Introduction to Inverse Problems for Differential Equations

Description

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. Front Matter ....Pages i-xiii Introduction Ill-Posedness of Inverse Problems for Differential and Integral Equations (Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov)....Pages 1-19 Front Matter ....Pages 21-21 Functional Analysis Background of Ill-Posed Problems (Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov)....Pages 23-62 Inverse Source Problems with Final Overdetermination (Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov)....Pages 63-119 Front Matter ....Pages 121-121 Inverse Problems for Hyperbolic Equations (Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov)....Pages 123-143 One-Dimensional Inverse Problems for Electrodynamic Equations (Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov)....Pages 145-162 Inverse Problems for Parabolic Equations (Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov)....Pages 163-203 Inverse Problems for Elliptic Equations (Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov)....Pages 205-218 Inverse Problems for the Stationary Transport Equations (Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov)....Pages 219-225 The Inverse Kinematic Problem (Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov)....Pages 227-237 Back Matter ....Pages 239-261