2013
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Commuting $pi$regular rings
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2
R is called commuting regular ring (resp. semigroup) if for each x,y $in$ R thereexists a $in$ R such that xy = yxayx. In this paper, we introduce the concept of commuting$pi$regular rings (resp. semigroups) and study various properties of them.
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67
70


Sh
Sahebi
Department of Mathematics, Faculty of Science, Islamic Azad University,
Central Tehran Branch, PO. Code 1416894351,
Tehran, Iran
Department of Mathematics, Faculty of Science,
Iran


M
Azadi
Department of Mathematics, Faculty of Science, Islamic Azad University,
Central Tehran Branch, PO. Code 1416894351,
Tehran, Iran
Department of Mathematics, Faculty of Science,
Iran
Regular
Commuting $pi$regular
[[1] M. Azadi, H. Doostie, L. Pourfaraj, Certain rings and semigroups examining the regularity property, Journal##of mathematics, statistics and allied elds., 29(2008), 1:16.##[2] J. W. Fisher, R. L. Snider, On the Von Neumann regularity of rings with regular prime factor rings, Pacic##J. Math., 54(1974),1: 135144.##[3] H. Doostie, L. Pourfaraj, On the minimal of commuting regular rings and semigroups, Intarnal, J. Appl.##Math. 19(2006), 2: 201216.##[4] J. M. Howie, Fundamentals of semigroup Theory, Clarendon Press. Oxford, New York (1995).##[5] Sh. A. Safari Sabet, Commutativity conditions for rings with unity, Internal. J. Appl. Math. 15(2004), 1:##[6] A. H. Yamini, Sh. A. Safari Sabet, Commuting regular rings, Internal. J. Appl. Math. 14(2003) 4: 357364.##]
On strongly Jclean rings associated with polynomial identity g(x) = 0
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2
In this paper, we introduce the new notion of strongly Jclean rings associatedwith polynomial identity g(x) = 0, as a generalization of strongly Jclean rings. We denotestrongly Jclean rings associated with polynomial identity g(x) = 0 by strongly g(x)Jcleanrings. Next, we investigate some properties of strongly g(x)Jclean.
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71
76


H
Haj Seyyed Javadi
Department of Mathematics, Shahed University, Tehran, Iran.
Department of Mathematics, Shahed University,
Iran


S
Jamshidvand
Department of Mathematics, Shahed University, Tehran, Iran.
Department of Mathematics, Shahed University,
Iran
jamshidvand1367@gmail.com


M
Maleki
Department of Mathematics, Shahed University, Tehran, Iran.
Department of Mathematics, Shahed University,
Iran
Full element
uniquely clean ring
nil clean ring
[[1] M. Y. Ahn, (2003). Weakly clean rings and almost clean rings. Ph.D. Thesis, University of Lowa. ##[2] D. D. Anderson, V. P. Camillo, Commutative rings whose elements are a sum of unit and idempotent. ##Comm. Algebra 30 (2002), pp. 3327{3336. ##[3] B. Li, L. Feng, Fclean rings and rings having many full elements. J. Korean Math. Soc. 2 (2010), pp. ##[4] J. Che, W. K. Nicholson, Y. Zhou, Group rings in which every element is uniquely the sum of a unit ##and idempotent. J. Algebra. 306 (2006), pp. 453{460. ##[5] H. Chen, Morita contexts with many units. Comm. Algebra. 30 (3) (2002), pp. 1499{1512. ##[6] A. J. Diesl, Classes of strongly clean rings. Ph.D. Thesis, University of California, Berkeley, (2006). ##[7] A. Haghany, Hopcity and cohopcity for Morita Contexts. Comm. Algebra. 27(1)(1999), pp. 477{ ##[8] W. K. Nicholson, Y. Zhou, Rings in which elements are uniquely the some of an idempotent and unit. ##Clasy. Math. J. 46(2004), pp. 227{236.##]
A note on unique solvability of the absolute value equation
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It is proved that applying sucient regularity conditions to the interval matrix[A jBj;A + jBj], we can create a new unique solvability condition for the absolute valueequation Ax + Bjxj = b, since regularity of interval matrices implies unique solvability oftheir corresponding absolute value equation. This condition is formulated in terms of positivedeniteness of a certain point matrix. Special case B = I is veried too as an application.
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77
81


