The semi-inversion method for summation dual equations is discussed in this monograph. The method is applied to numerical solution of many diffraction problems of practical interest, such as diffraction on metallic cylinders with slots and spherical screens with holes.
The book consists of three parts. In the first part the theory of the summation dual equations is discussed. In the second part various diffraction problems for one or several cylinders with slots are treated. In the third part diffraction problems for spherical screens with holes are investigated. A typical example of summation dual equations is the equation:
(1) \(\sum^{\infty}_{n=-\infty}x_ n\exp (in\phi) = 0\), \(0<\alpha \leq | \phi | \leq \pi\), (2) \(\sum^{\infty}_{n=-\infty}C_ nx_ n\exp (in\phi) = f(\phi)\), \(| \phi | \leq \alpha\).
Here f(\(\phi)\) and \(\{C_ n\}\) are the data, \(x_ n\) are to be found. Similar equations with functions other than trigonometric are studied. These equations remind one of dual integral equations. The idea of the semi-inversion method is as follows. Suppose that \(C_ n=\gamma_ n+\epsilon_ n\), where \(\epsilon_ n\) are in some sense small and the problem (1), (2) with \(\gamma_ n\) in place of \(C_ n\) can be solved. Then one writes (1), (2) as an operator equation \((\Gamma +\epsilon)x=f\), passes to the equation \((I+\Gamma^{-1}\epsilon)x=\Gamma^{-1}f\), and solves the last equation by iterations, provided that \(\| \Gamma^{- 1}\epsilon \| <1\).
The bibliography contains 156 entries. The book will be of interest to specialists in computational electrodynamics.