15-88 Michael V. Klibanov

On a problem of S.L. Sobolev (141K, pdf) Aug 30, 15

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Abstract. In 1930 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 3-d. Although Sobolev did not construct the fundamental solution, his construction was modified in 1986 by V.G. Romanov to obtain that 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.

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