On a problem of S.L. Sobolev

Abstract

In his famous works of 1930 [1,2] Sergey L. Sobolev has proposed a construction of the solution of the Cauchy problem for the hyperbolic equation of the second order with variable coefficients in Rş. Although Sobolev did not construct the fundamental solution, his construction was modified later by Romanov (2002) and Smirnov (1964) to obtain the fundamental solution. However, these works impose a restrictive assumption of the regularity of geodesic lines in a large domain. In addition, it is unclear how to realize those methods numerically. In this paper a simple construction of a function, which is associated in a clear way with the fundamental solution of the acoustic equation with the variable speed in 3-d, is proposed. Conditions on geodesic lines are not imposed. An important feature of this construction is that it lends itself to effective computations.