Problems of convergence of the direct methods of numerical solution of singular integral equations with Cauchy-type kernels

• Main Author: Ioakimidis, Nikolaos
• Other Authors: Ιωακειμίδης, Νικόλαος
• Format: Technical Report
• Language: English
• Published: 2018
• Subjects: Cauchy-type singular integral equations; Direct methods; Numerical solution; Convergence; Galerkin method; Collocation method; Quadrature–collocation method; Quadrature method; Ιδιόμορφες ολοκληρωτικές εξισώσεις τύπου Cauchy; Άμεσες μέθοδοι; Αριθμητική επίλυση; Σύγκλιση; Μέθοδος Galerkin; Μέθοδος συντοπισμού; Μέθοδος αριθμητικής ολοκληρώσεως–συντοπισμού; Μέθοδος αριθμητικής ολοκληρώσεως

In this technical report, general directions are given for the proof of the convergence of four direct methods of numerical solution of real Cauchy-type singular integral equations on a finite open interval. The methods under consideration are (i) the Galerkin method, (ii) the collocation method, (iii) the quadrature–collocation method and (iv) the quadrature method. It is believed that the given directions can really be applied to the detailed and rigorous proof of the convergence of the aforementioned methods as well as of additional related or even more general methods.