Integral Geometry & Inverse
Problems for Kinetic Equations

By A.Kh. Amirov
December 2001
VSP
ISBN: 90-6764-352-1
210 pages
$182.00 hardcover

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In this monograph a new method for proving the solvability of integral geometry problems and inverse problems for kinetic equations is presented. The application of this method has led to interesting problems of the Dirichlet type for third order differential equations, the solvability of which appears to depend on the geometry of the domain for which the problem is stated.

Another subject of the book is the problem of integral geometry on paraboloids, in particular the uniqueness of solutions to the Goursat problem for a differential inequality, which implies new theorems on the uniqueness of solutions to this problem for a class of quasilinear hyperbolic equations. A class of multidimensional inverse problems associated with problems of integral geometry and the inverse problem for the quantum kinetic equations are also included.

This monograph will be of value and interest to mathematicians who deal with problems of integral geometry, direct and inverse problems of mathematical physics and geophysics and for specialists in computerized tomography.

Contents:

Introduction

SOLVABILITY OF PROBLEMS OF INTEGRAL GEOMETRY

Two-dimensional inverse problem for the transport equation

Three-dimensional inverse problem for the transport equation

Solvability of the problem of integral geometry along geodesics

A planar problem of integral geometry

Certain problems of tomography

INVERSE PROBLEMS FOR KINETIC EQUATIONS

The problem of integral geometry and an inverse problem for the kinetic equation

Linear kinetic equation

A modification of problem 2.2.1

One-dimensional kinetic equation

Equations of the Boltzmann type

The Vlasov system

Some inverse and direct problems for the kinetic equation

EVOLUTIONARY EQUATIONS

The Cauchy problem for an integro-differential equation

The problems (3.1.1) -- (3.1.2) for m = 2k + 1, p = 1 (the case of nonperiodic solutions)

Boundary value problems

The Cauchy problem for an evolutionary equation

Inverse problem for an evolutionary equation

INVERSE PROBLEMS FOR SECOND ORDER DIFFERENTIAL EQUATIONS

Quantum kinetic equation

Ultrahyperbolic equation

On a class of multidimensional inverse problems

Inverse problems with concentrated data

Appendix

Bibliography

Mathematics
Inverse & Ill-Posed Problems Series

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