Inverse Problems of Wave Processes

A. S. Blagoveshchenskii

• Publisher: De Gruyter
• Year: 2014
• ISBN:
  9783110940893
• Edition: 1
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

This monograph covers dynamical inverse problems, that is problems whose data are the values of wave fields. It deals with the problem of determination of one or more coefficients of a hyperbolic equation or a system of hyperbolic equations. The desired coefficients are functions of point. Most attention is given to the case where the required functions depend only on one coordinate.

The first chapter of the book deals mainly with methods of solution of one-dimensional inverse problems. The second chapter focuses on scalar inverse problems of wave propagation in a layered medium. In the final chapter inverse problems for elasticity equations in stratified media and acoustic equations for moving media are given.

Table of contents

• Introduction
• Chapter 1. One-dimensional inverse problems
• 1.1. Setting of a problem for string equation
• 1.1.1. General remarks
• 1.1.2. Mathematical setting of the problem
• 1.1.3. Physical interpretation of inverse problem data
• 1.1.4. Reformulation of the problem in terms of a hyperbolic system
• 1.1.5. Determination of the string parameters from σ(y) and additional information
• 1.2. Peculiarities of solution. Formulation of the direct problem
• 1.2.1. Correct formulation of the direct problem
• 1.2.2. Singularities of solution of the hyperbolic system
• 1.2.3. Singularities of solution of the string equation
• 1.2.4. Singularities of solution in case of discontinuous coefficients
• 1.3. The first method of solution of inverse problem
• 1.3.1. Derivation of the system of integral equations
• 1.3.2. Investigation of the system of integral equations
• 1.3.3. Continuous dependence of solution on inverse problem data
• 1.3.4. Solution of the inverse problem by successive steps
• 1.3.5. The case of discontinuous σ(y)
• 1.4. Method of linear integral equations
• 1.4.1. Fundamental system of solutions of equation (1.1.12)
• 1.4.2. Derivation of linear integral equations
• 1.4.3. Recovery of the coefficient q(y) from a solution of the linear integral equation
• 1.4.4. Structure of equations (1.4.10). Existence and uniqueness of solution
• 1.4.5. Modification of equations (1.4.14), (1.4.15) under a change of the source
• 1.4.6. Proof of necessity. Conditions for solvability of the inverse problem
• 1.4.7. Proof of sufficiency
• 1.5. The case of discontinuous s(y)
• 1.5.1. The first method
• 1.5.2. The second method
• 1.6. The Gel’fand-Levitan equation for second-order hyperbolic equations
• 1.6.1. Derivation of a linear integral equation
• 1.6.2. Recovery of coefficients by solution of the Gel’fand - Levitan equation
• 1.6.3. The scattering problem
• 1.7. Some cases of explicit solution of inverse problem
• 1.7.1. Description of inverse problem data
• 1.7.2. Construction of the solution
• 1.8. On connection of inverse problems with nonlinear ordinary differential equations
• 1.8.1. Connection between ordinary differential equations and inverse problems
• 1.8.2. Use of the connection between differential equations and inverse problems
• 1.9. The case of more general system of equations
• 1.9.1. Setting of a problem
• 1.9.2. Linear integral equations
• 1.9.3. Determination of q1 and q2
• Chapter 2. Theory of inverse problems for wave processes in layered media
• 2.1. Inverse problems of acoustics
• 2.1.1. General remarks
• 2.1.2. Setting of an inverse problem for the acoustic equation
• 2.1.3. Solving the inverse problem of acoustics by the Fourier transformation
• 2.1.4. Solution of the inverse problem of acoustics by the Radon transformation
• 2.1.5. The inverse problem of scattering a plane wave
• 2.1.6. The inverse problem of wave propagation in wave guides
• 2.1.7. The inverse problem for a layered ball
• 2.1.8. The method of moments. Formulation of a problem and its reduction to an integral equation
• 2.1.9. Construction of the Green function
• 2.1.10. Investigation of integral equation (2.1.34)
• 2.1.11. The inversion formula for the operator T̂
• 2.1.12. Once more about the scattering problem
• 2.2. General second-order hyperbolic equation. Problem in a half-space
• 2.2.1. Setting of a problem
• 2.2.2. Transformation of the problem
• 2.3. The scattering problem for the general hyperbolic equation
• 2.3.1. Setting of the problem, reformulation in terms of a system
• 2.3.2. Reduction to an inverse problem investigated above
• 2.3.3. The maximum possible information on the coefficients of equation (2.2.1)
• Chapter 3. Inverse problems for vector wave processes
• 3.1. Inverse problem for elasticity equation
• 3.1.1. General remarks
• 3.1.2. Setting of the problem
• 3.1.3. Solution of the inverse Lamb problem
• 3.2. Inverse problem of sound propagation in a moving layered medium
• 3.2.1. Acoustic equations in a moving medium
• 3.2.2. Transformation of the system for the layered medium
• 3.3. The case of one-dimensional sound propagation
• 3.3.1. General description of the problems in question
• 3.3.2. Mathematical setting of the problems
• 3.3.3. Formulation of the results
• 3.3.4. Integral equations for Problem 1
• 3.3.5. Integral equations for Problem 2
• 3.3.6. The model problem
• 3.4. Inverse problems for hyperbolic systems
• 3.4.1. General remarks. Setting of a problem
• 3.4.2. Formulation of the direct problem
• 3.4.3. Setting of the inverse problem. Formulation of the result
• 3.4.4. Proof
• 3.5. Second-order hyperbolic system
• 3.5.1. Setting of a problem
• 3.5.2. Proof
• 3.5.3. The inverse problem with a fixed interval of nonhomogeneities
• Bibliography