Multidimensional Inverse Problems
"in Ray Formulations" for Some
Hyperbolic Equations and SystemS

This project deals with the Cauchy problems for acoustical equation,
equations systems of electrodynamic and elasticity. The right-hand sides of
these equation and systems are generalized functions (vector-functions)
describing impulsed point sources of acoustical, electromagnetic or
elastic waves respectively.

First step of the investigation consists in the description of the
structures and properties of the generalized solutions of the
Cauchy problems for considered equation and systems. These
properties of the generalized solutions are described in the forms
which are necessary for further consideration of some inverse
problems.

Second step of study deals with multidimensional inverse problems
"in ray formulation". These problems consist in finding the coefficients
of the considered equtions and systems as the functions depending on
space variables. From physical point of view the coefficients
of these equation and systems are unknown characteristics of
ingomogeneous media in which the wave processes spread. The additional
information for the inverse problems solving is the following. The
dynamic wave fields are observed at the poins of the boudary of
a fixed domain in time interval of arriving the wave fronts from
sources which are concentrated on the boundary of this domain.
It suggests to solve these problems by means of reducing them to
successive solving inverse kinematic, integral geometry,
tomography problems and others.

From mathematical point of view the main results of the investigation
will consist in uniqueness, stability theorems and for some special
inverse problems the existence theorems also. This research
continues the investigation of the inverse problems which are described in
monographies Romanov (1984), Yakhno(1990, 1998).

References

Lavrent'ev M.M., Romanov V.G., Shishatskii S.P., (1986)
Ill-Posed Problems of Mathematical Physics and Analysis,
American Mathematical Society Translations of Mathematical
Monography,Vol.64.

Romanov V.G., (1984) Inverse Problems of Mathematical Physics,
Moscow, Nauka,(in Russian).

Yakhno V.G., (1990) Inverse Problems for Differential
Equations Systems of Elastisity, Novosibirsk, Nauka, (in Russian).

Yakhno V.G., (1998) Multidimensional Inverse Problems in Ray
Formulations for Hyperbolic
Equations, Inverse and Ill-posed Problems, v. 6, 4, VSP, Utrecht,
The Netherlands.

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