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On Taylor-series expansion methods for the second kind integral equations. (English) Zbl 1190.65195
Summary: We comment on the recent papers by {\it Y. Ren, B. Zhang} and {\it H. Qiao} [ibid. 110, No. 1, 15--24 (1999; Zbl 0936.65146)] and {\it K. Maleknejad, N. Aghazadeh}, and {\it M. Rabbani} [Appl. Math. Comput. 175, No. 2, 1229--1234 (2006; Zbl 1093.65124)] concerning the use of the Taylor series to approximate a solution of the Fredholm integral equation of the second kind as well as a solution of a system of Fredholm equations. The technique presented in Ren et al. [loc. cit.] takes advantage of a rapidly decaying convolution kernel $k(|s - t|)$ as $|s - t|$ increases. However, it does not apply to equations having other types of kernels. We present in this paper a more general Taylor expansion method which can be applied to approximate a solution of the Fredholm equation having a smooth kernel. Also, it is shown that when the new method is applied to the Fredholm equation with a rapidly decaying kernel, it provides more accurate results than the method by Ren et al. [loc. cit.]. We also discuss an application of the new Taylor-series method to a system of Fredholm integral equations of the second kind.

##### MSC:
 65R20 Integral equations (numerical methods) 45B05 Fredholm integral equations 45F05 Systems of nonsingular linear integral equations 45F15 Systems of singular linear integral equations
Full Text:
##### References:
 [1] Ren, Yuhe; Zhang, Bo; Qiao, Hong: A simple Taylor-series expansion method for a class of second integral equations, J. comput. Appl. math. 110, 15-24 (1999) · Zbl 0936.65146 · doi:10.1016/S0377-0427(99)00192-2 [2] Atkinson, K. E.: The numerical solution of integral equations of the second kind, Cambridge monographs on applied and computational mathematics (1997) [3] Kaneko, H.; Xu, Y.: Gauss-type quadratures for weakly singular integrals and their applications to Fredholm integral equations of the second kind, Math. comp. 62, No. 206, 739-753 (1994) · Zbl 0799.65023 · doi:10.2307/2153534 [4] Kaneko, H.; Noren, R.: An application of approximation theory to the numerical solution of Fredholm integral equations of the second kind, Numer. funct. Anal. optim. 12, No. 5--6, 517-523 (1991) · Zbl 0757.65143 · doi:10.1080/01630569108816447 [5] Kaneko, H.; Xu, Y.: Superconvergence of the iterated Galerkin method for Hammerstein equations, SIAM J. Numer. anal. 33, 1048-1064 (1996) · Zbl 0860.65138 · doi:10.1137/0733051 [6] P. Huabsomboon, B. Novaprateep, H. Kaneko, Taylor-series expansion method for volterra integral equations of the second kind, Appl. Math. Comput. (submitted for publication). · Zbl 1217.65238 · http://www.jams.or.jp/scm/abstract/e-2010/2010-60a.txt [7] Maleknejad, K.; Aghazadeh, N.; Rabbani, M.: Numerical omputational solution of second kind Fredholm integral equations system by using a Taylor-series expansion method, Appl. math. Comput. 175, 1229-1234 (2006) · Zbl 1093.65124 · doi:10.1016/j.amc.2005.08.039 [8] P. Huabsomboon, B. Novaprateep, H. Kaneko, Taylor-series expansion methods for nonlinear Hammerstein equations, Appl. Math. Comput. (submitted for publication). · Zbl 1271.65153