Geodesic
A polynomial collocation method for Cauchy singular integral equations over the interval
Electronic transactions on numerical analysis, Tome 14 (2002), pp. 79-126
In this paper we consider a polynomial collocation method for the numerical solution of a singular integral equation over the interval. More precisely, the operator of our integral equation is supposed to be of the form $aI + \mu - 1bS\mu I + K$ with S the Cauchy integral operator, with piecewise continuous coefficients a and b , with a regular integral operator K , and with a Jacobi weight $\mu $. To the equation $[aI + \mu - 1bS\mu I + K]$u = f we apply a collocation method, where the collocation points are the Chebyshev nodes of the second kind and where the trial space is the space of polynomials multiplied by another Jacobi weight. For the stability and convergence of this collocation in weighted L2 spaces, we derive necessary and sufficient conditions.
Classification : 45L10, 65R20, 65N38
Keywords: Cauchy singular integral equation, polynomial collocation method, stability 
@article{ETNA_2002__14__a7,
     author = {Junghanns,  P. and Rathsfeld,  A.},
     title = {A polynomial collocation method for {Cauchy} singular integral equations over the interval},
     journal = {Electronic transactions on numerical analysis},
     pages = {79--126},
     year = {2002},
     volume = {14},
     zbl = {1024.65131},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2002__14__a7/}
}
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Junghanns,  P.; Rathsfeld,  A. A polynomial collocation method for Cauchy singular integral equations over the interval. Electronic transactions on numerical analysis, Tome 14 (2002), pp. 79-126. http://geodesic.mathdoc.fr/item/ETNA_2002__14__a7/