×

zbMATH — the first resource for mathematics

Legendre wavelets method for the nonlinear Volterra—Fredholm integral equations. (English) Zbl 1205.65342
Summary: A numerical method for solving the nonlinear Volterra—Fredholm integral equations is presented. The method is based upon Legendre wavelet approximations. The properties of Legendre wavelet are first presented. These properties together with the Gaussian integration method are then utilized to reduce the Volterra—Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

MSC:
65R20 Numerical methods for integral equations
65T60 Numerical methods for wavelets
45G10 Other nonlinear integral equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chui, C.K., Wavelets: A mathematical tool for signal analysis, (1997), SIAM Philadelphia, PA · Zbl 0903.94007
[2] Beylkin, G.; Coifman, R.; Rokhlin, V., Fast wavelet transforms and numerical algorithms, I. commun. pure appl. math., 44, 141-183, (1991) · Zbl 0722.65022
[3] Tricomi, F.G., Integral equations, (1982), Dover Publications, New York
[4] Lardy, L.J., A variation of nystrom’s method for Hammerstein equations, J. integral equat., 3, 43-60, (1981) · Zbl 0471.65096
[5] Kumar, S.; Sloan, I.H., A new collocation-type method for Hammerstein integral equations, J. math. comput., 48, 123-129, (1987)
[6] Brunner, H., Implicitly linear collocation method for nonlinear Volterra equations, J. appl. num. math., 9, 235-247, (1992) · Zbl 0761.65103
[7] Guoqiang, H., Asymptotic error expansion variation of a collocation method for volterra – hammerstein equations, J. appl. num. math., 13, 357-369, (1993) · Zbl 0799.65150
[8] Yalcinbas, S., Taylor polynomial solution of nonlinear volterra – fredholm integral equations, Appl. math. comput., 127, 195-206, (2002) · Zbl 1025.45003
[9] Constantinides, A., Applied numerical methods with personal computers, (1987), McGraw-Hill New York · Zbl 0653.65001
[10] Gu, J.S.; Jiang, W.S., The Haar wavelets operational matrix of integration, Int. J. syst. sci., 27, 623-628, (1996) · Zbl 0875.93116
[11] Razzaghi, M.; Yousefi, S., Legendre wavelets direct method for variational problems, Math. comp. simulat., 53, 185-192, (2000)
[12] Razzaghi, M.; Yousefi, S., Legendre wavelets method for constrained optimal control problems, Math. meth. appl. sci., 25, 529-539, (2002) · Zbl 1001.49033
[13] Wazwaz, A.M., A first course in integral equations, (1997), World scientific Publishing Company New Jersey
[14] Kalman, R.E.; Kalaba, R.E., Quasilinearization and nonlinear boundary-value problems, (1996), Elsevier New York · Zbl 0165.18103
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.