Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, Urmia University, P.O.Box 165, Urmia-Iran

2 Department of Mathematics and Computer Science, University of Tabriz, Tabriz-Iran

Abstract

In this paper, we studied the numerical solution of nonlinear weakly singular Volterra-Fredholm integral equations by using the product integration method. Also, we shall study the convergence behavior of a fully discrete version of a product integration method for numerical solution of the nonlinear Volterra-Fredholm integral equations. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.

Keywords

[1] P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia(1985).
[2] N. Levinson, A nonlinear Volterra equations arising in the theory of superuidity, J. Math. Anal. Appl. 1 (1960) 1-11.
[3] A. Pedas and E.Tamme, Fully discrete Galerkin method for Fredholm integro-di erential equations with weakly singular kernels, Computational Methods in Applied Mathematics, Vol 8 No 3 (2008) 294-308.
[4] T. Tang, S. McKee and T. Diogo, product integration method for an integral equation with logarithmic singular kernel, App. Numer. Math. 9 (1992), 259-266.
[5] M. Rasty and M. Hadizadeh, A Product integration approach on new orthogonal poly- nomials for nonlinear weakly singular integral equations, Acta Appl. Math. 109 (2010), 861-873.
[6] A.P. Orsi, Product integration for Volterra integral equations of the second kined with weakly singular kernels, Math. Comput. 212 (1996), 1201-1212.
[7] H. Brunner, High-order collocation methods for singular Volterra functional equations of neutral typr, Applied Numerical Mathematics, 57 (2007) 533-548.
[8] H. Brunner, The numerical solution of weakly singular Volterra functional integro-di erential equations with variable delays, Comm. Pure Appl. Anal. 5 (2006) 261-276.
[9] J.B. Keller and W.E. Olmstead, Temperature of nonlinear radiating semi-in nite solid, Q. Appl. Math. 29 (1972) 559-566.
[10] V.S. Chelyshkov, Alternative orthogonal polynomials and quadratures, Electron. Trans. Numer. Anal. 25(7), (2006), 17-26.
[11] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge Monographs on Applied and Computational Mathematics, vol. 15, Cambridge University Press, Cambridge, 2004.
[12] R. P. Kanwal and K. C. Liu, A Taylor expansion approach for solving integral equations, Int. J. Math. Educ. Sci. Technol. 3 (1989), 411-414.
[13] G. Criscuolo, G. Mastroianni and G. Monegato, Convergence properties of a class of product formulas for weakly singular integral equations, Math. Comput. 55 (1990), 213-230.
[14] P. Nevai, Mean convergence of Lagrange interpolation. III. Trans. Am. Math. Soc. 282 (1984), 669-689.
[15] H. Kaneko and Y.Xu, Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of second kind. Math. Comput. 62, (1994) 739-753.
[16] V.I. Krylov, Approximate Calculation of Integrals. Macmillan Company, New York (1962).
[17] P.K. Kythe and P. Puri, Computational Methods for Linear Integral Equations. Birkhuser, Boston (2002).