T
Lotfi
Department of Mathematics, Hamedan Branch, Islamic Azad University,
Hamedan, Iran.
Department of Mathematics, Hamedan Branch,
Iran


H
Vieseh
Department of Mathematics, Hamedan Branch, Islamic Azad University,
Hamedan, Iran.
Department of Mathematics, Hamedan Branch,
Iran
Eigenvalue
Generalized eigenvalue
Quadratic eigenvalue
Numerical computation
Iterative method
[[10] J. Rohn, A theorem of the alternatives for the equation Ax + Bjxj = b, Linear, Multilinear Algebra, 52(6)##(2004) 421426.##[11] J. Rohn, A theorem of the alternatives for the equation jAxj jBjjxj = b, Optim Lett, 6, (2012) 585591.##[12] J. Rohn, A residual existence theorem for linear equations, Optim Lett, 4(2), (2010) 287292.##[13] J. Rohn, On unique solvability of the absolute value equation, Optim, Lett., 3, (2003) 603606.##[14] J. Rohn, Checking properties of interval matrices, Technical Report 686, Institute of Computer Science,##Academy of Sciences of the Czech Republic, Prague, September 1996.##[15] J. Rohn, V. Hooshyarbakhsh and R. Farhadsefat, An iterative method for solving Absolute value equations##and sucient conditions for unique solvability, Optimization Letters, 2012.##[16] S. M. Rump, Verication methods for dence and sparse systems of equations, In: Topics in Validated Com##putations Oldenburg, 1993, Stud. Comput. Math., 5, NorthHolland, Amsterdam, (1994) 63135.##]
On edge detour index polynomials
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The edge detour index polynomials were recently introduced for computing theedge detour indices. In this paper we nd relations among edge detour polynomials for the2dimensional graph of TUC4C8(S) in a Euclidean plane and TUC4C8(S) nanotorus.
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83
89


Sh
Safari Sabet
Department of Mathematics, Islamic Azad University, Central Tehran Branch,
Tehran, Iran
Department of Mathematics, Islamic Azad University
Iran


M
Farmani
Department of Mathematics, Islamic Azad University, Central Tehran Branch,
Tehran, Iran
Department of Mathematics, Islamic Azad University
Iran


O
Khormali
Mathematics and Informatics Research Group, ACECR, Tarbiat Modares University,
P. O. Box: 14115343, Tehran, Iran
Mathematics and Informatics Research Group,
Iran


A
Mahmiani
Department of Mathematics, Payame Noor University, 193954797,
Tehran, Iran
Department of Mathematics, Payame Noor University,
Iran


Z
Bagheri
Islamic Azad University Branch of Azadshaher, Azadshaher, Iran
Islamic Azad University Branch of Azadshaher,
Iran
Heun equation
Wiener process
Stochastic differential equation
Linear equations system
[[1] H. Wiener, Structural determination of paran boiling points, J. Am. Chem. Soc., 69 (1997) 1720.##[2] A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index for trees: theory and applications, Acta Appl.##Math., 66 (3) (2001) 211249.##[3] F. Harary, Graph Theory, AddisonWesley, Reading, Massachusetts, 1969.##[4] M. V. Diudea, G. Katona, I. Lukovitz and N. Trinajstic, Detour and ClujDetour Indices, Croat. Chem. Acta,##71 (1998) 459471.##[5] P. E. John, Ueber die Berechnung des Wiener index fuerausgewaehlte DeltadimensionaleGitterstrukturen,##MATCH Commun. Math. Comput. Chem., 32 (1995) 207219.##[6] R. Jalal Shahkoohi, O. Khormali and A. Mahmiani, The polynomial of detour index for a graph, World##Applied Sciences Journal, 15 (10) (2011) 14731483.##[7] A. Mahmiani, O. Khormali and A. Iranmanesh, The edge versions of detour index, MATCH Commun. Math.##Comput. Chem., 62 (2) (2009) 419431.##[8] Sh. Safari Sabet, A. Mahmiani, O. Khormali, M. Farmani and Z. Bagheri, On the edge detour index polyno##mials, MiddleEast Journal of Scientic Research, 10 (4) (2011) 539548.##[9] Sh. Safari Sabet, M. Farmani, O. Khormali, A. Mahmiani and Z. Bagheri, Relations among the edge detour##index polynomials in nanotubes, Journal of Mathematical Nanoscience, 3 (1) (2013) 312.##]
Operational matrices with respect to Hermite polynomials and their applications in solving linear dierential equations with variable coecients
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In this paper, a new and ecient approach is applied for numerical approximationof the linear dierential equations with variable coecients based on operational matriceswith respect to Hermite polynomials. Explicit formulae which express the Hermite expansioncoecients for the moments of derivatives of any dierentiable function in terms of theoriginal expansion coecients of the function itself are given in the matrix form. The mainimportance of this scheme is that using this approach reduces solving the linear dierentialequations to solve a system of linear algebraic equations, thus greatly simplifying the problem.In addition, two experiments are given to demonstrate the validity and applicability of themethod.
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91
103


