Integral Geometry and Inverse Problems for Hyperbolic Equations
1st Edition
Published on 23. July 1974
Book
Hardback
VI, 154 pages
978-3-540-06429-9 (ISBN)
Description
There are currently many practical situations in which one wishes to determine the coefficients in an ordinary or partial differential equation from known functionals of its solution. These are often called "inverse problems of mathematical physics" and may be contrasted with problems in which an equation is given and one looks for its solution under initial and boundary conditions. Although inverse problems are often ill-posed in the classical sense, their practical importance is such that they may be considered among the pressing problems of current mathematical re search. A. N. Tihonov showed [82], [83] that there is a broad class of inverse problems for which a particular non-classical definition of well-posed ness is appropriate. This new definition requires that a solution be unique in a class of solutions belonging to a given subset M of a function space. The existence of a solution in this set is assumed a priori for some set of data. The classical requirement of continuous dependence of the solution on the data is retained but it is interpreted differently. It is required that solutions depend continuously only on that data which does not take the solutions out of M.
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Content
I. Some Problems in Integral Geometry.- 1. Problem of Finding a Function from its Integrals over Ellipsoids of Revolution.- 2. Generalization to the Case of Analytic Curves.- 3. Existence Theorem for the Case of Ellipses.- 4. Determination of a Function from its Integrals over a Family of Curves Invariant to Displacement.- 5. The Integral-Geometric Problem for m Functions.- 6. Determination of a Function in a Circle from its Integrals over a Family of Curves Invariant to Rotation about Center of the Circle.- 7. Integral-Geometric Problem for Surfaces Invariant to Displacement.- 8. Integral-Geometric Problems for a Family of Curves Generated by a Riemannian Metric.- II. Inverse Problems for Hyperbolic Linear Differential Equations.- 1. General Information Concerning the Solution of the Cauchy Problem for Linear Hyperbolic Equations.- 2. One-Dimensional Inverse Problem for the Telegraph Equation in Three-Dimensional Space.- 3. Linearized Inverse Problem for the Telegraph Equation.- 4. The Problem of Finding the Coefficients of the Lower Order Derivatives in a Second-Order Equation.- 5. Linearized Inverse Kinematic Problem for the Wave Equation in Variable Isotropic Media.- 6. One-Dimensional Inverse Kinematic Problem for the Wave Equation in Anisotropic Media.- 7. Multidimensional Linearized Inverse Kinematic Problem for the Wave Equation in Anisotropic Media.- III. Application of the Linearized Inverse Kinematic Problem to Geophysics.- 1. The Earth's Structure from a Geophysical Standpoint and the Problem of Determining the Velocity Structure of the Earth's Mantle.- 2. Numerical Solution of the Linearized Inverse Kinematic Problem.- 3. Some Numerical Results.