Legendre wavelet method for solving Hammerstein integral equations of the second kind

Document Type : Research Articles


1 دانشگاه خوارزمی تهران

2 دانشکده ریاضی دانشگاه خوارزمی تهران


An ecient method, based on the Legendre wavelets, is proposed to solve the
second kind Fredholm and Volterra integral equations of Hammerstein type.
The properties of Legendre wavelet family are utilized to reduce a nonlinear
integral equation to a system of nonlinear algebraic equations, which is easily
handled with the well-known Newton's method. Examples assuring eciency
of the method and its superiority are presented.

[1] K. E. Atkinson, The Numerical Solution of Integral Equations of The
Second Kind, Cambridge University Press, Cambridge, 1997.
[2] S. M. Berman, A. L. Stewart, A Nonlinear Integral Equation for Visual
Impedance, Biol. Cybernetics 33 (1979) 137{141.
[3] A.M. Bica, M. Curila, S. Curila, About a numerical method of successive
interpolations for functional Hammerstein integral equations, J. Comput.
Appl. Math. 236 (2012) 2005{2024.
[4] A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen,
Acta Math. 54 (1930) 117{176.
[5] H. R. Thieme, On a class of Hammerstein integral equations, Manuscripta
Math. 29 (1979) 49{84.
[6] J. Banas, J. Rocha Martin, K. Sadarangani, On solutions of a quadratic
integral equation of Hammerstein type, Math. Comput. Model. 43 (2006)
[7] J. Banas, Integrable solutions of Hammerstein and Uryshon integral
equations, J. Austral. Math. Soc. (A) 46 (1989) 61{68.
[8] D. ORegan, Existence results for nonlinear integral equations, J. Math.
Anal. Appl. 192 (1995) 705{726.
[9] D. ORegan, M. Meehan, Existence Theory for Nonlinear Integral and
Integro-di erential Equations, Kluwer Academic Publishers, Dordrecht,
[10] K. Atkinson, A survey of numerical methods for solving nonlinear integral
equations, J. Int. Eqns. Applics. 4 (1992) 15{46.
[11] M. M. Shamivand, A. Shahsavaran, Numerical solution of Hammerstein
Fredholm and Volterra integral equations of the second kind using block
pulse functions and collocation method, Math. Sci. J.,7 (2011) 93-103.
[12] A. Shahsavaran, E. Babolian, Computational method for solving nonlinear
Fredholm integral equations of Hammerstein type based on Lagrange
interpolation and quadrature method, Math. Sci. J.,5 (2009) 137-145.
[13] S. Kumar, I. Sloan, A new collocation-type method for Hammerstein
equations, Math. Comp. 48 (1987) 585{593.
[14] G. N. Elnagar, M. Kazemi, Chebyshev spectral solution of nonlinear
Volterra-Hammerstein integral equations, J. Comput. Appl. Math. 76
(1996) 147{158.
[15] G. N. Elnagar, M. Kazemi, A cell-averaging chebyshev spectral method
for nonlinear fredholm-hammerstein integral equations, Int. J. Comput.
Math. 60 (1996) 91{104.
[16] H. Kaneko, R. D. Noren, B. Novaprateep, Wavelet applications to the
PetrovGalerkin method for Hammerstein equations, Appl. Numer. Math.
45 (2003) 255-273.
[17] K. Maleknejad, H. Derili, The collocation method for Hammerstein
equations by Daubechies wavelets, Appl. Math. Comput. 172 (2006) 846{
[18] S. Youse , M. Razzaghi, Legendre wavelets method for the nonlinear
VolterraFredholm integral equations, Math. Comput. Simul. 70 (2005) 1{
[19] E. Babolian, F. Fattahzadeh, E. Golpar Raboky, A Chebyshev
approximation for solving nonlinear integral equations of Hammerstein
type, Appl. Math. Comput. 189 (2007) 641{646.
[20] Y. Ordokhani, Solution of nonlinear VolterraFredholmHammerstein
integral equations via rationalized Haar functions, Appl. Math. Comput.
180 (2006) 436{443.
[21] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992.
[22] M. Razzaghi, S. Youse , The Legendre wavelets operational matrix of
integration, Int. J. Syst. Sci. 32 (2001) 495{502.
[23] M. Rehman, R. A. Khan, The Legendre wavelet method for solving
fractional di erential equations, Commun. Nonlinear Sci. Numer.
Simulat. 16 (2011) 4163{4173.
[24] M. Razzaghi, S. Youse , Legendre wavelets method for the solution of
nonlinear problems in the calculus of variations, Math. Comput. Model.
34 (2001) 45{54.
[25] F. Awawdeh, A. Adawi, A numerical method for solving nonlinear integral
equations, Int. Math. Forum 4 (2009) 805{817.
[26] Y. Mahmoudi, Wavelet Galerkin method for numerical solution of
nonlinear integral equation, Appl. Math. Comput. 167 (2005) 1119{1129.
[27] E. Babolian, A. Shahsavaran, Numerical solution of nonlinear Fredholm
and Volterra integral equations of the second kind using Haar wavelets and
collocation method, J. Sci. Tarbiat Moallem University, 7 (2007) 213{222.
[28] K. Maleknejad, K. Nedaiasl, Application of Sinc-collocation method for
solving a class of nonlinear Fredholm integral equations, Comput. Math.
Appl. 62 (2011) 3292{3303.