A numerical method for solving nonlinear Fredholm integral equations, based on the Haar wavelet approach, is presented. Its efficiency is tested by solving four examples for which the exact solution is known. This allows us to estimate the exactness of the obtained numerical results. High accuracy of the results even in the case of a small number of grid points is observed.
1. Atkinson, K. E. A survey of numerical methods for solving nonlinear integral equations. J. Integral Equations Appl., 1992, 4, 15–40.
https://doi.org/10.1216/jiea/1181075664
2. Atkinson, K. and Flores, J. The discrete collocation method for nonlinear integral equations. IMA J. Numer. Anal., 1993, 13, 195–213.
https://doi.org/10.1093/imanum/13.2.195
3. Kaneko, H., Noren, R. D. and Padilla, P. A. Superconvergence of the iterated collocation methods for Hammerstein equations. J. Comput. Appl. Math., 1997, 80, 335–349.
https://doi.org/10.1016/S0377-0427(97)00040-X
4. Pedas, A. and Vainikko, G. Superconvergence of piecewise polynomial collocations for nonlinear weakly singular integral equations. J. Integral Equations Appl., 1997, 9, 379–405.
https://doi.org/10.1216/jiea/1181076030
5. Vainikko, G., Kivinukk, A. and Lippus, J. Fast solvers of integral equations of the second kind: wavelet methods. J. Complexity, 2005, 21, 243–273.
https://doi.org/10.1016/j.jco.2004.07.002
6. Chen, W.-S. and Lin, W. Galerkin trigonometric wavelet methods for the natural boundary integral equations. Appl. Math. Comput., 2005, 121, 79–92.
https://doi.org/10.1016/S0096-3003(99)00263-5
7. Anioutine, A. P. and Kyurkchan, A. G. Application of wavelets technique to the integral equations of the method of auxiliary currents. J. Quant. Spectrosc. Radiat. Transf., 2003, 79–80, 495–508.
https://doi.org/10.1016/S0022-4073(02)00303-5
8. Sánchez-Ávila, C. Wavelet domain signal deconvolution with singularity-preserving regularization. Math. Comput. Simul., 2003, 61, 165–176.
https://doi.org/10.1016/S0378-4754(02)00073-3
9. Liang, X.-Z., Liu, M.-C. and Che, X.-J. Solving second kind integral equations by Galerkin methods with continuous orthogonal wavelets. J. Comput. Appl. Math., 2001, 136, 149–161.
https://doi.org/10.1016/S0377-0427(00)00581-1
10. Kaneko, H., Noren, R. D. and Novaprateep, B. Wavelet applications of the Petrov–Galerkin method for Hammerstein equations. Appl. Numer. Math., 2003, 45, 255–273.
https://doi.org/10.1016/S0168-9274(02)00173-3
11. Lepik, Ü. and Tamme, E. Application of the Haar wavelets for solution of linear integral equations. In Dynamical Systems and Applications, Proceedings, 5--10 July 2004, Antalya, Turkey. 2005, 395–407.
12. Maleknejad, K. and Mirzaee, F. Using rationalized Haar wavelet for solving linear integral equations. Appl. Math. Comput., 2005, 160, 579–587.
https://doi.org/10.1016/j.amc.2003.11.036
13. Chen, C. F. and Hsiao, C. H. Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proc. Control Theory Appl., 1997, 144, 87–94.
https://doi.org/10.1049/ip-cta:19970702
