse problem of electrocardiography, In: Biomedical and Life Physics, Biomedical and Life Physics, Vieweg-Verlag, Braunschweig (1996), 169–172.
18. [18] F. Greensite, The mathematical basis for imaging cardiac electrical function, CRC Crit. Rev. Biomed. Eng., 22 (1994), 347–399.
19. [19] Y. Rudy, The electrocardiogram and its relationship to excitation of the heart, In: N. Sperelakis (Ed.), Physiology and Pathophysiology of the Heart, 3rd Ed., Kluwer Academic Publishers Group, Chapter 11 (1995), 201–239.
20. [20] S.I. Kabanikhin, Inverse and Ill-Posed Problems: Theory and Applications of Inverse and Ill-posed Problems Series, Walter de Gruyter, Berlin (2012).
21. [21] A.G. Megrabov, Forward and Inverse Problems for Hyperboliic, Elliptic, and Mixed Type Equations, Inverse and Ill-Posed Problems Series, VSP, Utrecht (2003).
22. [22] A.G. Ramm, Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, NY (2005).
23. [23] H.T. Banks and H.T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, Boca Raton, FL (2009).
24. [24] I. Graham, U. Langer, J. Melenk and M. Sini, Direct and Inverse Problems in Wave Propagation and Applications, Radon Ser. on Computational and Applied Mathematics (2011).
25. [25] C.A. Richard, B. Borchers, C.H. Thurber, Parameter Estimation and Inverse Problems, Academic Press Inc., 2nd Ed. (2013).