Geodesic
The solution of singular integral equations by approximate projection methods
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 19 (1979) no. 5, pp. 1149-1161

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The approximate solution of singular equations of the 1st and 2nd kinds, when the line of integration is a segment, is considered. By contraction of the domain of definition or range of values of the operator, the one-to-one property of the mapping is established. Versions of the Bubnov-Galerkin method are used for the approximation solution. Chebyshev and Jacobi polynomials are used as coordinate elements. It is shown that the algebraic system is uniquely solvable for fairly large $n$, and that the approximate solutions converge to the exact solution in spaces with a weight. The process is stable. 
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     author = {A. V. Dzhishkariani},
     title = {The solution of~singular integral equations by~approximate projection methods},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     publisher = {mathdoc},
     volume = {19},
     number = {5},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_1979_19_5_a7/}
}
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A. V. Dzhishkariani. The solution of singular integral equations by approximate projection methods. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 19 (1979) no. 5, pp. 1149-1161. http://geodesic.mathdoc.fr/item/ZVMMF_1979_19_5_a7/