From 1934 Sokolov taught at the Institute of Mathematics at the Academy of Sciences of Ukraine. He also taught at several other higher educational institutions in Kiev.
Sokolov's main work was on the n-body problem, which he worked on for nearly 50 years and he summarised his contributions in the book Singular trajectories of a system of free material points (Russian) which was published in 1951. He also worked on functional equations and on such practical problems as the filtration of groundwater. Examples of papers where he makes practical applications to water flow are On the flow of ground water into a drainage ditch of trapezoidal section (Russian) (1951), Filtration without backwater from an unlined canal of trapezoidal section in homogeneous ground (Russian) (1952), On a problem of the theory of unsteady motion of ground water (Russian) (1953), On the theory of plane unsteady filtration of ground water (Russian) (1954), On an axially symmetric problem of the theory of unsteady motion of ground water (Russian) (1955). Other applications include On the determination of dynamic pull in shaft-lifting cables (Ukrainian) (1955) and On approximate solution of the basic equation of the dynamics of a hoisting cable (Ukrainian) (1955). Sokolov also published on celestial mechanics and hydromechanics.
One of the topics which will always be associated with Sokolov's name is his method for finding approximate solutions to differential and integral equations. The method which he introduced is now sometimes called 'the averaging method with functional corrections' or sometimes called 'the Sokolov method'. His methods were highly practical and useful in many applications to mathematical physics, but they were also studied with the highest degree of mathematical rigour. Examples of his papers on this topic are On a method of approximate solution of linear integral and differential equations (Ukrainian) (1955), Sur la méthode du moyennage des corrections fonctionnelles Ⓣ (Russian) (1957), Sur l'application de la méthode des corrections fonctionnelles moyennes aux équations intégrales non linéaires Ⓣ (Russian) (1957), On a method of approximate solution of systems of linear integral equations (Russian) (1961), On a method of approximate solution of systems of nonlinear integral equations with constant limits (Russian) (1963), and On sufficient tests for the convergence of the method of averaging of functional corrections (Russian) (1965).
His many papers in this area were brought together in the important book The method of averaging of functional corrections (1967) which he wrote at an elementary level. E L Albasiny, reviewing Sokolov's textbook, first formally describes the approximations from averaging of functional corrections. He then writes:-
The approximations may converge although Picard iteration diverges. This basic approach is developed by the author and applied to the approximate solution of Fredholm and Volterra-type integral equations of the second kind, to their nonlinear counterparts, to integral equations of mixed type, to linear and nonlinear one-dimensional boundary value problems, to initial-value problems in ordinary differential equations and to certain elliptic, hyperbolic and parabolic equations.The first part of Sokolov's book discusses applications of his method to problems which can be modelled by linear integral equations with constant limits. He gives a number of different sufficient conditions for the approximations to converge and presents error estimates. The next three parts look first at problems which can be modelled by nonlinear integral equations with constant limits and then extend the analysis to the situation where the upper limit is variable. In the final part Sokolov examines applications of his method to integral equations of mixed type, then in a number of appendices he presents some generalisations of the method.
Sokolov died only a few months before his 75th birthday and only a few months short of completing 50 years of scientific work in the Ukrainian Academy of Sciences.
Article by: J J O'Connor and E F Robertson
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