Integral Methods in Science and Engineering : Theoretical and Practical Aspects - M. Zuhair Nashed

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Integral Methods in Science and Engineering

Theoretical and Practical Aspects

By: M. Zuhair Nashed (Editor), D. Rollins (Editor)

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Hardcover | 20 October 2005

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332 Pages

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The quantitative and qualitative study of the physical world makes use of many mathematical models governed by a great diversity of ordinary, partial differential, integral, and integro-differential equations. An essential step in such investigations is the solution of these types of equations, which sometimes can be performed analytically, while at other times only numerically. This edited, self-contained volume presents a series of state-of-the-art analytic and numerical methods of solution constructed for important problems arising in science and engineering, all based on the powerful operation of (exact or approximate) integration.

The book, consisting of twenty seven selected chapters presented by well-known specialists in the field, is an outgrowth of the Eighth International Conference on Integral Methods in Science and Engineering, held August 2a "4, 2004, in Orlando, FL. Contributors cover a wide variety of topics, from the theoretical development of boundary integral methods to the application of integration-based analytic and numerical techniques that include integral equations, finite and boundary elements, conservation laws, hybrid approaches, and other procedures.

The volume may be used as a reference guide and a practical resource. It is suitable for researchers and practitioners in applied mathematics, physics, and mechanical and electrical engineering, as well as graduate students in these disciplines.

