Mathematics does not exist in isolation but is linked inextricably to the physical world. At the 2003 International Congress of Industrial and Applied Mathematics, leading mathematicians from around the globe gathered for a symposium on the "Mathematics of Real World Problems," which focused on furthering the establishment and dissemination of those links. Presented in four parts, Mathematical Models and Methods for Real World Systems comprises chapters by those invited to this symposium. The first part examines mathematics for technology, exploring future challenges of mathematical technology, offering a wide-ranging definition of industrial mathematics, and explaining the mathematics of type-II superconductors. After lucid discussions on theoretical and applied aspects of wavelets, the book presents classical and fractal methods for physical problems, including a fractal approach to porous media textures and using MATLAB® to model chaos in the motion of a satellite. The final section surveys recent trends in variational methods, focusing on areas such as elliptic inverse problems, sweeping processes, and the BBKY hierarchy of quantum kinetic equations. By virtue of its abstraction, mathematics allows the transfer of ideas between fields of applications. Mathematical Models and Methods for Real World Systems clearly demonstrates this and promotes the kind of cross-thinking that nurtures creativity and leads to further innovation.
Author(s): K.M. Furati, Abul Hasan Siddiqi
Series: Pure and Applied Mathematics, a Series of Monographs and Tex
Edition: illustrated edition
Publisher: Crc Pr Inc
Year: 2005
Language: English
Pages: 448
Mathematical Models and Methods for Real World Systems......Page 1
CONTENTS......Page 6
PREFACE......Page 9
CONTRIBUTING AUTHORS......Page 11
Part I: Mathematics for Technology......Page 14
1 Introduction......Page 15
2 Simulation of Processes and the Behavior of Products......Page 17
3 Optimization, Control, and Design......Page 31
5 Management and Exploitation of Data......Page 36
6 Virtual Material Design......Page 40
7 Biotechnology, Food, and Health......Page 43
8 Conclusions......Page 47
References......Page 49
1 Introduction......Page 50
2 Categories of Research......Page 51
3 What Is Industrial Mathematics?......Page 53
4 Structures to Undertake Industrial Mathematics......Page 54
References......Page 56
Abstract......Page 57
1 Introduction......Page 58
2 Introduction to Parabolic Variational Inequalities......Page 60
2.2 Existence of Solutions, Uniqueness and Numerical Analysis of QVI......Page 62
2.3 Parabolic Quasi-Variational Inequality......Page 63
3 Fast Algorithm for the Bean Critical-State Model for Superconductivity......Page 64
4 Quasi-Variational Inequality for the Kim Model of Type-II Superconductivity......Page 67
5 Existence of Solutions of the Kim Model......Page 70
6 Parallel Algorithms for the Kim Model......Page 73
6.1 Formulation of Parabolic Quasi-Variational Inequalities......Page 74
6.2 Decomposition Method......Page 75
6.3 Decomposition Approximation......Page 76
7 Superconducting Fault Current Limiters......Page 77
References......Page 78
Part II: Wavelet Methods for Real-World Problems......Page 80
1 Introduction......Page 81
2.1 General Frame Theory......Page 82
2.2 Wavelet Frames......Page 85
2.3 Irregular Wavelet Frames......Page 87
3 Dyadic Wavelet Frames......Page 88
3.1 The Duals of a Wavelet Frame......Page 89
3.2 Multiresolution Analysis......Page 92
4 Frame Multiresolution Analysis......Page 95
5 The Unitary Extension Principle......Page 98
6 The Oblique Extension Principle......Page 103
7 Construction of Dual Wavelet Pairs......Page 109
References......Page 112
Abstract......Page 115
1 Introduction......Page 116
2 Wavelet-Galerkin Method......Page 117
3 Algebraic Problem......Page 119
4 Some Numerical Comparisons......Page 123
References......Page 128
Abstract......Page 133
1 Introduction......Page 134
2 Basic Tools......Page 135
2.1 Multiresolution Analysis (MRA)......Page 138
2.2 Wavelet Decomposition......Page 140
2.3 Wavelet Packets......Page 141
3 Image Processing......Page 142
3.1 Image Compression......Page 144
Compression Results......Page 145
3.2 Denoising......Page 146
3.3 Miscellaneous Issues – Quantization and Blowup of Solutions......Page 147
4 Partial Differential Equations......Page 149
4.1 Error Estimation by the Coifman Orthogonal System......Page 150
4.2 Preconditioning Based on Wavelet Properties......Page 152
Preconditioning of Helmholtz Problem......Page 157
4.3 Illustration of Adaptive Wavelet Methods for Elliptic Operators......Page 158
4.4 Evolution Equation......Page 159
5 Optimization Problems of Chemical Engineering......Page 161
6 Black-Scholes Model of Option Pricing......Page 164
6.1 Wavelet-Based Method for European Option......Page 165
7 Wavelets for Financial Time-Series Analysis......Page 167
8 Maxwell's Equations......Page 168
9 Turbulence Analysis......Page 170