Z
Kalateh Bojdi
Department of Mathematics, Birjand University, Birjand, Iran;
Department of Mathematics, Birjand University,
Iran


S
AhmadiAsl
Department of Mathematics, Birjand University, Birjand, Iran;
Department of Mathematics, Birjand University,
Iran


A
Aminataei
Faculty of Mathematics, K. N. Toosi University of Technology,
P.O. Box 163151618, Tehran, Iran.
Faculty of Mathematics, K. N. Toosi University
Iran
Operational matrices
Hermite polynomials
Linear dierential equations with variable coecients
[[1] R.P. Agraval, and D.O. Oregan, Ordinary and Partial Dierential Equations, Springer, 2009.##[2] A. Aminataei, and S.S. Hussaini, The comparison of the stability of decomposition method with numerical##methods of equation solution, Appl. Math. Comput. 186 (2007), pp. 665{669.##[3] A. Aminataei, and S.S. Hussaini, The barrier of decomposition method, Int. J. Contemp. Math. Sci. 5 (2010),##pp. 2487{2494.##[4] R. Askey, Orthogonal Polynomials and Special Functions, SIAMCBMS, Philadelphia, 1975.##[5] T. Akkaya, and S. Yalcinbas, Boubaker polynomial approach for solving highorder linear dierential##dierence equations, AIP Conference Proceedings of 9th international conference on mathematical problems##in engineering, 56 (2012), PP. 26{33.##[6] G. Benyu, The State of Art in Spectral Methods. Hong Kong University, 1996.##[7] J.P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications, Inc, New York, 2000.##[8] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Method in Fluid Dynamics, Prentice Hall,##Engelwood Clis, NJ, 1984.##[9] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Methods: Fundamentals in Single Do##mains, SpringerVerlag, 2006.##[10] H. Danfu, and S. Xufeng, Numerical solution of integrodierential equations by using CAS wavelet opera##tional matrix of integration, Appl. Math. Comput. 194 (2007), pp. 460{466.##[11] C.F. Dunkl, and Y. Xu, Orthogonal Polynomials of Several Variables, Cambridge University Press, 2001.##[12] K. Erdem, and S. Yalcinbas, Bernoulli polynomial approach to highorder linear dierentialdierence equa##tions, AIP Conference Proceedings of Numerical Analysis and Applied Mathematics, 73 (2012), 360{364.##[13] M.R. Eslahchi, and M. Dehghan, Application of Taylor series in obtaining the orthogonal operational matrix,##Computers and Mathematics with Applications, 61 (2011), PP. 2596{2604.##[14] D. Funaro, Polynomial Approximations of Dierential Equations, SpringerVerlag, 1992.##[15] W. Gautschi, Orthogonal Polynomials (Computation and Approximation), Oxford University Press, 2004.##[16] D. Gottlieb, and S.A. Orszag,Numerical Analysis of Spectral Methods: Theory and Applications, SIAM##CBMS, Philadelphia, 1977.##[17] M. Gulsu, and M. Sezer, A method for the approximate solution of the highorder linear dierence equations##in terms of Taylor polynomials, Int. J. Comput. Math. 82 (2005), pp. 629{642.##[18] M. Gulsu, M. Sezer, and Z. Guney, Approximate solution of general highorder linear nonhomogenous dif##ference equations by means of Taylor collocation method, Appl. Math. Comput. 173 (2006), pp. 683{693.##[19] M. Gulsu, and M. Sezer, A Taylor polynomial approach for solving dierentialdierence equations, Comput.##Appl. Math. 186 (2006), pp. 349{364.##[20] J.S. Hesthaven, S. Gottlieb, and D. Gottlieb, Spectral Methods for TimeDependent Problems, Cambridge##University, 2009.