Inverse Problems for Kinetic and Other Evolution Equations

This monograph deals 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.

Autorentext

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

Zusammenfassung

This monograph deals 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.

Inhalt

Chapter 1 Formulas for solutions and coefficients of kinetic and other equations: kinetic equations; several formulas for solutions and coefficients of kinetic equations; formulas in the inverse problems for kinetic equations with a potential; formulas in tomography problems; formulas of inverse problems for kinetic equation and integral geometry involving integration along geodesics; differential and functional equations of inverse problems for nonlinear equations. Chapter 2 Theorems of uniqueness for inverse problems for kinetic equations: inverse problem for a system of kinetic equations; inverse problem for a system of quantum kinetic equations; on uniqueness of determination of a form by its integrals along geodesics; dynamical model of the ethnic system; formulas in direct and inverse problems. Chapter 3 Spherical harmonic method and inverse problem for kinetic equations: spherical harmonic method; steady-state transfer equation; determining the dispersion index in the case of the P1-approximation; definition of the dispersion index in the case of the P2-approximation; reconstruction of the dispersion index and the source function. Chapter 4 Inverse problems for evolution equations of determining two coefficients; nonlocal boundary-value problems for nonlinear equations and inverse problems of determining two coefficients; recurrent formulas on derivatives of solutions; integrodifferential equations in inverse problems of determining two coefficients for evolution equations; inverse problem for a system of Maxwell equations; determining two unknown coefficients of the parabolic-type equation; inhomogeneous conditions of overdetermination; representation of solutions and coefficients of partial differential equations of the second order. Chapter 5 Some results of multidimensional inverse problems theory: formulas for coefficients in inverse problems for general evolutionary equations; formulas in inverse problems for difference-differential equations; inverse problems for evolutionary equations with degeneration and others; grou p analysis and formulas in inverse problems of mathematical physics; uniqueness of the solution of an integral equation of the first kind over real algebras with division of the finite dimension; methods of geometry in the inverse seismic problem; problems associated with projections of convex bodies onto planes.