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

Afendikova, N. G.; Lifanov, I. K.; Matveev, A. F.

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

Keywords:

Carleman-Vekua regularization method; generalized polynomial approximation; convergence; direct method; collocation; uniform metric