Introduction .................................................... 1
Chapter 1. Inverse problems for semibounded string with the
directional derivative condition given in the end ... 19
1.1. Formulation of the direct problem ...................... 20
1.2. The form of solution of the direct problem convenient
for solving the inverse problems ....................... 22
1.3. The inverse problem with the data u(0,ξ) and
du/dz|z=o .............................................. 36
1.4. The inverse problem for semibounded string which
has no analog in the case χ = 0 ........................ 39
Chapter 2. Inverse problems for the elliptic equation
in the half-plane ................................... 41
2.1. Formulation of the direct problem ...................... 42
2.2. The form of solution of the direct problem applied
for solution of the inverse problems ................... 45
2.3. The setting and solution of the inverse problems ....... 49
Chapter 3. Inverse problems of scattering plane waves from
inhomogeneous transition layers (half-space) ........ 55
3.1. The direct problem ..................................... 59
3.2. Determination of properties of inhomogeneous layer
by the forms of incident and reflected waves given
for a single angle Θ0 .................................. 64
3.3. 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 ............................ 66
3.4. The algorithm of numerical solution of the inverse
problem 3.3 (determination of v(z) and ρ(z) by the
forms φ0(ξ,Θ0), #966;1(ξ,Θ0) of incident and
reflected waves for three angles Θ0 .................... 70
3.5. Derivation of the speed v(z) and the density ρ(z)
in the numerical experiments ........................... 76
Chapter 4. Inverse problems for finite string with the
condition of directional derivative in one end ...... 85
4.1. Formulation of the direct problem ...................... 86
4.2. Solution of the direct problem ......................... 87
4.3. The inverse problem with the data in the free end of
the string ............................................. 93
4.4. The inverse problem with the data set in the boundary
z = 0 .................................................. 96
4.5. Inverse problems for the string with the fixed end
z = H .................................................. 97
Chapter 5. Inverse problems for the elliptic equation in
the strip ........................................... 99
5.1. Setting of the direct problem ......................... 100
5.2. Solution of the direct problem ........................ 101
5.3. The inverse problem with the data in the boundary
z = H ................................................. 104
5.4. The inverse problem with the data in the boundary
z = 0 ................................................. 106
5.5. Problems with the condition u(H, ξ) = 0 ............... 107
Chapter 6. Inverse problems of scattering the plane waves
from inhomogeneous layers with a free or fixed
boundary ........................................... 111
6.1. The direct problem .................................... 112
6.2. Determination of properties of the inhomogeneous
layer given the data for a single angle of
incidence ............................................. 116
6.3. Determination of the depth of inhomogeneous layer,
the density ρ(z) and the speed v(z) in this layer
if the form of incident wave φ0(ξ, θ0) is known ....... 117
6.4. Determination of the depth of inhomogeneous layer,
the density ρ0(z), the speed v(z) in the layer and
the form of incident wave ............................. 119
Chapter 7. Direct and inverse problems for the equations
of mixed type ...................................... 125
7.1. Formulation and the uniqueness theorem for the
direct problem ........................................ 126
7.2. The representation of solution of the direct
problem 7.1. The case of K(h+0) = 0, K(h-0) = 0 ....... 130
7.3. The case of Lavrentiev—Bitsadze equation. The
formulas for solution of the direct problem 7.1 ....... 141
7.4. Inverse problems. The case K(h+0) ≠ 0, K(h-0) ≠ 0 ..... 145
7.5. The general case ...................................... 150
7.6. The other problems .................................... 173
7.7. The physical content .................................. 175
Chapter 8. Inverse problems connected with determination of
arbitrary set of point sources ..................... 179
8.1. Direct problem and its solution ....................... 180
8.2. Some auxiliary geometrical definitions ................ 184
8.3. The auxiliary results for the case 1 connected with
the T-systems ......................................... 186
8.4. Preliminary remarks on solutions of the inverse
problems .............................................. 195
8.5. The static inverse problem with the data on the
strait line ........................................... 196
8.6. The nonstationary inverse problem with the data
given in the straight line for the case 1 ............. 198
8.7. The inverse static and nonstationary problems ......... 203
8.8. On zeros of the field u(x,y,z) of form (8.1.13) ....... 208
8.9. The zeros of the function u(x,y,z,t) of form (8.1.4)
or (8.1.5) in the plane z = 0 ......................... 209
8.10.Solution of the nonstationary inverse problem 8.1
in the case 2 for E = E1, E2, E3, E4, E5, E6 ........... 210
8.11.Stationary inverse problem ............................ 216
8.12.Possible applications ................................. 218
8.11.Bibliography .......................................... 221
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