Integral Geometry and Inverse Problems for Kinetic Equations

The Inverse and Ill-Posed Problems Series is a series of monographs publishing postgraduate level information on inverse and ill-posed problems for an international readership of professional scientists and researchers. The series aims to publish works which involve both theory and applications in, e.g., physics, medicine, geophysics, acoustics, electrodynamics, tomography, and ecology.

Autorentext

Anvar Kh. Amirov, Institute of High Temperatures, Russian Academy of Sciences, Moscow, Russia.

Zusammenfassung

In this monograph a 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 considered subject 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.

Inhalt

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.