##[21] C.H. Hsiao, Hybrid function method for solving Fredholm and Volterra integral equations of the second kind,##Comput. Appl. Math. 230 (2009), pp. 59{68.##[22] A. Imani, A. Aminataei, and A. Imani, Collocation method via Jacobi polynomials for solving nonlinear##ordinary dierential equations, Int. J. Math. Math. Sci., Article ID 673085, 11P, 2011.##[23] F. Khellat, S. A. Youse, The linear Legendre wavelets operational matrix of integration and its application,##J. Frank. Inst. 343 (2006), PP. 181{190.##[24] A.C. King, J. Bilingham, and S.R. Otto, Dierential Equations (Linear, Nonlinear, Integral, Partial), Cam##bridge University, 2003.##[25] E.L. Ortiz, and L. Samara, An operational approach to the Tau method for the numer ical solution of##nonlinear dierential equations, Computing, 27 (1981), pp. 15{25.##[26] E.L. Ortiz, On the numerical solution of nonlinear and functional dierential equa tions with the Tau##method, in: Numerical Treatment of Dierential Equations in Applications, in: Lecture Notes in Math. 679##(1978), pp. 127{139.##[27] F. Marcellan, and W.V. Assche, Orthogonal Polynomials and Special Functions (a Computation and Appli##cations), SpringerVerlag Berlin Heidelberg, 2006.##[28] K. Maleknejad, and F. Mirzaee, Numerical solution of integrodierential equations by using rationalized##Haar functions method, Kyber. Int. J. Syst. Math. 35 (2006), pp. 1735{1744.##[29] M. Razzaghi, and Y. Ordokhani, Solution of nonlinear Volterra Hammerstein integral equations via ratio##nalized Haar functions, Math. Prob. Eng. 7 (2001), PP. 205{219.##[30] M. Razzaghi, and S.A. Youse, Legendre wavelets method for the nonlinear Volterra Fredholm integral##equations, Math. Comput. Simul. 70 (2005), pp. 1{8.##[31] M.H. Reihani, and Z. Abadi, Rationalized Haar functions method for solving Fredholm and Volterra integral##equations, Comput. Appl. Math. 200 (2007), pp. 1220.##[32] M. Sezer, and A.A. Dascioglu, Taylor polynomial solutions of general linear dierentialdierence equations##with variable coecients, Appl. Math. Comput. 174 (2006), pp. 1526{1538.##[33] M. Sezer, and M. Gulsu, Polynomial solution of the most general linear Fredholm integrodierential##dierence equation by means of Taylor matrix method, Int. J. Complex Variables. 50 (2005), pp. 367{382.##[34] J. Shen, T. Tang, and L.L. Wang, Spectral Methods Algorithms, Analysis and Applications, Springer, 2011.##[35] L.N. Trefethen,Spectral Methods in Matlab, SIAM, Philadelphia, PA, 2000. ##[36] A.M. Wazwaz, The combined Laplace transformAdomian decomposition method for handling nonlinear##Volterra integrodierential equations, Appl. Math. Comput. 216 (2010), pp. 1304{1309.##]
A new approach to solve fuzzy system of linear equations by Homotopy perturbation method
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In this paper, we present an ecient numerical algorithm for solving fuzzy systemsof linear equations based on homotopy perturbation method. The method is discussed indetail and illustrated by solving some numerical examples.
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105
115


M
Paripour
Department of Mathematics, Hamedan University of Technology,
Hamedan, 65156579, Iran;
Department of Mathematics, Hamedan University
Iran
paripour@hut.ac.ir, paripour@gmail.com


J
Saeidian
Faculty of Mathematical Sciences and Computer, Kharazmi University,
50 Taleghani Avenue, Tehran 1561836314, Iran;
Faculty of Mathematical Sciences and Computer,
Iran


