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The article provides an overview of mathematical studies performed by Vladimir Alexandrovich Kondratiev, on the 70th anniversary of his birth. Included are studies on: the oscillation and nonoscillation conditions for solutions of ordinary differential equations; elliptic equations in domains with angular and conical points; and a problem with oblique derivatives.

We study the two-dimensional eikonal equation ψ²x + ψ²y = 1/v²(x, y). We carry out the group analysis of the equation, establish a connection between the group properties and geometric characteristics of the Riemannian space with the metric ds² = [dx² + dy²]/v² (x, y). We select the most important classes of equations and derive some conditions for reducibility of a given equation to an equation of one of those classes. We find a condition for two equations to be equivalent (the theorem of seven invariants). For the equations corresponding to Riemannian spaces of constant curvature, we obtain explicit formulas for the solutions describing the wave front for a point source and also the ray equations. [ABSTRACT FROM AUTHOR]

We present a group classification of the family of equations of the form (∇ψ)2 = 1/ v2( x, y, z), called the eikonal equations, which are equations of characteristics for equations describing acoustic and electromagnetic waves in inhomogeneous media. For some cases of equations with linear and quadratic functions v( x, y, z), having the sense of the speed of waves in a medium, we present explicit solutions (eikonals of point sources) and completely describe the geometry of rays. [ABSTRACT FROM AUTHOR]

Discusses the moving source phantom in geometric optics of inhomogenous media. Examination of the center-of-curvature motion phenomenon for a wave front propagating in an inhomogenous medium. Dependency of velocity on the curvature radius; Illustration of velocity functions; Presentation of integral theorems.

Differential equations on networks is a relatively new branch (it has existed about 20 years) in the theory of differential equations. This issue is the result of work in this direction performed in Russia within the framework of the seminar of Yu. V. Pokornyi (Voronezh), in which models connected with oscillations in various systems consisting of elastic elements were at the center of attention. In this review, we describe the fundamental problem statements, the specific features connected with their analysis, the basic approaches, and the methods for studying the corresponding differential equations. [ABSTRACT FROM AUTHOR]

We present a method for analyzing the Green's function, adapted to nonclassical problems for ordinary differential equations (including those on spatial graphs). We analyze the evolution of the view of the Green's function—from a Hilbert system of axioms to a direct analysis of the influence function. [ABSTRACT FROM AUTHOR]

The article presents information about Voronezh Spring Mathematical School that imparts education about modern methods in qualitative theory of boundary value problems. The school was organized by Voronezh State University in cooperation with Moscow State University and the Mathematical Institute of the Russian Academy of Sciences. The school comprised about 80 participants from 20 cities. The main scientific directions of the school program were the following: modern aspects of the theory of boundary value problems, computer science and control, geometry and analysis, plasticity, related problems of applied and engineering mathematics, problems of mathematical education.