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Solution methods for bisingular integral equations with internal coefficients. (English. Russian original) Zbl 1100.65124
Russ. Math. 48, No. 8, 9-23 (2004); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2004, No. 8, 11-25 (2004).
From the introduction: The theory of equations of the form $a_0(s,\sigma) \varphi(s,\sigma)+\frac{1}{2\pi}\int_0^{2\pi}a_1(s, \sigma;\xi)\text{ctg}\frac {\xi-s}{2}\varphi(\xi,\sigma)d\xi+$ $+ \frac{1}{2\pi}\int_0^{2\pi}a_2(s,\sigma; \eta)\text{ctg}\frac{\eta-\sigma}{2}\varphi(s,\eta)d\eta+\tag{1}$ $+\frac{1} {4\pi^2} \int_0^{2\pi}\int_0^{2\pi}a_{12}(s,\sigma;\xi,\eta)\text{ctg}\frac{\xi-s}{2}\text{ctg}\frac{\eta-\sigma}{2}\varphi(\xi,\eta)d\xi\,d\eta=f(s, \sigma),\;-\infty<s,\;\sigma<\infty,$ including approximative methods for their solution, is very complicated and is far from being completed. Below we formulate simple sufficient conditions of existence, uniqueness and stability of solutions of equation, and we propose practically efficient (including optimal) approximative solution methods.

##### MSC:
 65R20 Numerical methods for integral equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 45M10 Stability theory for integral equations