Inverse Problems for Kinetic and Other Evolution Equations

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Overview

This monographdeals with methods of studying multidimensional inverse problems for kinetic and other evolution equations, in particular transfer equations. The methods used are applied to concrete inverse problems, especially multidimensional inverse problems applicable in linear and nonlinear statements. A significant part of the book contains formulas and relations for solving inverse problems, including formulas for the solution and coefficients of kinetic equations, differential-difference equations, nonlinear evolution equations, and second order equations.

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Editorial Reviews

From The Critics
Inverse problems for differential equations are problems of determining, besides the solution, the coefficients of equations, given additional information on the solution. Inverse problems are linked to problems of scattering and potential theories; heat- and radiation-transfer equations; integral geometry; tomography; and identification, control, and image recovery. In this text for mathematicians and engineers, Anikonov (affiliation not cited ) describes methods of studying multidimensional inverse problems for kinetic and other evolution equations, and, in particular, for transfer equations. Distributed in the U.S. by Brill. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

Meet the Author

Yurii E. Anikonov, Sobolev Institute of Mathematics, Russian Academy of Sciences, Russia.

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Table of Contents

Chapter 1. Formulas for solutions and coefficients of kinetic and other equations 1
1.1. Kinetic equations 1
1.2. Several formulas for solutions and coefficients of kinetic equations 5
1.3. Formulas in the inverse problems for kinetic equations with a potential 9
1.4. Formulas in tomography problems 11
1.5. Formulas of inverse problems for kinetic equation and integral geometry involving integration along geodesics 15
1.6. Differential and functional equations of inverse problems for nonlinear equations 21
Chapter 2. Theorems of uniqueness for inverse problems for kinetic equations 35
2.1. Inverse problem for a system of kinetic equations 35
2.2. Inverse problems for a system of quantum kinetic equations 40
2.3. On uniqueness of determination of a form by its integrals along geodesics 55
2.4. Dynamical model of the ethnic system. Formulas in direct and inverse problems 59
Chapter 3. Spherical harmonic method and inverse problem for kinetic equations 73
3.1. Spherical harmonics method 73
3.2. Steady-state transfer equation 79
3.3. Determining the dispersion index in the case of the P[subscript 1]-approximation 93
3.4. Definition of the dispersion index in the case of the P[subscript 2]-approximation 101
3.5. Reconstruction of the dispersion index and the source function 121
Chapter 4. Inverse problems for evolution equations of determining two coefficients 127
4.1. Nonlocal boundary-value problems for nonlinear equations and inverse problems of determining two coefficients 127
4.2. Recurrent formulas on derivatives of solutions 139
4.3. Integrodifferential equations in inverse problems of determining two coefficients for evolution equations 144
4.4. Inverse problem for a system of Maxwell equations 152
4.5. Determining two unknown coefficients of the parabolic-type equation 160
4.6. Inhomogeneous conditions of overdetermination 172
4.7. Representation of solutions and coefficients of partial differential equations of the second order 184
Chapter 5. Some results of multidimensional inverse problems theory 199
5.1. Formulas for coefficients in inverse problems for general evolutionary equations 199
5.2. Formulas in inverse problems for difference-differential equations 208
5.3. Inverse problem for evolutionary equations with degeneration and others 212
5.4. Group analysis and formulas in inverse problems of mathematical physics 216
5.5. Uniqueness of the solution of an integral equation of the first kind over real algebras with division of the finite dimension 237
5.6. Methods of geometry in the inverse seismic problem 244
5.7. Problems associated with projections of convex bodies onto planes 252
Bibliography 267
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