# One-Dimensional Inverse Problems of Mathematical Physics

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*M. M. Lavrent′ev; K. G. Reznitskaya; V. G. Yakhno*

This monograph deals with the inverse problems of determining a
variable coefficient and right side for hyperbolic and parabolic
equations on the basis of known solutions at fixed points of space for
all times. The problems are one-dimensional in nature since the
desired coefficient of the equation is a function of only one
coordinate, while the desired right side is a function only of time.
The authors use methods based on the spectral theory of ordinary
differential operators of second order and also methods which make it
possible to reduce the investigation of the inverse problems to the
investigation of nonlinear operator equations. The problems studied
have applied importance, since they are models for interpreting data
of geophysical prospecting by seismic and electric means.

In the first chapter the authors prove the one-to-one correspondence
between solutions of direct Cauchy problems for equations of different
types, and they present the solution of an inverse problem of heat
conduction. In the second chapter they consider a second-order
hyperbolic equation describing a wave process in three-dimensional
half-space. The third chapter investigates formulations of
one-dimensional inverse problems for the wave equation in
multidimensional space.