Geodesic
Uniform convergence of the rectangle method for singular integral equation with the H\"older density
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2012), pp. 3-12.

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We investigate an approximate method of solving singular integral equation. The method consists in approximation of singular equation with the use of compound formula of rectangles type. The corresponding systems of linear algebraic equations are uniquely solvable if integral equation is solvable, and coefficients of an equation satisfy the strong ellipticity condition. Under these conditions we estimate the rate of the convergence of solutions of systems of linear equations to the solution of the integral equation in the uniform vector norm.
Keywords: singular integral equation, discretization of integral operators by the method of rectangles, Hölder function, Hölder density, convergence in the uniform vector norm. 
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M. E. Abramyan. Uniform convergence of the rectangle method for singular integral equation with the H\"older density. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2012), pp. 3-12. http://geodesic.mathdoc.fr/item/IVM_2012_2_a0/

[1] Belotserkovskii S. M., Lifanov I. K., Chislennye metody v singulyarnykh integralnykh uravneniyakh, Nauka, M., 1985 | MR | Zbl

[2] Lifanov I. K., Metod singulyarnykh integralnykh uravnenii i chislennyi eksperiment (v matematicheskoi fizike, aerodinamike, teorii uprugosti i difraktsii voln), Yanus, M., 1995 | MR | Zbl

[3] Rathsfeld A., “Quadraturformelverfahren für eindimensionale singuläre Integralgleichungen”, Semin. Anal. Operator equal. and num. analysis, Karl Weierstrass-Inst. Math. Akad. Wiss. DDR, Berlin, 1986, 147–186 | MR

[4] Prößdorf S., Rathsfeld A., “Stabilitätskriterien für Näherungsverfahren bei singulären Integralgleichungen in $L^p$”, Z. Anal. Anwend., 6 (1987), 539–558 | MR | Zbl

[5] Prößdorf S., Silbermann B., Numerical analysis for integral and related operator equations, Birkhäuser Verlag, Basel, 1991 | MR | Zbl

[6] Muskhelishvili N. I., Singulyarnye integralnye uravneniya, Nauka, M., 1968 | MR | Zbl

[7] Voevodin V. V., Kuznetsov Yu. A., Matritsy i vychisleniya, Nauka, M., 1984 | MR | Zbl

[8] Abramyan M. E., “Obosnovanie skhodimosti v ravnomernoi norme metoda pryamougolnikov dlya polnogo singulyarnogo integralnogo uravneniya s nepreryvnymi koeffitsientami na okruzhnosti”, Izv. vuzov. Sev.-Kavk. region. Estestv. nauki, 2008, no. 5, 5–10 | Zbl

[9] Gokhberg I. Ts., Krupnik N. Ya., Vvedenie v teoriyu odnomernykh singulyarnykh integralnykh operatorov, Shtiintsa, Kishinev, 1973 | MR

[10] Abramyan M. E., “Obosnovanie skhodimosti metoda pryamougolnikov dlya polnogo singulyarnogo integralnogo uravneniya s nepreryvnymi koeffitsientami na okruzhnosti”, Matem. zametki, 77:2 (2005), 163–175 | MR | Zbl

[11] Abramyan M. E., Pilidi V. S., Obosnovanie metoda pryamougolnikov dlya singulyarnogo integralnogo uravneniya s postoyannymi koeffitsientami na okruzhnosti, VINITI, No 2385-B98, Rostovsk. gos. un-t, 1998

[12] Zabreiko P. P., Koshelev A. I., Krasnoselskii M. A. i dr., Integralnye uravneniya, Nauka, M., 1968 | MR

[13] Pilidi V. S., Lokalnyi metod issledovaniya lineinykh operatornykh uravnenii tipa bisingulyarnykh integralnykh uravnenii, Diss. $\dots$ kand. fiz.-matem. nauk, RGU, Rostov n/D, 1972

[14] Pilidi V. S., “Kriterii ravnomernoi obratimosti regulyarnykh approksimatsii odnomernykh singulyarnykh integralnykh operatorov s kusochno nepreryvnymi koeffitsientami”, Izv. AN SSSR. Seriya matem., 54:6 (1990), 1270–1294 | MR | Zbl