Megrabov A.G. Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations

Berlin: de Gruyter, 2003. — 242 p.Inverse problems are an important and rapidly developing direction in mathematics,mathematical physics, differential equations, and various applied technologies (geophysics, optic, tomography, remote sensing, radar-location, etc.). In this monographdirect and inverse problems for partial differential equations are considered. The type of equations focusedare hyperbolic, elliptic, and mixed (elliptic-hyperbolic). The direct problems arise as generalizations of problems of scattering plane elastic or acoustic waves from inhomogeneous layer (or from half-space). The inverse problems are those of determination ofmedium parameters by giving the forms of incident and reflected waves or the vibrations of certain points of the medium. The method of researchof all inverse problems is spectral-analytical, consisting in reducing the considered inverse problems to the known inverse problems for the Sturm-Liouville equation or the string equation. Besides the book considers discrete inverse problems. In these problems an arbitrary set of point sources (emissive sources, oscillators, point masses) is determined.Introduction
Inverse problems for semibounded string with the directional derivative condition given in the end
Formulation of the direct problem
The form of solution of the direct problem convenient for solving the inverse problems
The inverse problem with the data u (0, .)and .u/.z|z=q
The inverse problem for semibounded string which has no analog in the case . = 0
Inverse problems for the elliptic equation in the half-plane
Formulation of the direct problem
The form of solution of the direct problem applied for solution of the inverse problems
The setting and solution of the inverse problems
Inverse problems of scattering plane waves from inhomogeneous transition layers (half-space)
The direct problem
Determination of properties of inhomogeneous layer by the forms of incident and reflected waves given for a single angle .0
The method of recovery of the density and the speed in the inhomogeneous layer as the functions of the depth given the set of plane waves reflected from the layer at various angles
The algorithm of numerical solution of the inverse problem 3.3 (determination of v(z) and .(z) by the forms .1{...0) of incident and reflected waves for three angles .0)
Derivation of the speed v(z) and the density .(z) in the numerical experiments
Inverse problems for finite string with the condition of directional derivative in one end
Formulation of the direct problem
Solution of the direct problem
The inverse problem with the data in the free end of the string
The inverse problem with the data set in the boundary z = 0
Inverse problems for the string with the fixed end z = H
Inverse problems for the elliptic equation in the strip
Setting of the direct problem
Solution of the direct problem
The inverse problem with the data in the boundary z = H
The inverse problem with the data in the boundary z = 0
Problems with the condition u(H, .)=0
Inverse problems of scattering the plane waves from inhomogeneous layers with a free or fixed boundary
The direct problem
Determination of properties of the inhomogeneous layer given the data for a single angle of incidence
Determination of the depth of inhomogeneous layer, the density .(z) and the speed v(z) in this layer if the form of incident wave .0(...0) is known is known
Determination of the depth of inhomogeneous layer, the density .(z), the speed v(z) in the layer and the form of incident wave
Direct and inverse problems for the equations of mixed type
Formulation and the uniqueness theorem for the direct problem
The representation of solution of the direct problem 7.1. The case of K(h + 0) . 0, K(h - 0) . 0
The case of LavrentievBitsadze equation. The formulas for solution of the direct problem 7.1
Inverse problems. The case K(h + 0) . 0, K(h - 0) . 0
The general case
The other problems
The physical content
Inverse problems connected with determination of arbitrary set of point sources
Direct problem and its solution
Some auxiliary geometrical definitions
The auxiliary results for the case 1 connected with the T-systems
Preliminary remarks on solutions of the inverse problems
The static inverse problem with the data on the strait line
The nonstationary inverse problem with the data given in the straight line for the case 1
The inverse static and nonstationary problems
On zeros of the field u(x, y, z) of form (8.1.13)
The zeros of the function u(x,y,z,t) of form (8.1.4) or (8.1.5) in the plane z = 0
Solution of the nonstationary inverse problem 8.1 in the case 2 for E = E1, E2, E3, E5, E6
Stationary inverse problem
Possible applications
Bibliography