Hybrid of Rationalized Haar Functions Method for Mixed Hammerstein Integral Equations

Document Type : Research Article



A numerical method for solving nonlinear mixed Hammerstein integral equations is presented in this paper. The method is based upon hybrid of rationalized Haar functions approximations. The properties of hybrid functions which are the combinations of block-pulse functions and rationalized Haar functions are first presented. The Newton-Cotes nodes and Newton-Cotes integration method are then utilized to reduce the nonlinear mixed Hammerstein integral equations to the solutions algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.


[1]     F. G. Tricomi, "Integral equations, "Dover, 1982.
[2]     L. J. Lardy, "A variation of Nystrom's method for Hammerestein equations," J. Integral Equations, vol. 3,  p.p. 123-129 , 1982.
[3]     S. Kumar and I. H. Sloan, "A new collocation-type method for Hammerstein integral equations," J. Math. Comp., vol. 48, p.p.  123-129, 1987.
[4]     H. Brunner, "Implicitly linear collocation method for nonlinear Volterra equations," J. Appl. Num. Math., vol. 9, p.p. 235-247, 1982.
[5]     H. Guoqiang, "Asymptotic error expansion variation of a collocation method for Volterra- Hammerstein equations," J. Appl. Num. Math., vol. 13, p.p. 357-369, 1993.
[6]     C. H. Hsiao and C. F. Chen," Solving integral equation via Walsh functions,"  Comput. Elec. Engng., vol. 6, p.p. 279-292, 1979.
[7]     C. H. Wang and Y. P. Shih, "Explicit solutions of integral equations via block -pulse functions," Int. J. Syst. Sci., vol  13, p.p. 773-782, 1982.
[8]     C. Hwang and Y. P. Shih, "Solution of integral equations via Laguerre polynomials," Comp. and Elect. Engng., vol. 9, p.p.  123-129, 1982.
[9]     R. Y. Chang and M. L. Wang, "Solutions of integral equations via shifted Legendre polynomials, " Int. J. Syst. Sci., vol 16, p.p. 197-208, 1985.
[10]  J. H. Chou and I. R. Horng, "Double shifted chebyshev series for convoluation integral and integral equations," Int. J. Contr., vol. 42, p.p. 225-232, 1985.
[11]  M. Razzaghi, M. Razzaghi and A. Arabshahi, "Solution of convolution integral and Fredholm integral equations via double Fourier series, " Appl. Math. Comp., vol. 40, p.p. 215-224, 1990.
[12]  M. Razzaghi and J. Nazarzadeh, "Walsh functions," Wiley Encyclopedia of Electrical and Electronics Engineering, vol. 23 , p.p. 429-440, 1999.
[13]  K. G. Beauchamp, "Walsh functions and their applications," 1975.
[14]  R. T. Lynch and J. J. Reis, "Haar transform image coding," National Telecommun. Conf., Dallas, TX, 44.3-1-44.3, 1976.
[15]  J. J. Reis and R. T. Lynch and J. Butman, "Adaptive Haar transform video bandwidth reduction system for RPV's,"  Ann. Meeting Soc. Photo Optic Inst. Eng. (SPIE), San Dieago, CA.  p.p. 24-35, 1976.
[16]  M. Ohkita and Y. Kobayashi, "An application of rationalized Haar functions to solution of linear differential equations," IEEE Trans, on Circuit and systems, vol. 19, p.p. 853-862, 1986.
[17]  M. Ohkita and Y. Kobayashi, "An application of rationalized Haar functions to solution of linear partial differential equations," Mathematics and Computers in Simulations, vol. 30, p.p. 419-428, 1988.
[18]  M.Razzaghi and H. Marzban, "A hybrid analysis direct method in the calculus of variations," Intern. J. Computer Math., vol. 75,  p.p. 259-269, 1999.
[19]  S. Yalcinbas, "Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations,"Applied Mathematics and Computation, vol. 127, p.p. 195-206, 2002.
[20]  G. M. Phillips and P. J. Taylor, "Theory and Application of Numerical Analysis," Academic Press , New York,  1973.
[21]  M. Razzaghi and Y. Ordokhani, "Solution of nonlinear Volterra-Hammerstein integral equations via rationalized Haar functions, " Mathematical Problems in Engineering, vol. 7, p.p. 205-218,  2001.
[22]  A.M. Wazwaz, "A first course in integral equations," World scientific Publishing Company, New Jersey, 1997.
[23]  M. Razzaghi and Y.Ordokhani, "An application of rationalized Haar functions for variational problems," Applied Mathematics and Computation, vol. 122, p.p. 353-364, 2001.
[24]  G.N. Elnagar, and M. Kazemi, "Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations," J. Computational and Applied Mathematics, vol. 76, p.p. 147-158, 1996.