The developments of mathematical methods for the study of direct and inverse problems in electrodynamics

Since the discovery of electromagnetic waves and the formulation of Maxwell's equations, the theory of electromagnetic waves has become one of the most important branches of mathematical physics. The variety of problems in electrodynamics has often stimulated the raising and development of new problems in mathematical physics. Examples are the study of the interior structure of the earth by electromagnetic methods, which has promoted the development of the general theory of inverse problems; the propagation of electromagnetic waves in non-homogeneous media, which has led to the development of the mathematical theory of diffraction; problems of the transmission of ultra-high frequency electromagnetic waves, which has stimulated the development of the mathematical theory of wave-guide propagation of oscillations; problems of synthesizing systems of antennae and various electromagnetic apparatuses, effective solution of which is associated with the development of methods of mathematical projection, and a number of other problems. The development of mathematical models for the class of problems quoted and the creation of effective methods of studying them has long been connected with the name of Andrei Nikolaevich Tikhonov. This paper is a survey of the basic results obtained in this area during the last decade, and is a logical continuation of [1].