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arXiv · 1508.07232

On a problem of S.L. Sobolev

Michael V. Klibanov
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Abstract

In 1930 Sergey L. Sobolev [7,8] 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 later by Romanov [4,5] 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.

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Michael V. Klibanov. 2015-08-26. On a problem of S.L. Sobolev. https://arxiv.org/abs/1508.07232

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