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**The method of singular integral equations and a numerical experiment in mathematical physics, aerodynamics and the theory of elasticity and wave diffraction.
(Метод сингулярных интеграл’ных уравнений и численныйѐксперимент в математической физике, аѐродинамике, теории упругости и дифракции волн.)**
*(Russian)*
Zbl 0904.73001

Moskva: TOO Yanus. 520 p. (1995).

The first part of the book contains elements of singular integral equation theory in classes of absolutely integrable and nonintegrable functions. Equations with multiple integrals of Cauchy and Hilbert type are also considered.

In the second part the author gives elements of potential theory for Helmholtz equation. It is shown that Dirichlet and Neumann problems can be reduced to singular integral equations. For the planar case, singularities of solutions of such equations are studied. Stationary and nonstationary, planar and spatial aerohydrodynamic problems are formulated as boundary value problems which are then reduced to boundary singular integral equations. For these integral equations, some peculiarities of their solving are indicated. Such a way is made for electrostatic problems, for dual summatory equations, and some planar problems of elasticity theory.

The third part contains calculation methods for one- and two-dimensional singular integrals from the second part. For singular integrals arising in problems of wave diffraction, the author derives for the first time quadrature formulae of discrete vortex pair type in planar case and of closed vortex frame type in spatial case. Concrete calculations of some singular integrals are given.

These quadrature formulae are applied in the fourth part to numerical solution of singular integral equations of the first and second kinds, with constant and variable coefficients. Here equations with both Cauchy and Hilbert kernels are considered.

The final part contains discrete mathematical models for problems of aerodynamics, electrodynamics and elasticity theory described in the second part of the book. Basing on these models, the author performs some numerical experiments.

In the second part the author gives elements of potential theory for Helmholtz equation. It is shown that Dirichlet and Neumann problems can be reduced to singular integral equations. For the planar case, singularities of solutions of such equations are studied. Stationary and nonstationary, planar and spatial aerohydrodynamic problems are formulated as boundary value problems which are then reduced to boundary singular integral equations. For these integral equations, some peculiarities of their solving are indicated. Such a way is made for electrostatic problems, for dual summatory equations, and some planar problems of elasticity theory.

The third part contains calculation methods for one- and two-dimensional singular integrals from the second part. For singular integrals arising in problems of wave diffraction, the author derives for the first time quadrature formulae of discrete vortex pair type in planar case and of closed vortex frame type in spatial case. Concrete calculations of some singular integrals are given.

These quadrature formulae are applied in the fourth part to numerical solution of singular integral equations of the first and second kinds, with constant and variable coefficients. Here equations with both Cauchy and Hilbert kernels are considered.

The final part contains discrete mathematical models for problems of aerodynamics, electrodynamics and elasticity theory described in the second part of the book. Basing on these models, the author performs some numerical experiments.

Reviewer: P.A.Velmisov (Ul’yanovsk)

### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65R20 | Numerical methods for integral equations |

45Exx | Singular integral equations |

74B05 | Classical linear elasticity |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

35Q72 | Other PDE from mechanics (MSC2000) |

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |