An efficient method for the numerical solution of functional integral equations

Subject Areas : Differential and Integral equations

M. Nili Ahmadabadi ¹ (Department of Mathematics, Najafabad Branch, Islamic Azad University, Najafabad, Iran)

Received: 2019-01-21 Accepted : 2020-04-02 Published : 2020-06-01

Keywords: collocation method, Radial basis functions, Functional integral equations, mesh-less method,

Abstract :

We propose an efficient mesh-less method for functional integral equations. Its convergence analysis has been provided. It is tested via a few numerical experiments which show the efficiency and applicability of the proposed method. Attractive numerical results have been obtained.

References:

[1] A. Alipanah, M. Dehghan, Numerical solution of the nonlinear Fredholm integral equations by positive definite functions, Appl. Math. Comput. 190 (2007), 1754-1761.

[2] I. Babuska, U. Banerjee, J.E. Osborn, Q. Zhang, Effect of numerical integration on meshless methods, Comput. Methods Appl. Mech. Engrg. 198 (2009), 2886-2897.

[3] C. Canuto, M. Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods, Fundamentals in Single Domains, Springer-Verlag, 2006.

[4] G.E. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific Publishing. Co. Pte. Ltd., Hackensack, NJ, 2007.

[5] R. L. Hardy, Theory and applications of the multiquadric-biharmonic method. 20 years of discovery 1968-1988, Comput. Math. Appl. 19 (8-9) (1990), 163-208.

[6] T. Ismaeelpour, A. Askari Hemmat, H. Saeedi, B-spline operational matrix of fractional integration, Optik. 130 (2017), 291-305.

[7] E. J. Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl. 19 (8-9) (1990), 147-161.

[8] K. Parand, M. Delkhosh, Operational matrices to solve nonlinear Volterra-Fredholm integro-differential equations of multi-arbitrary order, Gazi J. of Sci. 29 (2016), 895-907.

[9] K. Parand, M. Delkhosh, Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions, Ricerche mat. 65 (2016), 307-328.

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

[11] C. Vetro, F. Vetro, On the existence of at least a solution for functional integral equations via measure of noncompactness, Banach J. Math. Anal. 11 (3) (2017), 497-512.

[12] H. Wendland, Scattered Data Approximation, Cambridge University Press, 2005.

[13] S. Yousefi, A. Lotfi, M. Dehghan, The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems. J. Vibration and Control. 17 (13) (2011), 2059-2065.

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