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**Spaces of fractional quotients, discrete operators, and their applications. II.**
*(English.
Russian original)*
Zbl 0985.47028

Sb. Math. 190, No. 11, 1623-1687 (1999); translation from Mat. Sb. 190, No. 11, 67-134 (1999).

The authors continue the investigation started in the previous paper [Sb. Math. 190, No. 9, 1267-1323 (1999); translation from Mat. Sb. 190, No. 9, 41-98 (1999; Zbl 0953.46021)].

Authors’ abstract: The theory of discrete operators in spaces of fractional quotients is developed. A theorem on the stability of discrete operators under smooth perturbations is proved. On this basis, using special quadrature formulae of rectangular kind, the convergence of approximate solutions of hypersingular integral operators to their exact solutions is demonstrated and mathematical substantiation of the method of closed discrete vortex frameworks is obtained. The same line of arguments is also applied to difference equations arising in the solution of the homogeneous Dirichlet problem for a general second-order elliptic equations with variable coefficients.

Authors’ abstract: The theory of discrete operators in spaces of fractional quotients is developed. A theorem on the stability of discrete operators under smooth perturbations is proved. On this basis, using special quadrature formulae of rectangular kind, the convergence of approximate solutions of hypersingular integral operators to their exact solutions is demonstrated and mathematical substantiation of the method of closed discrete vortex frameworks is obtained. The same line of arguments is also applied to difference equations arising in the solution of the homogeneous Dirichlet problem for a general second-order elliptic equations with variable coefficients.

Reviewer: Roland Duduchava (Tbilisi)

### MSC:

47B39 | Linear difference operators |

39A70 | Difference operators |

46E99 | Linear function spaces and their duals |

65N06 | Finite difference methods for boundary value problems involving PDEs |

65N38 | Boundary element methods for boundary value problems involving PDEs |