## Approximate solution of singular integral equations.(English. Russian original)Zbl 0645.65092

Differ. Equations 23, No. 8, 953-962 (1987); translation from Differ. Uravn. 23, No. 8, 1392-1402 (1987).
The paper deals with the construction of a direct method for the numerical solution of the following singular integral equations $(1)\quad a(t_ 0)\phi (t_ 0)+\frac{b(t_ 0)}{\pi_ i}\int_{L}\frac{\phi (t)dt}{t-t_ 0}+\int_{L}K(t_ 0,t)\phi (t)dt=f(t_ 0)$
$(2)\quad a(\vartheta_ 0)\phi (\vartheta_ 0)- \frac{b(\vartheta_ 0)}{2\pi}\int^{2\pi}_{0}ctg\frac{\vartheta - \vartheta_ 0}{2}\phi (\vartheta)d\vartheta +\int^{2\pi}_{0}K(\vartheta_ 0,\vartheta)\phi (\vartheta)d\vartheta =f(\vartheta_ 0)$ where L is a piecewise smooth curve, the functions a, b, K and f are Hölder continuous on L or [0,2$$\pi$$ ], $$2\pi$$- periodic for the second equation, and such that $$a^ 2-b^ 2\neq 0.$$
The approximate solution $$\psi_{n,\kappa}$$ with the index $$\kappa$$ is supposed to be of the form $$w_{\kappa}^{(+)}(t)\psi_{n_ 1-1;n_ 2}(t)$$, where $$\psi_{n_ 1-1;n_ 2}(t)$$ is a generalized polynomial approximant of $$\psi$$ of the form $\psi_{n_ 1-1;n_ 2}(t)=\sum^{n}_{k=1}\psi_ n(t_ k)(_ kP_{n_ 1-1;n_ 2}(t))/(t-t_ k),$ $$w_{\kappa}^{(+)}(t)=Z(t)/[a^ 2(t)-b^ 2(t)]$$ and Z(t) is the canonical function. Then, under suitable assumptions, the corresponding system of linear equations is non-singular for sufficiently large n and arbitrary collocation points $$t_ k$$, and its solution converges to the solution of (1) in the uniform metric. The procedure for the equation (2) is analogous.
Reviewer: P.Polcar

### MSC:

 65R20 Numerical methods for integral equations 45E05 Integral equations with kernels of Cauchy type