Abstract
Direct methods for solving Cauchy-type singular integral equations (S.I.E.) are based on Gauss numerical integration rule [1] where the S.I.E. is reduced to a linear system of equations by applying the resulting functional equation at properly selected collocation points. The equivalence of this formulation with the one based on the Lagrange interpolatory approximation of the unknown function was shown in the paper.
Indirect methods for the solution of S. I. E. may be obtained after a reduction of it to an equivalent Fredholm integral equation and an application of the same numerical technique to the latter.
It was shown in this paper that both methods are equivalent in the sense that they give the same numerical results. Using these results the error estimate and the convergence of the methods was established.
Zusammenfassung
Direkte Methoden zur Lösung von singulären Integralgleichungen vom Cauchy-Typus (S. I. G.) beruhen auf der Gaussschen Regel für numerische Integration, wobei die S. I. G. durch Anwendung der resultierenden Funktionalgleichung an geeignet gewählten Kollokationspunkten auf ein lineares Gleichungssystem reduziert wird. In diesem Artikel wurde die Äquivalenz dieser Methode mit derjenigen, welche auf der Lagrangeschen Interpolations-Approximation der unbekannten Funktion, beruht, gezeigt.
Indirekte Methoden zur Lösung von S. I. G. können durch Anwendung derselben numerischen Regel an der Fredholmschen Integralgleichung, auf welche die S. I. G. reduziert wird, erhalten werden.
In diesem Artikel wurde gezeigt, daß beide Methoden, im Sinne, daß sie dieselben numerischen Resultate liefern, äquivalent sind. Schließlich wurde mit Hilfe dieser Resultate der, Fehler und die Konvergenz der Methoden festgestellt.
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Tsamasphyros, G., Theocaris, P.S. Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations. Computing 27, 71–80 (1981). https://doi.org/10.1007/BF02243439
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DOI: https://doi.org/10.1007/BF02243439