Nazari, J., Almasieh, H. (2014). A meshless technique for nonlinear Volterra-Fredholm integral equations via hybrid of radial basis functions. Theory of Approximation and Applications, 10(2), 43-59.

Jinoos Nazari; Homa Almasieh. "A meshless technique for nonlinear Volterra-Fredholm integral equations via hybrid of radial basis functions". Theory of Approximation and Applications, 10, 2, 2014, 43-59.

Nazari, J., Almasieh, H. (2014). 'A meshless technique for nonlinear Volterra-Fredholm integral equations via hybrid of radial basis functions', Theory of Approximation and Applications, 10(2), pp. 43-59.

Nazari, J., Almasieh, H. A meshless technique for nonlinear Volterra-Fredholm integral equations via hybrid of radial basis functions. Theory of Approximation and Applications, 2014; 10(2): 43-59.

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

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

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

Abstract

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.

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

Authors [Persian]

Jinoos Nazari^{1}; Homa Almasieh^{2}

^{}

^{}

Abstract [Persian]

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{ 179. [2] A. Alipanah, M. Dehghan, Numerical solution of the nonlinear Fredholm integral equations by positive denite 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 Scientic, 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 denite 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. 5