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 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.
Author(s): Christian Constanda, M. Zuhair Nashed, D. Rollins (Editors)
Edition: 1
Publisher: Birkhäuser Boston
Year: 2005
Language: English
Pages: 329
Tags: Математика;Вычислительная математика;
Copyright
......Page 4
Table of Contents
......Page 6
Preface......Page 12
Contributors......Page 14
1.1 The General Framework......Page 18
1.2 Nonlinear Boundary Value Problems......Page 23
1.3 Spectral Differential Problems......Page 26
1.4 Newton Method for the Matrix Eigenvalue Problem
......Page 30
References......Page 31
2.1 Introduction......Page 34
2.2 Nodal Method in Multi-layer Heat Conduction......Page 35
2.2.1 Lumped Analysis: Standard Approach......Page 36
2.2.2 Lumped Analysis: Improved Approach......Page 37
2.2.3. Time Integration: Laplace Transformation Method......Page 39
2.3 Numerical Results......Page 41
2.4. Final Remarks......Page 43
References......Page 44
3.2 Prerequisites......Page 46
3.3 Homogeneous System......Page 49
3.4 Homogeneous Initial Data......Page 50
References......Page 52
4.2 Prerequisites......Page 54
4.3 The Parameter-dependent Problems......Page 56
4.4 The Main Results......Page 60
References......Page 62
5.1 Introduction and Statement of the Problem......Page 64
5.2 Asymptotics in the Case r = 1......Page 67
5.3 Asymptotics in the Case r > 1......Page 73
References......Page 75
6.1 Introduction......Page 78
6.2 Background......Page 80
6.3 The Tikhonov–Morozov Method......Page 81
6.4 An Abstract Finite Element Method......Page 82
References......Page 87
7.1 Introduction......Page 88
7.2 The Tikhonov–Morozov Method......Page 90
7.3 Operators with Compact Resolvent......Page 91
7.4 The General Case......Page 93
References......Page 94
8.2 Boundary Integral Formulation......Page 96
8.3 Numerical Methods......Page 98
8.4 Numerical Results......Page 100
References......Page 103
9.1 Introduction......Page 106
9.2 The Parameter Choice Problem......Page 107
9.3 Advantages of CREF......Page 108
9.4.1.1 CLS Example......Page 109
9.4.1.3 Conjugate Gradient and CLS/Identity Example......Page 110
9.4.2.2 Conjugate Gradient and CLS/Laplacian Example......Page 111
References......Page 112
10.1 Introduction......Page 116
10.2 Taylor Series......Page 117
10.3 Integrals of Oscillatory Type......Page 118
10.4 Numerical Examples......Page 120
References......Page 121
11.2 Main Definitions and Preliminaries......Page 122
11.3 Stability of Periodic Systems......Page 124
11.4 Stability of Almost Periodic Systems......Page 127
References......Page 132
12.1 Introduction......Page 134
12.2.1 Three-dimensional Nonlinear Heat Conduction......Page 135
12.2.2 Transient Heat Conduction......Page 137
12.2.2.1 Governing Equation and the Laplace Transformation......Page 139
12.2.2.2 BEM for the Modified Helmholtz Equation......Page 140
12.2.2.3 Numerical Inversion of the Laplace-transformed Solution
......Page 141
12.3 Explicit Domain Decomposition......Page 142
12.4 Iterative Solution Algorithm......Page 144
12.6 Numerical Validation and Examples......Page 147
12.7 Conclusions......Page 149
References......Page 150
13.1 Introduction and Statement of the Main Results......Page 154
13.2 Estimates for Singular Integral Operators......Page 158
13.3 Traces and Conormal Derivatives......Page 163
13.4 Boundary Integral Operators and Proofs of the Main Results
......Page 169
13.5 Regularity of Green Potentials in Lipschitz Domains
......Page 170
13.6 The Two-dimensional Setting......Page 175
References......Page 176
14.1 Introduction......Page 178
14.2 Formulation of the Boundary Value Problem......Page 179
14.3 Parametrix and Potential-type Operators......Page 180
14.4 Green Identities and Integral Relations......Page 182
14.5.1 Integral Equation System (GG)......Page 183
14.5.2 Integral Equation System (GT )......Page 186
14.6.1 Integro-differential Problem (GD)......Page 188
14.7 Concluding Remarks......Page 191
References......Page 192
15.1 Introduction......Page 194
15.2 Statement of the Main Result......Page 198
15.3 Prerequisites......Page 200
15.4 Proof of Theorem 1......Page 201
References......Page 205
16.1 Introduction and Perspectives......Page 206
16.2 Sampling Solutions of Integral Equations of the First Kind
......Page 209
16.3 Wavelet Sampling Solutions of Integral Equations of the First Kind
......Page 211
References......Page 212
17.1 Introduction......Page 216
17.2 Qualitative Analysis of the Asymptotic Behavior of the NSL’s PDE
......Page 218
17.3 Determination of the Spectral Coefficients of the Density Function and Temperature
......Page 221
17.4 Computation of the Friction Drag Coefficient of the Wedged Delta Wing
......Page 222
References......Page 224
18.2 Solution Techniques......Page 226
18.3 Results for Five of Each of Laplace and Poisson Neumann BC Problems
......Page 228
18.4 Discussion......Page 229
18.5 Closure......Page 231
References......Page 233
19.1 Introduction......Page 236
19.2 Solution Methodologies......Page 237
19.3 3D and 4D Laplace Dirichlet BVPs......Page 238
19.4 Linear and Nonlinear Helmholtz Dirichlet BVPs
......Page 240
19.5 Coding Considerations......Page 241
19.6 Some Remarks on DFI Methodology......Page 242
19.7 Discussion......Page 243
19.8.1 DFI Conceptual Features......Page 245
19.8.2 Physical Features......Page 246
19.8.4 Numerical Features......Page 247
19.9 Closure......Page 248
References......Page 249
20.2 The Boundary Value Problem......Page 252
20.3 Numerical Method......Page 254
20.4 Convergence......Page 256
20.5 Computational Results......Page 259
References......Page 261
21.2 Torsion of Micropolar Beams......Page 262
21.3 Generalized Fourier Series......Page 263
21.4 Example: Torsion of an Elliptic Beam......Page 264
References......Page 266
22.1 Introduction......Page 268
22.2 Results......Page 270
References......Page 273
23.1 Introduction......Page 274
23.2 Rigid-Body Dynamics Model......Page 275
23.3 Continuous Contact Model......Page 277
23.4 Discrete Contact Model for a Falling Rod......Page 278
23.5 Numerical Simulation of a Falling Rigid Rod......Page 280
23.6 Discussion and Conclusion......Page 285
References......Page 286
24.1 Introduction......Page 288
24.2 Some Comparison Results......Page 290
24.3 Problem (E1). Blow-up Solutions......Page 292
References......Page 294
25.1 Introduction......Page 296
25.2 Compact-support Solutions......Page 297
25.3 Dead-core and Blow-up Solutions......Page 301
References......Page 305
26.1 Introduction......Page 306
26.2.1 Computation of the gk......Page 307
26.2.2 Evaluation of the Fourier Series......Page 309
26.3.1 Smooth Part......Page 310
26.3.2 Local Part......Page 311
26.4 Numerical Example and Conclusions......Page 312
References......Page 314
27.1 Introduction......Page 316
27.2 GILTT Formulation......Page 317
27.2.2.1 Type-1 ODE System......Page 318
27.2.2.2 Type-2 ODE System......Page 319
27.3 GILTT in Atmospheric Pollutant Dispersion......Page 320
References......Page 325
Index......Page 326
