This monograph covers a wide range of topics in the theory of hypersingular integral equations and the associated areas of numerical analysis, aerodynamics, diffraction of waves and theory of elasticity.
Various methods for solving singular and hypersingular integral equations are presented in the book. In particular, the authors construct new exact analytical solutions of some two-dimensional hypersingular integral equations and obtain an analytical solution in quadratures for the hypersingular equation on the sphere. They present theorems for the existence and uniqueness of solutions for hypersingular integral equations, and the theory they develop in this monograph can be used to investigate the convergence of approximate solutions obtained by finite-difference methods for elliptic-type partial differential equations.
Numerous examples demonstrate some of the many applications of hypersingular integral equations. These include solutions to problems involving flow past the deck of a ship, the diffraction of acoustic waves on a cube, and flow past an airfoil with suction on its surface. This is a lucid monograph written in a deceptively easy style. The book will be useful to researchers and graduate students in various areas of mathematics, mechanics and physics.
Table of Contents: Singular integrals and integral equations. Sobolev-Slobodetskii spaces. Hypersingular integral equations. Neumann problem and integral equations with double layer potential. Spaces of fractional quotients and their properties. Discrete operators in quotient spaces. Stability of discrete operators in quotient spaces. Asymptotic estimates of discrete Green function. Quadrature formulas for singular and hypersingular integrals. Numerical analysis of hypersingular integral equations. Problems in aerodynamics. Some problems of physics.