Equations of Mathematical Diffraction Theory

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CRC Press, 29 сент. 2004 г. - Всего страниц: 312
Equations of Mathematical Diffraction Theory focuses on the comparative analysis and development of efficient analytical methods for solving equations of mathematical diffraction theory. Following an overview of some general properties of integral and differential operators in the context of the linear theory of diffraction processes, the authors provide estimates of the operator norms for various ranges of the wave number variation, and then examine the spectral properties of these operators. They also present a new analytical method for constructing asymptotic solutions of boundary integral equations in mathematical diffraction theory for the high-frequency case.

Clearly demonstrating the close connection between heuristic and rigorous methods in mathematical diffraction theory, this valuable book provides you with the differential and integral equations that can easily be used in practical applications.
 

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Содержание

Some Preliminaries from Analysis and the Theory of Wave Processes
1
12 Convolution Integral Equations and the WienerHopf Method
6
13 Summation of Divergent Series and Integrals
9
14 Asymptotic Estimates of Integrals
12
15 Fredholm Theory for Integral Equations of the Second Kind
21
16 Fredholm Integral Equations of the First Kind
24
17 Singular Integral Equations with a CauchyType Singularity in the Kernel
29
18 HyperSingular Integrals and Integral Equations
35
57 Integral Equation of the Mixed Boundary Value Problem for Elastic Layer
160
ShortWave Asymptotic Methods on the Basis of Multiple Integrals
165
62 HighFrequency Wave Fields in Elastic HalfSpace
169
63 Asymptotic Nature of the Geometrical Diffraction Theory
171
64 HighFrequency Diffraction with ReReflections
175
Examples of HighFrequency Multiple Diffraction
180
Physical Diffraction Theory for Nonconvex Obstacles
184
67 ShortWave Integral Operator in Diffraction by a Flaw in Elastic Medium
186

19 Governing Equations of Hydroaeroacoustics Electromagnetic Theory and Dynamic Elasticity
38
Integral Equations of Diffraction Theory for Obstacles in Unbounded Medium
45
22 Basic Integral Equations of the Diffraction Theory
52
General Case and Low Frequencies
57
24 Full LowFrequency Solution for Spherical Obstacle Acoustically hard obstacle
61
Scattering Diagram for Obstacles of Canonical Shape
65
26 Asymptotic Character of the Kirchhoff Physical Diffraction Theory
68
Wave Fields in a Layer of Constant Thickness
73
32 Principles of Selection of Unique Solution in Unbounded Domain
76
33 Waves in Elastic Layer
82
34 Generalized Riemanns Zeta Function and Summation of Some Oscillating Series
87
Efficient Calculation of Wave Fields in a Layer of Constant Thickness
91
36 Waves in the Stratified HalfPlane
94
Analytical Methods for Simply Connected Bounded Domains
101
42 Explicit Formulas for Eigenfrequencies of Round Disc
107
43 Some Variational Principles for Eigenvalues
110
44 WeylCarleman Theory of Asymptotic Distribution of Large Eigenvalues
115
45 Exact Explicit Results for Some Polygons
119
46 Explicit Analytical Results for Some Polyhedra
125
Integral Equations in Diffraction by Linear Obstacles
133
52 Operator Equation in Diffraction Problem on a Crack in Unbounded Elastic Medium
138
53 HighFrequency Asymptotics in Diffraction by Linear Obstacles in Unbounded Medium
142
54 HighFrequency Asymptotics for Diffraction by Linear Obstacles in Open Waveguides
145
55 HighFrequency Diffraction by a Linear Discontinuity in the Waveguide
151
56 Waves in Elastic HalfSpace Factorization of the Rayleigh Function
157
68 HighFrequency Asymptotics of Integral Operator in a ThreeDimensional Diffraction Theory
190
Inverse Problems of the ShortWave Diffraction
195
72 Reducing Inverse Problem of the ShortWave Diffraction to Minkowski Problem
199
73 Explicit Results for a Differential Operator of the 2D Inverse Problem
201
74 Exact Explicit Inversion of the Basic Operator in the Case of Axial Symmetry
203
75 Nonlinear Differential Operator of the ThreeDimensional Inverse Problem
205
2D Case
208
3D Case
213
IllPosed Equations of Inverse Diffraction Problems for Arbitrary Boundary
219
82 Regularization of IllPosed Problems with the Help of Smoothing Functional
222
83 Iterative Methods for Operator Equations of the First Kind
226
84 Comparison of Various Methods for Reconstruction of the Scatterer Geometry
231
Combination of Iterations and Smoothing
235
86 A Correct Treatment of IllPosed Boundary Equations in Acoustics of Closed Regions
242
87 IllPosed Method of Auxiliary Sources in Diffraction Theory
248
88 A Method of Global Random Search in Inverse Problems
251
89 IllPosed Problem on Reconstruction of Convex Hull of the Obstacle in Acoustic Medium
253
Numerical Methods for Irregular Operator Equations
259
92 Galerkin Methods for Integral Equations of the First Kind with Weakly Singular Kernels
263
93 Integral Equations of the Physical Diffraction Theory in the Case of Nonconvex Obstacles
268
94 Numerical Methods in Singular Integral Equations with the CauchyType Kernel
272
95 Numerical Methods for HyperSingular Integral Equations
276
References
281
Index
287
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