Document Type : Research Article

**Authors**

Malayer University

**Abstract**

The main purpose of this article is to describe a numerical scheme for solving three-dimensional linear Fredholm integral equations of the second kind on the cubic domains. The method is based on interpolation by radial basis functions (RBFs) based on Gauss-Legendre nodes and weights. Error analysis is presented for this method. Finally, several examples are given and numerical examples are presented to demonstrate the validity and applicability of the method.

**Keywords**

[1] Guoqiang, H. and Jiong, W. Extrapolation of nystrom solution for two dimensional nonlinear Fredholm integral equations, J. Comput. Appl. Math.134 (2001), 259-268.

[2] Guoqiang, H., Hayami, K., Sugihara,K. and Jiong, W. Extrapolation method of iterated collocation solution for two-dimensional nonlinear Volterra integral equations, Appl. Math. Comput. 112 (2000), 49-61.

[3] Brunner, H. Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, 2004.

[4] Xie, W. J. and Lin, F. R. A fast numerical solution method for two dimensional Fredholm integral equations of the second kind, Applied Numerical Mathematics, 59(2009), 1709-1719.

[5] Salehi, R. and Dehghan, M. A moving least square reproducing polynomial meshless method, Appl. Numer. Math. 69 (2013), 34-58.

[6] Dehghan, M. and Salehi, R. The numerical solution of the non-linear integro-differential equations based on the meshless method, J. Comput. Appl. Math. 236, 2367-2377 (2012)

[7] Dehghan, M. and Mirzaei, D. Meshless Local Petrov-Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, App. Numer. Math. 59 (2009), 1043-1058.

[8] Kansa, E.J. Multiquadrics - A scattered data approximation scheme with applications to computational fluid dynamics-I, Comput. Math. Appl. 19(1990) 127-145.

[9] Kansa, E.J. Multiquadrics - A scattered data approximation scheme with applications to computational fluid dynamics-II, Comput. Math. Appl. 19(1990) 147-161.

[10] Hon, Y.C. and Mao, X.Z. An efficient numerical scheme for Burgers equation, Appl. Math. Comput. 95 (1) (1998) 37-50.

[11] Hon, Y.C., Cheung, K.F., Mao, X.Z. and Kansa, E.J. Multiquadric solution for shallow water equations, ASCE J. Hydraul. Eng. 125 (5)

(1999) 524-533.

[12] Zerroukat, M., Power, H. and Chen, C.S. A numerical method for heat transfer problem using collocation and radial basis functions, Internat. J. Numer. Methods Engrg. 42 (1992) 1263-1278.

[13] Hon, Y.C. and Mao, X.Z. A radial basis function method for solving options pricing model, Financ. Eng. 8 (1) (1999) 31-49.

[14] Marcozzi, M., Choi, S. and Chen, C.S. On the use of boundary conditions for variational formulations arising in financial mathematics, Appl. Math. Comput. 124 (2001) 197-214.

[15] Golbabai, A. and Seifollahi, S. Numerical solution of the second kind integral equations using radial basis function networks, Appl. Math. Comput. 174 (2006) 877-883.

[16] Golbabai, A. and Seifollahi, S. Radial basis function networks in the numerical solution of linear integro-differential equations, Appl. Math. Comput. 188 (2007) 427-432.

[17] Parand, K. and Rad, J.A. Numerical solution of nonlinear Volterra Fredholm-Hammerstein integral equations via collocation method based on radial basis functions, Appl. Math. Comput. 218 (2012) 5292-5309.

[18] Golbabai, A., Mammadov, M. and Seifollahi, S. Solving a system of nonlinear integral equations by an RBF network, Comput. Math. Appl. 57 (2009) 1651-1658.

[19] Alipanah, A. and Esmaeili, S. Numerical solution of the two-dimensional Fredholm integral equations using Gaussian radial basis function, J. Comput. Appl. Math. 235 (2011) 5342-5347.

[20] Hardy, R. L. Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 176 (1971) 1905-1915.

[21] Franke, R. Scattered data interpolation: Tests of some methods, Mathematics of Computation, 38(1982) 181-200.

[22] Wendland, H. Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press, Cambridge (2005).

[23] Fasshauer, G. E. Meshfree approximation methods with MATLAB, Interdisciplinary Mathematical Sciences, vol. 6. World Scientific Publishing Company, Singapore (2007).

[24] Schoenberg, I. J. Metric spaces and completely monotone functions, Ann. Math., 39: 811-841, 1938.

[25] Micchelli, C. A. Interpolation of scattered data: distance matrices and conditionally positive denite functions, Constr. Approx., 2: 11-22, 1986.

[26] Larsson, E. and Fornberg, B. A numerical study of some radial basis function based solution methods for elliptic PDEs, Comput. Math. Appl. 46 (2003) 891-902.

[27] Assari, P., Adibi, H. and Dehghan, M. A numerical method for solving linear integral equations of the second kind on the non-rectangular domains based on the meshless method, Applied Mathematical Modelling, vol. 37, pp. 9269-9294, 2013.

[28] Fasshauer, G. E. Newton iteration with multiquadrics for the solution of nonlinear PDEs, Computers and Mathematics with Applications, 43: 423-438, 2002.

[29] Atkinson, K. E. and Han, W. Theoretical Numerical Analysis: a Func tional Analysis Framework, Springer-Verlag New York, INC, 2001.

[30] Atkinson, K. E. The Numerical Solution of Integral Equations of the Second Kind, vol. 4, Cambridge University Press, Cambridge, UK, 1997.

Summer and Autumn 2017

Pages 15-38