A
Sadeghi
Department of Mathematics, Science and Research Branch,
Islamic Azad University, Arak, Iran.
Department of Mathematics, Science and Research
Iran
[[1] S. Abbasbandy, R. Ezzati, Homotopy method for solving fuzzy nonlinear equations, Appl. Sci. 8 (2006), pp.##[2] T. Allahviranloo, The Adomian decomposition method for fuzzy system of linear equations, Appl. Math.##Comput. 163 (2) (2005), pp. 553{563.##[3] T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Appl. Math. Comput. 155 (2)##(2004), pp. 493{502.##[4] T. Allahviranloo, M. Ghanbari, Solving Fuzzy Linear Systems by Homotopy Perturbation Method, Inter. J.##Comput. Cognition 8 (2) (2010), pp. 91{61.##[5] B. Asady, S. Abbasbandy, M. Alavi, Fuzzy general linear systems, Appl. Math. Comput. 169 (2005), pp.##[6] E. Babolian, J. Saeidian, M. Paripour, Computing the Fourier Transform via Homotopy perturbation method,##Z. Naturforsch. 64a (2009), pp. 671{675.##[7] D. Dubois, H. Prade, Fuzzy Set and Systems: Theory and Application, Academic Press, New York, 1980.##[8] M. Friedman, M. Ming, A. Kandel, Fuzzy linear systems, Fuzzy Sets Syst. 96 (1998), pp. 201{209.##[9] R. Goetschell, W. Voxman, Elementary calculus, Fuzzy Sets Syst. 18 (1986), pp. 31{43.##[10] J. H. He, Homotopy perturbation technique, Comp. Meth. Appl. Mech. Eng. 178 (1999), pp. 257{262.##[11] J. H. He, A coupling method of homotopy technique and a perturbation technique for nonlinear problems,##Int. J. NonLinear Mech. 35 (1) (2000), pp. 37{43.##[12] J. H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput. 135##(1) (2003), pp. 73{79.##[13] B. Keramati, An approach to the solution of linear system of equations by He; homotopy perturbation##method, Chaos, Solitons and Fractals. 37 (2006), pp. 1528{1537.##[14] H. Ku Liu, Application of homotopy perturbation methods for solving systems on linear equations, Appl.##Math. Comput. 217 (2011), pp. 5259{5264.##[15] S. J. Liao, Beyond perturbation: An introduction to homotopy analysis method, Chapman Hall/CRC Press,##Boca Raton, 2003.##[16] M. Ma, M. Friedman, A. Kandel, A new fuzzy arithmetic, Fuzzy Sets and Syst. 108 (1999), pp. 83{90.##[17] H. Saberi Naja, S. A. Edalatpanah, A. H. Refahi Sheikhani, Application of Homotopy Perturbation Method##for Fuzzy Linear Systems and Comparison with Adomians Decomposition Method, Chinese Journal of Math##ematics 2013 (2013), pp. 1{7.##[18] K. Wang, B. Zheng, Inconsistent fuzzy linear systems, Appl. Math. Comput. 181 (2006), pp. 973{981.##[19] B. Zheng, K. Wang, General fuzzy linear systems, Appl. Math. Comput. 181 (2006), pp. 1276{1286.##]
The method of fundamental solutions for transient heat conduction in functionally graded materials: some special cases
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In this paper, the Method of Fundamental Solutions (MFS) is extended to solvesome special cases of the problem of transient heat conduction in functionally graded materials. First, the problem is transformed to a heat equation with constant coecients usinga suitable new transformation and then the MFS together with the Tikhonov regularizationmethod is used to solve the resulting equation.
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117
127


M
Nili Ahmadabadi
Department of Mathematics, Islamic Azad University, Najafabad Branch, Najafabad, Iran;
Department of Mathematics, Islamic Azad University
Iran
nili@phu.iaun.ac.ir


M
Arab
Department of Mathematics, Yazd University, Yazd, Iran.
Department of Mathematics, Yazd University,
Iran


