Introduction to Inverse Problems for Differential Equations

ISBN-13: 9783319873992 / Angielski / Miękka / 2018 / 261 str.

Alemdar Hasanov Hasanoğlu; Vladimir G. Romanov

Recenzje

"This monograph provides a well-written, easy-to-read, and basically self-contained survey on a wide range of inverse problems related with initial-boundary problems for partial deferential equations. It addresses many relevant topics in the theory of such problems, with a focus on existence, uniqueness and stability of inverse coefficient and source problems. ... The text can be used for self-study and will be of interest to experts in the field as well as graduate students." (Boris Rubin, zbMATH 1385.65053, 2018)

"This book is an important contribution to the theory of inverse problems. It gives a complete picture of inverse problems and their applications. ... It is a good research monograph for people working on inverse problems and related issues; a useful state-of-the-art reference guide for researchers and students; and a fine textbook for graduate students in mathematics and engineering." (Srinivasan Natesan, Computing Reviews, September, 2018)

Spis treści

Introduction: Ill-Posedness of Inverse Problems for Differential and Integral Equations.- PART I. INTRODUCTION TO INVERSE PROBLEMS: 1. Functional Analysis Background of Ill-Posed Problems.- 2. Inverse Source Problems With Final Overdetermination.- PART II. INVERSE PROBLEMS FOR DIFFERENTIAL EQUATIONS: 3. Inverse Problems for Hyperbolic Equations.- 4. One-dimensional Inverse Problems for Electrodynamics Equations.- 5. Inverse Problems for Parabolic Equations.- 6. Inverse Problems for Elliptic Equations.- 7. Inverse Problems for the Stationary Transport Equations.- 8. The Inverse Kinematic Problem.- Appendix A: Invertibility of Linear Operators.- Appendix B: Some Estimates For One-dimensional Parabolic Equation.- Bibliography.- Index.

Blurb

This book presents a systematic exposition of the main ideas and methods in treating inverse problems for PDEs arising in basic mathematical models, though it makes no claim to being exhaustive. Mathematical models of most physical phenomena are governed by initial and boundary value problems for PDEs, and inverse problems governed by these equations arise naturally in nearly all branches of science and engineering.

The book’s content, especially in the Introduction and Part I, is self-contained and is intended to also be accessible for beginning graduate students, whose mathematical background includes only basic courses in advanced calculus, PDEs and functional analysis. Further, the book can be used as the backbone for a lecture course on inverse and ill-posed problems for partial differential equations.

In turn, the second part of the book consists of six nearly-independent chapters. The choice of these chapters was motivated by the fact that the inverse coefficient and source problems considered here are based on the basic and commonly used mathematical models governed by PDEs. These chapters describe not only these inverse problems, but also main inversion methods and techniques. Since the most distinctive features of any inverse problems related to PDEs are hidden in the properties of the corresponding solutions to direct problems, special attention is paid to the investigation of these properties.