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Preface	p. xi
Contributors	p. xiii
Newton-type Methods for Some Nonlinear Differential Problems	p. 1
The General Framework	p. 1
Nonlinear Boundary Value Problems	p. 6
Spectral Differential Problems	p. 9
Newton Method for the Matrix Eigenvalue Problem	p. 13
References	p. 14
Nodal and Laplace Transform Methods for Solving 2D Heat Conduction	p. 17
Introduction	p. 17
Nodal Method in Multi-layer Heat Conduction	p. 18
Numerical Results	p. 24
Final Remarks	p. 26
References	p. 27
The Cauchy Problem in the Bending of Thermoelastic Plates	p. 29
Introduction	p. 29
Prerequisites	p. 29
Homogeneous System	p. 32
Homogeneous Initial Data	p. 33
References	p. 35
Mixed Initial-boundary Value Problems for Thermoelastic Plates	p. 37
Introduction	p. 37
Prerequisites	p. 37
The Parameter-dependent Problems	p. 39
The Main Results	p. 43
References	p. 45
On the Structure of the Eigenfunctions of a Vibrating Plate with a Concentrated Mass and Very Small Thickness	p. 47
Introduction and Statement of the Problem	p. 47
Asymptotics in the Case r = 1	p. 50
Asymptotics in the Case r > 1	p. 56
References	p. 58
A Finite-dimensional Stabilized Variational Method for Unbounded Operators	p. 61
Introduction	p. 61
Background	p. 63
The Tikhonov-Morozov Method	p. 64
An Abstract Finite Element Method	p. 65
References	p. 70
A Converse Result for the Tikhonov-Morozov Method	p. 71
Introduction	p. 71
The Tikhonov-Morozov Method	p. 73
Operators with Compact Resolvent	p. 74
The General Case	p. 76
References	p. 77
A Weakly Singular Boundary Integral Formulation of the External Helmholtz Problem Valid for All Wavenumbers	p. 79
Introduction	p. 79
Boundary Integral Formulation	p. 79
Numerical Methods	p. 81
Numerical Results	p. 83
Conclusions	p. 86
References	p. 86
Cross-referencing for Determining Regularization Parameters in Ill-Posed Imaging Problems	p. 89
Introduction	p. 89
The Parameter Choice Problem	p. 90
Advantages of CREF	p. 91
Examples	p. 92
Summary	p. 95
References	p. 95
A Numerical Integration Method for Oscillatory Functions over an Infinite Interval by Substitution and Taylor Series	p. 99
Introduction	p. 99
Taylor Series	p. 100
Integrals of Oscillatory Type	p. 101
Numerical Examples	p. 103
Conclusion	p. 104
References	p. 104
On the Stability of Discrete Systems	p. 105
Introduction	p. 105
Main Definitions and Preliminaries	p. 105
Stability of Periodic Systems	p. 107
Stability of Almost Periodic Systems	p. 110
References	p. 115
Parallel Domain Decomposition Boundary Element Method for Large-scale Heat Transfer Problems	p. 117
Introduction	p. 117
Applications in Heat Transfer	p. 118
Explicit Domain Decomposition	p. 125
Iterative Solution Algorithm	p. 127
Parallel Implementation on a PC Cluster	p. 130
Numerical Validation and Examples	p. 130
Conclusions	p. 132
References	p. 133
The Poisson Problem for the Lame System on Low-dimensional Lipschitz Domains	p. 137
Introduction and Statement of the Main Results	p. 137
Estimates for Singular Integral Operators	p. 141
Traces and Conormal Derivatives	p. 146
Boundary Integral Operators and Proofs of the Main Results	p. 152
Regularity of Green Potentials in Lipschitz Domains	p. 153
The Two-dimensional Setting	p. 158
References	p. 159
Analysis of Boundary-domain Integral and Integro-differential Equations for a Dirichlet Problem with a Variable Coefficient	p. 161
Introduction	p. 161
Formulation of the Boundary Value Problem	p. 162
Parametrix and Potential-type Operators	p. 163
Green Identities and Integral Relations	p. 165
Segregated Boundary-domain Integral Equations	p. 166
United Boundary-domain Integro-differential Equations and Problem	p. 171
Concluding Remarks	p. 174
References	p. 175
On the Regularity of the Harmonic Green Potential in Nonsmooth Domains	p. 177
Introduction	p. 177
Statement of the Main Result	p. 181
Prerequisites	p. 183
Proof of Theorem 1	p. 184
References	p. 188
Applications of Wavelets and Kernel Methods in Inverse Problems	p. 189
Introduction and Perspectives	p. 189
Sampling Solutions of Integral Equations of the First Kind	p. 192
Wavelet Sampling Solutions of Integral Equations of the First Kind	p. 194
References	p. 195
Zonal, Spectral Solutions for the Navier-Stokes Layer and Their Aerodynamical Applications	p. 199
Introduction	p. 199
Qualitative Analysis of the Asymptotic Behavior of the NSL's PDE	p. 201
Determination of the Spectral Coefficients of the Density Function and Temperature	p. 204
Computation of the Friction Drag Coefficient of the Wedged Delta Wing	p. 205
Conclusions	p. 207
References	p. 207
Hybrid Laplace and Poisson Solvers. Part III: Neumann BCs	p. 209
Introduction	p. 209
Solution Techniques	p. 209
Results for Five of Each of Laplace and Poisson Neumann BC Problems	p. 211
Discussion	p. 212
Closure	p. 214
References	p. 216
Hybrid Laplace and Poisson Solvers. Part IV: Extensions	p. 219
Introduction	p. 219
Solution Methodologies	p. 220
3D and 4D Laplace Dirichlet BVPs	p. 221
Linear and Nonlinear Helmholtz Dirichlet BVPs	p. 223
Coding Considerations	p. 224
Some Remarks on DFI Methodology	p. 225
Discussion	p. 226
Some DFI Advantages	p. 228
Closure	p. 231
References	p. 232
A Contact Problem for a Convection-diffusion Equation	p. 235
Introduction	p. 235
The Boundary Value Problem	p. 235
Numerical Method	p. 237
Convergence	p. 239
Computational Results	p. 242
Conclusions	p. 244
References	p. 244
Integral Representation of the Solution of Torsion of an Elliptic Beam with Microstructure	p. 245
Introduction	p. 245
Torsion of Micropolar Beams	p. 245
Generalized Fourier Series	p. 246
Example: Torsion of an Elliptic Beam	p. 247
References	p. 249
A Coupled Second-order Boundary Value Problem at Resonance	p. 251
Introduction	p. 251
Results	p. 253
References	p. 256
Multiple Impact Dynamics of a Falling Rod and Its Numerical Solution	p. 257
Introduction	p. 257
Rigid-Body Dynamics Model	p. 258
Continuous Contact Model	p. 260
Discrete Contact Model for a Falling Rod	p. 261
Numerical Simulation of a Falling Rigid Rod	p. 263
Discussion and Conclusion	p. 268
References	p. 269
On the Monotone Solutions of Some ODEs. I: Structure of the Solutions	p. 271
Introduction	p. 271
Some Comparison Results	p. 273
Problem (E1). Blow-up Solutions	p. 275
References	p. 277
On the Monotone Solutions of Some ODEs. II: Dead-core, Compact-support, and Blow-up Solutions	p. 279
Introduction	p. 279
Compact-support Solutions	p. 280
Dead-core and Blow-up Solutions	p. 284
References	p. 288
A Spectral Method for the Fast Solution of Boundary Integral Formulations of Elliptic Problems	p. 289
Introduction	p. 289
A Fast Algorithm for Smooth, Periodic Kernels	p. 290
Extension to Singular Kernels	p. 293
Numerical Example and Conclusions	p. 295
References	p. 297
The GILTT Pollutant Simulation in a Stable Atmosphere	p. 299
Introduction	p. 299
GILTT Formulation	p. 300
GILTT in Atmospheric Pollutant Dispersion	p. 303
Final Remarks	p. 308
References	p. 308
Index	p. 309
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Integral Methods in Science and Engineering

Theoretical and Practical Aspects

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