A meshless technique for nonlinear Volterra-Fredholm integral equations via hybrid of radial basis functions

Document Type : Research Articles


1 Department of Mathematics, Islamic Azad University, Khorasgan(Isfahan) Branch

2 Department of Mathematics, Khorasgan (Isfahan) Branch, Islamic Azad University


In this paper, an effective technique is proposed to determine the
numerical solution of nonlinear Volterra-Fredholm integral
equations (VFIEs) which is based on interpolation by the hybrid of
radial basis functions (RBFs) including both inverse multiquadrics
(IMQs), hyperbolic secant (Sechs) and strictly positive definite
functions. Zeros of the shifted Legendre polynomial are used as
the collocation points to set up the nonlinear systems. The
integrals involved in the formulation of the problems are
approximated based on Legendre-Gauss-Lobatto integration rule.
This technique is so convenience to implement and yields very
accurate results compared with the other basis. In addition a
convergence theorem is proved to show the stability of this
technique. Illustrated examples are included to confirm the
validity and applicability of the proposed method. The comparison
of the errors is implemented by the other methods in references
using both inverse multiquadrics (IMQs), hyperbolic secant (Sechs)
and strictly positive definite functions.


[1] H. Almasieh, J. Nazari Meleh, Numerical solution of a class of mixed
two-dimensional nonlinear Volterra-Fredholm integral equations using
multiquadric radial basis functions, Comput. Appl. Math. 260 (2014) 173{
[2] A. Alipanah, M. Dehghan, Numerical solution of the nonlinear Fredholm
integral equations by positive de nite functions, Appl. Math. Comput. 190
(2007) 1754{1761.
[3] B. J. C. Baxter, The interpolation theory of Radial Basis Functions,
Cambridge University, 1992.
[4] A. H. D. Cheng, M. A. Galberg, E. J. Kansa, Q. Zammito, Exponential
convergence and H-c multiquadratic collocation method for partial
di erential equations, Numer. Meth. Part. D. E. 19 (2003) 571{594.
[5] W. Cheney, W. Light, A course in approximation theory, New York, 1999.
[6] K. B. Datta, B. M. Mohan, Orthogonal Functions in System and Control,
World Scienti c, Singapore, 1995.
[7] G. N. Elnagar, M. A. Kazemi, Pseudospectral Legendre-based optimal
computaion of nonlinear constrained variational problems, J. Comput.
Appl. Math. 88 (1997) 363{375.
[8] G. N. Elnagar, M. Razzaghi, A collocation-type method for linear
quadratic optimal control problems, Optim. Control. Appl. Meth. 18
(1998) 227{235.
[9] R. E. Garlson, T .A. Foly, The parameter R2 in multiquadratic
interpolation, Comput. Math. Appl. 21 (1991) 29{42.
[10] M. A. Galberg, Some recent results and proposals for the use of radial basis
functions in the BEM, Eng. Anal. Bound. Elem. 23(4) (1999) 285{296.
[11] C. Kui-Fang, Strictly positive de nite functions, J. Approx. Theory 87
(1996) 148{15.
[12] K. Maleknejad, Y. Mahmoudi, Numerical solution of linear Fredholm
integral equations by using Hybrid Taylor and Block-Pulse functions,Appl.
Math. Comput. 149 (2004) 799{806.
[13] 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.
[14] M. Razzaghi, S. Youse , Legendre wavelets mehod for the nonlinear
Volterra-Fredholm integral equations, Math. Comput. Simul. 70 (2005)
[15] J. Rashidinia, M. Zarebnia, New approach for numerical solution of
Hammerstein integral equations, Appl. Math. Comput. 185 (2007) 147{
[16] M. H. Reihani, Z. Abadi, Rationalized Haar function method for solving
Fredholm and Volterra integral equations, J. Comp. Appl. Math. 200
(2007) 12{20.
[17] J. Shen, T. Tang, High order numerical Methods and Algorithms, Abstract
and Applied Analysis, Chinese Science Press, 2005.
[18] A. E. Tarwater, A parameter study of Hardy's multiquadratic method for
scattered data interpolation, Report UCRL - 53670, Lawrence Livermore
National Laboratory, 1985.
[19] S. Yalinbas, Taylor polynomial solution of nonlinear Volterra-Fredholm
integral equations, Appl. Math. Comput. 127 (2002) 195{206.