10 Limitations, Ridgelets, and Curvelets......Page 171
10.2 Curvelets......Page 172
10.3 Subband Filtering......Page 173
10.4 Definition of Curvelet Transform......Page 174
11 Concluding Remarks and Open Problems......Page 175
References......Page 177
Abstract......Page 186
2.1 Wavelet Methods......Page 187
Reconstruction Algorithm......Page 189
Time Series......Page 190
Singularity Spectrum......Page 191
Procedure for Numerical Calculation......Page 193
3 Wavelet Based Time Series Analysis of Indian Rainfall Data......Page 194
4 Main Features......Page 204
References......Page 216
Websites......Page 217
Abstract......Page 218
2.1 Material......Page 219
3.1 Time Series Analysis of Horizontal Wind Speed Components......Page 220
3.2 Wavelet Analysis of Horizontal Wind Speed Components Using Daubechies Wavelet......Page 225
3.3 Wavelet Analysis of Zonal Wind Speeds Performed for the Year 1997 by Using Morlet Wavelet......Page 228
References......Page 230
Abstract......Page 232
2.3 Methodology......Page 233
3.1 Statistical Analysis of Meteorological Parameters......Page 234
3.2 Time Series Analysis......Page 237
3.3 Harmonic Analysis......Page 240
4 Results and Conclusion......Page 249
References......Page 250
Abstract......Page 252
1 Introduction......Page 253
2.1 Time, Frequency, and Time-Frequency......Page 255
2.2 Short-Time Fourier Transform......Page 256
3.1 Continuous Wavelet Transform......Page 257
3.2 Multiresolution Analysis......Page 259
3.3 Discrete Wavelet Transform......Page 261
3.4 Wavelets Characteristics......Page 263
4.1 Geophysical Data Description......Page 265
4.2 Depth-Wavelength Analysis......Page 267
4.3 Time-Frequency Analysis......Page 268
5.1 Seismic Compression......Page 271
5.2 Coherent Noise Filtering......Page 272
6 Conclusions......Page 274
References......Page 275
Part III: Classical and Fractal Methods for Physical Problems......Page 277
1 Introduction......Page 278
2 Formation of Singularities......Page 281
References......Page 285
Abstract......Page 287
1 Introduction......Page 288
2 Formulation of the Problem......Page 289
3 Solution of the Problem......Page 291
4 Reflecting–Type Boundary Condition......Page 296
5 Numerical Results......Page 298
5.1 Parabolic Perturbation Sea Bottom......Page 299
5.2 Dipping Sea Bottom......Page 300
5.3 Quadratic Perturbation Sea Bottom......Page 301
References......Page 302
Abstract......Page 304
1 Introduction......Page 305
2 Equation of Motion......Page 306
2.1 Hamilton’s Equation......Page 307
2.3 Melnikov’s Function......Page 308
2.4 Graphical Representation of Melnikov’s Function......Page 309
3 Non-Resonant Planar Oscillations of A Satellite......Page 315
4.1 Asymptotic Solution near n Approximately Equal To c......Page 319
4.2 Asymptotic Solution near n Approximately Equal To 1......Page 325
5 Estimation of Resonance Width......Page 329
6 The Spin-Orbit Phase Space......Page 333
References......Page 337
Abstract......Page 340
1 Introduction......Page 341
2.1 Equation of Motion......Page 343
2.4 Evaluation of Melnikov's Integral......Page 347
Earth-Moon System......Page 348
3 Non-Resonant Planar Oscillations of a Satellite......Page 354
4 Resonant Planar Oscillation of a Satellite......Page 356
5 Estimation of Resonance Width......Page 367
7 Conclusion......Page 373
References......Page 374
Abstract......Page 376
1 Introduction......Page 377
2 Porosity and the Autocorrelation Function......Page 378
Second Step......Page 381
Fat Fractals: Synthetic Porous Media......Page 383
Thin Fractals: Structures of Porosity Zero......Page 385
References......Page 387
Part IV: Trends in Variational Methods......Page 389
1 Introduction......Page 390
2 Elliptic Inverse Problems......Page 392
2.1 Some Special Cases......Page 393
2.2 Differentiability of the Solution Operator......Page 395
2.3 A Smooth, Convex Objective Functional for Inversion......Page 398
2.4 BV-Regularization......Page 399
2.5 Some Assumptions on the Trilinear Form T(·,·,·)......Page 400
2.6 Finite-Dimensional Approximation......Page 401
3 Output Least-Squares Approach......Page 402
4 Augmented Lagrangian Approach......Page 403
5 Fixed Point Approach......Page 406
Algorithm 1......Page 410
Algorithm 2......Page 411
Standard Output Least-Square Functional......Page 412
Coefficient Dependent Output Least-Square Functional......Page 413
6 Numerical Results......Page 414
References......Page 418
1 Introduction......Page 421
2 Derivation of Hierarchy of Kinetic Equations for Correlation Functions......Page 422
References......Page 427
1 Introduction......Page 428
2 Convergence and the Choice of the Relaxation Parameter......Page 430
2.1 The Real Case......Page 432
2.2 The Complex Case......Page 434
3 Examples......Page 438
References......Page 439
1 Introduction......Page 441
2 A Special Class of Moreau’s Sweeping Process......Page 443
3 A Variant of a Special Class of Moreau’s Sweeping Process......Page 446
References......Page 447
Eagle Hill......Page 448
