Solution methods for bisingular integral equations with internal coefficients. (English. Russian original) Zbl 1100.65124

Gabdulkhaev, B. G.

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

Keywords:

two-dimensional singular integral equations; projection method; convergence; error estimates; quadrature-cubature method; stability