## A composite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equations.(English)Zbl 1221.65340

Summary: This paper presents a computational technique for the solution of the nonlinear mixed Volterra-Fredholm-Hammerstein integral equations. The method is based on the composite collocation method. The properties of hybrid of block-pulse functions and Lagrange polynomials are discussed and utilized to define the composite interpolation operator. The estimates for the errors are given. The composite interpolation operator together with the Gaussian integration formula are then used to transform the nonlinear mixed Volterra-Fredholm-Hammerstein integral equations into a system of nonlinear equations. The efficiency and accuracy of the proposed method is illustrated by four numerical examples.

### MSC:

 65R20 Numerical methods for integral equations 45G10 Other nonlinear integral equations
Full Text:

### References:

 [1] Tricomi, F. G., Integral equations (1982), Dover [2] Lardy, L. J., A variation of Nystroms method for Hammerstein integral equations, J Integ Equ, 3, 123-129 (1982) [3] Kumar, S.; Sloan, I. H., A new collocation-type method for Hammerstein integral equations, J Math Comput, 48, 123-129 (1987) [4] Brunner, H., Implicitly linear collocation method for nonlinear Volterra equations, J Appl Num Math, 9, 235-247 (1992) · Zbl 0761.65103 [5] Han, G., Asymptotic error expansion variation of a collocation method for Volterra-Hammerstein equations, J Appl Num Math, 13, 357-369 (1993) · Zbl 0799.65150 [6] Li, F.; Li, Y.; Liang, Z., Existence of solutions to nonlinear Hammerstein integral equations and applications, J Math Anal Appl, 323, 209-227 (2006) · Zbl 1104.45003 [7] Yalcinbas, S., Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Appl Math Comput, 127, 195-206 (2002) · Zbl 1025.45003 [8] Bildik, N.; Inc, M., Modified decomposition method for nonlinear Volterra-Fredholm integral equations, Chaos, Solitons Fractals, 33, 308-313 (2007) · Zbl 1152.45301 [9] Yousefi, S.; Razzaghi, M., Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Math Comput Simul, 70, 1-8 (2005) · Zbl 1205.65342 [10] Ordokhani, Y., Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via rationalized Haar functions, Appl Math Comput, 180, 436-443 (2006) · Zbl 1102.65141 [11] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral methods: fundamentals in single domains (2006), Springer · Zbl 1093.76002 [12] Maday, Y.; Quarteroni, A., Spectral and pseudo-spectral approximations of Navier-Stokes equations, SIAM J Numer Anal, 19, 761-780 (1982) · Zbl 0503.76035 [13] Westhaven, J. S.; Gottlieb, S.; Gottlieb, D., Spectral methods for time-dependent problems (2007), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1111.65093 [14] Davis, P. J.; Rabinowitz, P., Methods of numerical integration (1984), Academic Press · Zbl 0537.65020 [15] Bellman, R. E.; Kalaba, R. E., Quasilinearization and nonlinear boundary-value problems (1965), Elsevier · Zbl 0139.10702 [16] Madbouly, N.; Mc Ghee, D. F.; Roach, G. F., Adomian’s method for Hammerstein integral equations arising from chemical reactor theory, Appl Math Comput, 117, 241-249 (2001) · Zbl 1023.65143 [17] Poore AB, A tubular chemical reactor model, In: Collection of nonlinear model problems contributed to the proceeding of the AMS-SIAM. 1989. p. 28-31.; Poore AB, A tubular chemical reactor model, In: Collection of nonlinear model problems contributed to the proceeding of the AMS-SIAM. 1989. p. 28-31. [18] Madbouly N, Solutions of Hammerstein integral equations arising from chemical reactor theory. Ph.D. Thesis, University of Strathclyde; 1996.; Madbouly N, Solutions of Hammerstein integral equations arising from chemical reactor theory. Ph.D. Thesis, University of Strathclyde; 1996.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.