Inverse Boundary Spectral Problems

Inverse Boundary Spectral Problems

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Inverse boundary problems are a rapidly developing area of applied mathematics with applications throughout physics and the engineering sciences. However, the mathematical theory of inverse problems remains incomplete and needs further development to aid in the solution of many important practical problems. Inverse Boundary Spectral Problems develop a rigorous theory for solving several types of inverse problems exactly. In it, the authors consider the following: "Can the unknown coefficients of an elliptic partial differential equation be determined from the eigenvalues and the boundary values of the eigenfunctions?" Along with this problem, many inverse problems for heat and wave equations are solved. The authors approach inverse problems in a coordinate invariant way, that is, by applying ideas drawn from differential geometry. To solve them, they apply methods of Riemannian geometry, modern control theory, and the theory of localized wave packets, also known as Gaussian beams. The treatment includes the relevant background of each of these areas.
Although the theory of inverse boundary spectral problems has been in development for at least 10 years, until now the literature has been scattered throughout various journals. This self-contained monograph summarizes the relevant concepts and the techniques useful for dealing with them.
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Product details

  • Hardback | 260 pages
  • 157 x 239.8 x 22.6mm | 603.29g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 1584880058
  • 9781584880059

Review quote

"[This book] contains a wealth of important methods and ideas, and the presentation is always very clear. [A] very interesting and valuable contribution to the literature on inverse problems for partial differential equations." - Zentralblatt MATH, Vol. 1037
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Table of contents

INTRODUCTION ONE-DIMENSIONAL INVERSE PROBLEM The Problem and the Main Result Wave Equation Controllability and Projectors Gaussian Beams BASIC GEOMETRICAL AND ANALYTICAL METHODS FOR INVERSE PROBLEMS Basic Tools of Riemannian Geometry for Inverse Problems Elliptic Operators on Manifolds and Gauge Transformation Initial-Boundary Value Problem for Wave Equation Gaussian Beams Carleman Estimates and Unique Continuation GEL'FAND INVERSE BOUNDARY SPECTRAL PROBLEM FOR MANIFOLDS Formulation of the Problem and the Main Result Fourier Coefficients of Waves Domains of Influence Global Unique Continuation from the Boundary Gaussian Beams from the Boundary Domains of Influence and Gaussian Beams Boundary Distance Functions Reconstruction of the Riemannian Manifold Reconstruction of the Potential INVERSE PROBLEMS FOR WAVE AND OTHER TYPES OF EQUATIONS Inverse Problems with Different Types of Data Dynamical Inverse Problem for the Wave Equation Continuation of Data Inverse problems with Data Given on a Part of the Boundary Inverse Problems for Operators in Rm BIBLIOGRAPHY TABLE OF NOTATION
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