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

Subject Areas : Applied Mathematics

Jinoos Nazari ¹ (Department of Mathematics, Islamic Azad University, Khorasgan(Isfahan) Branch)
Homa Almasieh ² (Department of Mathematics, Khorasgan (Isfahan) Branch, Islamic Azad University)

Received: 2015-11-12 Accepted : 2016-03-20 Published : 2016-12-01

Keywords: Inverse multiquadrics, Strictly positive, Hyperbolic secant, definite functions, Nonlinear Volterra-Fredholm integral equation,

Abstract :

In this paper, an effective technique is proposed to determine thenumerical solution of nonlinear Volterra-Fredholm integralequations (VFIEs) which is based on interpolation by the hybrid ofradial basis functions (RBFs) including both inverse multiquadrics(IMQs), hyperbolic secant (Sechs) and strictly positive definitefunctions. Zeros of the shifted Legendre polynomial are used asthe collocation points to set up the nonlinear systems. Theintegrals involved in the formulation of the problems areapproximated based on Legendre-Gauss-Lobatto integration rule.This technique is so convenience to implement and yields veryaccurate results compared with the other basis. In addition aconvergence theorem is proved to show the stability of thistechnique. Illustrated examples are included to confirm thevalidity and applicability of the proposed method. The comparisonof the errors is implemented by the other methods in referencesusing both inverse multiquadrics (IMQs), hyperbolic secant (Sechs)and strictly positive definite functions.

References:

[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{
179.
[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
dierential 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)
1{8.
[15] J. Rashidinia, M. Zarebnia, New approach for numerical solution of
Hammerstein integral equations, Appl. Math. Comput. 185 (2007) 147{
154.
[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.
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