F.M
Maalek Ghaini
Department of Mathematics, Yazd University, Yazd, Iran.
Department of Mathematics, Yazd University,
Iran
Heat conduction
Functionally graded materials
Method of fundamental solutions
[[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;11:67581.##[3] C. S. Chen, H. A. Cho, M. A. Golberg, Some comments on the illconditioning of the method of fundamental##solutions. Eng. Anal. Boundary Elem. 2006;30:405410.##[4] F. de Monte, Transient heat conduction in onedimensional composite slab. A natural approach. Int. J. Heat##Mass Transf. 2000;43:360719.##[5] F. de Monte, Ananalytical approach to the unsteady heat conduction processes in onedimensional composite##media. Int. J. Heat Mass Transf. 2002;26:133343.##[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(12):6995.##[8] Z. G. Feng, E. E. Michaelides, The use of modied Greens functions in unsteady heat transfer. Int. J. Heat##Mass Transf. 1997;40:29973002.##[9] A. Friedman, Partial dierential equations of parabolic type. Englewood Clis, NJ: PrenticeHall Inc.; 1964##[10] M. A. Golberg, C. S. Chen, The method of fundamental solutions for potential, Hemholtz and diusion##problems. In: Boundary integral methods: numerical and mathematical aspects. Computational engineering,##vol. 1. Boston, MA: WIT Press, Computational Mechanics Publications; 1999. p. 10376.##[11] A. HajiSheikh, J. V. Beck, Greens function partitioning in Galerkinbase integral solution of the diusion##equation. J. Heat Transf. 1990;112:2834.##[12] P. C. Hansen, Regularization tools: a Matlab package for analysis and solution of discrete illposed problems.##Numer. Algorithms 1994;6:135.##[13] P. C. Hansen, Rankdecient and discrete illposed 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;102:7428.##[15] B. T. Johansson, D. Lesnic, A method of fundamental solutions for transient heat conduction. Eng. Anal.##Boundary Elem. 2008;32:697703.##[16] B. T. Johansson, D. Lesnic, A method of fundamental solutions for transient heat conduction in layered##materials. Eng. Anal. Boundary Elem. 2009;33:136267.##[17] A. Karageorghis, G. Fairweather, The method of fundamental solutions for the numerical solution of the##biharmonic equation, J. Comput. Phys. 1987;69:43459.##[18] R. Kress, A. Mohsen, On the simulation source technique for exterior problems in acoustics, Math. Methods##Appl. Sci. 1986;8:58597.##[19] V. D. Kupradze, A method for the approximate solution of limiting problems in mathematical physics. USSR##Comput. Math. Math. Phys. 1964;4:199205.##[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;4:82126.##[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 identication of piecewise homogeneous properties of rocks. In: Ingham DB, Wrobel LC,##editors. Boundary integral formulations for inverse analysis. Southampton: Comput. Mech. Publ.; 1997. p.##23757 [chapter 9].##[23] R. Mathon, H. Johnston, The approximate solution of elliptic boundaryvalue problems by fundamental##solutions, SIAM J. Numer. Anal. 1977;14(4):63850.##[24] M. D. Mikhailov, M. N. Ozisik, M. D. Vulchanov, Diusion in composite layers with automatic solution of##thee eigenvalue problem. Int. J. Heat Mass Transf. 1983;26:113141.##[25] M. Nili Ahmadabadi, M. Arab, F. M. Maalek Ghaini, The method of fundamental solutions for the inverse##spacedependent heat source problem. Eng. Anal. Boundary Elem. 2009;33:123135.##[26] P. A. Ramachandran, Method of fundamental solutions: singular value decomposition analysis. Commun.##Numer. Methods Eng. 2002;18:789801.##[27] J. N. Reddy, An introduction to the nite element method. NewYork: McGrawHill; 1984.##[28] M. Talha, B. N. Singh, Stochastic perturbationbased nite element for buckling statistics of FGM plates##with uncertain material properties in thermal environments, Composite Structures, Volume 108, February##2014, Pages 823833, ISSN 02638223, 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 8892, ISSN 09270256, http://dx.doi.org/10.1016/j.commatsci.2014.01.032.##[30] V. Vodicka, Warmeleitung in geschichteten kugel und zylinderkorpern. Schweiz Arch 1950;10:297304.##[31] Y. Yener, M. N. Ozisik, On the solution of unsteady heat conduction in multiregion nite media with timedependent##heat transfer coecient. In: Proceedings of the fth international heat transfer conference, vol. 1,##JSME, Tokyo; 1974. p. 18892.##[32] Zakaria Belabed, Mohammed Sid Ahmed Houari, Abdelouahed Tounsi, Mahmoud SR., Anwar Bg O., An##ecient and simple higher order shear and normal deformation theory for functionally graded material##(FGM) plates, Composites Part B: Engineering, Volume 60, April 2014, Pages 274283, ISSN 13598368,##http://dx.doi.org/10.1016/j.compositesb.2013.12.057.##]