The method of radial basis functions for the solution of nonlinear Fredholm integral equations system.

Subject Areas : History and biography

J. Nazari ¹ , M. Nili Ahmadabadi ² , H. Almasieh ³

1 - Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, Iran
2 - Department of Mathematics, Najafabad Branch, Islamic Azad University, Najafabad, Iran
3 - Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, Iran

Received: 2016-11-03 Accepted : 2017-04-05 Published : 2017-03-01

Keywords: Radial basis functions, Fredholm integral equations system,

Abstract :

In this paper, An effective and simple numerical method is proposed for solving systems of integral equations using radial basis functions (RBFs). We present an algorithm based on interpolation by radial basis functions including multiquadratics (MQs), using Legendre-Gauss-Lobatto nodes and weights. Also a theorem is proved for convergence of the algorithm. Some numerical examples are presented and results are compared to the analytical solution and Triangular functions (TF), Delta basis functions (DFs), block-pulse functions , sinc fucntions, Adomian decomposition, computational, Haar wavelet and direct methods to demonstrate the validity and applicability of the proposed method.

References:

[1] H. Almasieh, J. N. Meleh, A meshless method for the numerical solution of nonlinear Volterra-Fredholm integro-differential equations using radial basis functions, International Conference on Mathematical Sciences & computer Engineering (ICMSCE 2012).

[2] H. Almasieh, J. N. Meleh, Hybrid functions method based on radial basis functions for solving nonlinear Fredholm integrad equations, Journal of Mathematical Extention. 7 (3) (2013), 29-38.

[3] H. Almasieh, J. N. Meleh, Numerical solution of a class of mixed two-dimensional nonlinear Volterra-Fredholm integral equations using multiquadric radial basis functions, J. Comput. Appl. Math. 260 (2014), 173-179.

[4] H. Almasieh, M. Roodaki, Triangular functions method for the solution of Fredholm integral equations system, Ain Shams Engineering Journal 3 (2012), 411-416.

[5] K. E. Atkinson, The numerical solution of integral equations of the second kind, Cambridge University Press, Cambridge, 1997.

[6] E. Babolian, Z. Mansouri, A. Hatamzadeh-Varmarzgar, A direct method for numerically solving integral equations systems using orthogonal traingular functions, International Journal of Industrial Mathematics. 1 (2) (2009), 135-145.

[7] E. Babolian, J. Biazar, A. R. Vahidi, The decomposition method applied to systems of Fredholm integral equations of the second kind, Appl. Math. Comput. 148 (2004), 443-52.

[8] A. H.-D. Cheng, M. A. Golberg, E. J. Kansa, G. Zammito, Exponential convergence and H-C multiquadric collocation method for partial differential equations, Numer. Meth. Partial. Differ. Equa. 19 (5) (2003), 571-594.

[9] L. M. Delves, J. L. Mohammed, Computational methods for integral equations, Cambridge University Press, 1985.

[10] M. C. De Bonis, C. Laurita, Numerical treatment of second kind integral equation systems on bounded intervals, J. Comput. Appl. Math. 217 (2008), 67-87.

[11] K. Maleknejad, M. Shahrezaee, H. Khatami, Numerical solution of integral equations system of the second kind by blockpulse functions, Appl. Math. Comput. 166 (1) (2005), 15-24.

[12] O. M. Ogunlaran, O. F. Akinlotan, A computational method for system of linear Fredholm integral equations, Mathematical Theory and Modeling. 3 (4) (2013), 1-6.

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

[14] J. Rashidinia, M. Zarebnia, Convergence of approximate solution of system of Fredholm integral equations, J. Math. Anal. Appl. 333 (2007), 1216-1227.

[15] M. Roodaki, H. Almasieh, Delta basis functions and their applications to systems of integral equations, Comput. Math. Appl. 63 (1) (2012), 100-109.

[16] A. R. Vahidi, M. Mokhtari, On the decomposition method for system of linear Fredholm integral equations of the second kind, Appl. Math. Sci. 2 (2) (2008), 57-62.

[17] H. A. Zedan, E. Alaidarous, Haar wavelet method for the system of integral equations, Abstract and Applied Analysis. 2014 (2014), 1-9.