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One-Dimensional Inverse Problems of Mathematical Physics
M. M. Lavrent'ev, K. G. Reznitskaya, and V. G. Yakhno
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American Mathematical Society Translations--Series 2
1986; 70 pp; softcover
Volume: 130
Reprint/Revision History:
reprinted 1992
ISBN-10: 0-8218-3099-6
ISBN-13: 978-0-8218-3099-4
List Price: US$57
Member Price: US$45.60
Order Code: TRANS2/130
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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.

Table of Contents

  • Introduction
  • Solutions of direct and inverse problems and some of their relations
  • Source problems
  • A one-dimensional inverse problem for the wave equation
  • Appendix
  • Bibliography
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