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2017 | Buch

# Introduction to Inverse Problems for Differential Equations

verfasst von: Prof. Dr. Alemdar Hasanov Hasanoğlu, Prof. Dr. Vladimir G. Romanov

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### Über dieses Buch

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.

### Inhaltsverzeichnis

##### Chapter 1. Introduction Ill-Posedness of Inverse Problems for Differential and Integral Equations
Abstract
Inverse problems arise in almost all areas of science and technology, in modeling of problems motivated by various physical and social processes.
Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov

#### Introduction to Inverse Problems

##### Chapter 2. Functional Analysis Background of Ill-Posed Problems
Abstract
The main objective of this chapter is to present some necessary results of functional analysis, frequently used in study of inverse problems.
Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov
##### Chapter 3. Inverse Source Problems with Final Overdetermination
Abstract
Inverse source problems for evolution PDEs $$u_t=Au+F$$, $$t\in (0,T_f]$$, represent a well-known area in inverse problems theory and has many engineering applications.
Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov

#### Inverse Problems for Differential Equations

##### Chapter 4. Inverse Problems for Hyperbolic Equations
Abstract
In the first part of this chapter we study two inverse source problems related to the second order hyperbolic equations $$u_{tt}-u_{xx}=\rho (x, t)g(t)$$ and $$u_{tt}-u_{xx}=\rho (x, t)\varphi (x)$$ for the quarter plane $$\mathbb {R}^2_+=\{(x, t)|\, x>0, t>0\}$$, with Dirichlet type measured output data $$f(t):=u(x, t)\vert _{x=0}$$.
Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov
##### Chapter 5. One-Dimensional Inverse Problems for Electrodynamic Equations
Abstract
Here $$H=(H_1,H_2,H_3)$$ and $$E=( E_1,E_2,E_3)$$ are vectors of electric and magnetic strengths, $$\varepsilon >0$$, $$\mu >0$$ and $$\sigma$$ are the permittivity, permeability and conductivity coefficients, respectively, which define electro-dynamical parameters of a medium.
Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov
##### Chapter 6. Inverse Problems for Parabolic Equations
Abstract
This Chapter deals with inverse coefficient problems for linear second-order 1D parabolic equations. We establish, first, a relationship between solutions of direct problems for parabolic and hyperbolic equations.
Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov
##### Chapter 7. Inverse Problems for Elliptic Equations
Abstract
This chapter is an introduction to the basic inverse problems for elliptic equations.
Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov
##### Chapter 8. Inverse Problems for the Stationary Transport Equations
Abstract
Inverse problems related to the transport equations arise in many areas of applied sciences and have various applications in medical imaging and tomography.
Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov
##### Chapter 9. The Inverse Kinematic Problem
Abstract
The problem of recovering the sound speed (or index of refraction) from travel time measurements is an important issue in determining the substructure of the Earth.
Alemdar Hasanov Hasanoğlu, Vladimir G. Romanov
##### Backmatter
Titel
Introduction to Inverse Problems for Differential Equations
verfasst von
Prof. Dr. Alemdar Hasanov Hasanoğlu