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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 Y. Ren, B. Zhang and H. Qiao [ibid. 110, No. 1, 15–24 (1999; Zbl 0936.65146)] and K. Maleknejad, N. Aghazadeh, and 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 $k\left(|s-t|\right)$ 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