A nodal spline collocation method for the solution of Cauchy singular integral equations. (English) Zbl 1176.65153

Dagnino, C.; Demichelis, V.; Lamberti, Paola

JNAIAM, J. Numer. Anal. Ind. Appl. Math. 3, No. 3-4, 211-220 (2008).

The authors present a collocation method, based on optimal nodal spline approximation, for solving the following singular integral equation with constant coefficients
\[ aw_{\alpha ,\beta } (x)f(x) + \frac{b}{\pi }\int_{ - 1}^1 {\frac{w_{\alpha ,\beta } (t)f(t)}{t - x}dt} + \int_{ - 1}^1 {w_{\alpha ,\beta } (t)k(x,t)f(t)dt = g(x),\quad - 1 < x < 1}, \]
where the first integral is defined in the sense of Cauchy principal value, \(k\) is a Fredholm kernel and \(w_{\alpha ,\beta } \) is the Jacobi weight function
\[ w_{\alpha ,\beta } (t) = (1 - t)^\alpha (1 + t)^\beta ,\,\,\alpha ,\beta > - 1. \]

Reviewer: Temuri A. Jangveladze (Tbilisi)

MSC:

65R20	Numerical methods for integral equations
45E05	Integral equations with kernels of Cauchy type

Keywords:

Nodal splines; Cauchy singular integral equations; collocation method; Cauchy principal value; Fredholm kernel; Jacobi weight function