Document Type: Research Paper

Authors

1 Department of Mathematics, Islamic Azad University, Najafabad Branch, Najafabad, Iran

2 Department of Mathematics, Yazd University, Yazd, Iran

Abstract

In this paper, the Method of Fundamental Solutions (MFS) is extended to solve some special cases of the problem of transient heat conduction in functionally graded materials. First, the problem is transformed to a heat equation with constant coefficients using a suitable new transformation and then the MFS together with the Tikhonov regularization method is used to solve the resulting equation.

Keywords

Main Subjects

[1] H. S. Carslaw, J. C. Jaeger, Conduction of heat in solids, 2nd ed. London: Oxford University Press; 1959.
[2] C. S. Chen, The method of fundamental solutions for nonlinear thermal explosion, Commun. Numer. Methods Eng. 1995;11:675-81.
[3] C. S. Chen, H. A. Cho, M. A. Golberg, Some comments on the ill-conditioning of the method of fundamental solutions. Eng. Anal. Boundary Elem. 2006;30:405-410.
[4] F. de Monte, Transient heat conduction in one-dimensional composite slab. A natural approach. Int. J. Heat Mass Transf. 2000;43:3607-19.
[5] F. de Monte, Ananalytical approach to the unsteady heat conduction processes in one-dimensional composite media. Int. J. Heat Mass Transf. 2002;26:1333-43.
[6] H. W. Engl, M. Hanke, A. Neubauer, Regularization of inverse problems. Mathematics and its applications, vol. 357. Dordrecht: Kluwer Academic Publishers; 1996.
[7] G. Fairweather, A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math. 1998;9(1-2):69-95.
[8] Z. G. Feng, E. E. Michaelides, The use of modi ed Greens functions in unsteady heat transfer. Int. J. Heat Mass Transf. 1997;40:2997-3002.
[9] A. Friedman, Partial di erential equations of parabolic type. Englewood Cli s, NJ: Prentice-Hall Inc.; 1964
[10] M. A. Golberg, C. S. Chen, The method of fundamental solutions for potential, Hemholtz and diffusion problems. In: Boundary integral methods: numerical and mathematical aspects. Computational engineering, vol. 1. Boston, MA: WIT Press, Computational Mechanics Publications; 1999. p. 103-76.
[11] A. Haji-Sheikh, J. V. Beck, Greens function partitioning in Galerkin-base integral solution of the diffusion equation. J. Heat Transf. 1990;112:28-34.
[12] P. C. Hansen, Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 1994;6:1-35.
[13] P. C. Hansen, Rank-de cient and discrete ill-posed problems. Philadelphia: SIAM; 1998.
[14] S. C. Huang, Y. P. Chang, Heat conduction in unsteady, periodic and steady states in laminated composites. J. Heat Transf. 1980;102:742-8.
[15] B. T. Johansson, D. Lesnic, A method of fundamental solutions for transient heat conduction. Eng. Anal. Boundary Elem. 2008;32:697-703.
[16] B. T. Johansson, D. Lesnic, A method of fundamental solutions for transient heat conduction in layered materials. Eng. Anal. Boundary Elem. 2009;33:1362-67.
[17] A. Karageorghis, G. Fairweather, The method of fundamental solutions for the numerical solution of the biharmonic equation, J. Comput. Phys. 1987;69:434-59.
[18] R. Kress, A. Mohsen, On the simulation source technique for exterior problems in acoustics, Math. Methods Appl. Sci. 1986;8:585-97.
[19] V. D. Kupradze, A method for the approximate solution of limiting problems in mathematical physics. USSR Comput. Math. Math. Phys. 1964;4:199-205.
[20] V. D. Kupradze, M. A. Aleksidze, The method of functional equations for the approximate solution of certain boundary value problems, USSR Comput. Math. Math. Phys. 1964;4:82-126.
[21] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'ceva, Linear and quasilinear equations of parabolic type. Translational of mathematical monographs, vol. 23. Providence, RI: AMS; 1968.
[22] D. Lesnic, The identi cation of piecewise homogeneous properties of rocks. In: Ingham DB, Wrobel LC, editors. Boundary integral formulations for inverse analysis. Southampton: Comput. Mech. Publ.; 1997. p. 237-57 [chapter 9].
[23] R. Mathon, H. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. Numer. Anal. 1977;14(4):638-50.
[24] M. D. Mikhailov, M. N.  Ozisik, M. D. Vulchanov, Di usion in composite layers with automatic solution of thee eigenvalue problem. Int. J. Heat Mass Transf. 1983;26:1131-41.
[25] M. Nili Ahmadabadi, M. Arab, F. M. Maalek Ghaini, The method of fundamental solutions for the inverse space-dependent heat source problem. Eng. Anal. Boundary Elem. 2009;33:1231-35.
[26] P. A. Ramachandran, Method of fundamental solutions: singular value decomposition analysis. Commun. Numer. Methods Eng. 2002;18:789-801.
[27] J. N. Reddy, An introduction to the nite element method. NewYork: McGraw-Hill; 1984.
[28] M. Talha, B. N. Singh, Stochastic perturbation-based nite element for buckling statistics of FGM plates with uncertain material properties in thermal environments, Composite Structures, Volume 108, February 2014, Pages 823-833, ISSN 0263-8223, http://dx.doi.org/10.1016/j.compstruct.2013.10.013.
[29] Z. Tobias, K. Torsten, Functionally graded materials with a soft surface for improved indentation resistance: Layout and corresponding design principles, Computational Materials Science, Volume 86, 15 April 2014, Pages 88-92, ISSN 0927-0256, http://dx.doi.org/10.1016/j.commatsci.2014.01.032.
[30] V. Vodicka, Warmeleitung in geschichteten kugel -und zylinderkorpern. Schweiz Arch 1950;10:297-304.
[31] Y. Yener, M. N.  Ozisik, On the solution of unsteady heat conduction in multi-region nite media with timedependent
heat transfer coefficient. In: Proceedings of the fth international heat transfer conference, vol. 1, JSME, Tokyo; 1974. p. 188-92.
[32] Zakaria Belabed, Mohammed Sid Ahmed Houari, Abdelouahed Tounsi, Mahmoud SR., Anwar Bg O., An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates, Composites Part B: Engineering, Volume 60, April 2014, Pages 274-283, ISSN 1359-8368, http://dx.doi.org/10.1016/j.compositesb.2013.12.057.