Integral methods in science and engineering : theoretical and practical aspects

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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 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.

Mario Ahues and Alain LargillierIvanilda B. Aseka and Marco T. Vilhena and Haroldo F. Campos VelhoIgor Chudinovich and Christian ConstandaIgor Chudinovich and Christian ConstandaDelfina Gomez and Miguel Lobo and Eugenia PerezCharles W. GroetschCharles W. GroetschPaul J. Harris and Ke Chen and Jin ChengJohn W. Hilgers and Barbara S. BertramHiroshi HirayamaAlexander O. Ignatyev and Oleksiy A. IgnatyevAlain J. Kassab and Eduardo A. DivoSvitlana Mayboroda and Marius MitreaSergey E. MikhailovDorina MitreaZuhair NashedAdriana NastaseFred R. PayneFred R. PayneShirley Pomeranz and Gilbert Lewis and Christian ConstandaStanislav PotapenkoSeppo Seikkala and Markku HihnalaHua Shan and Jianzhong Su and Florin Badiu and Jiansen Zhu and Leon XuTadieTadieJohannes TauschSergio Wortmann and Marco T. Vilhena and Haroldo F. Campos Velho and Cynthia F. Segatto

Preface	p. xi
Contributors	p. xiii
1 Newton-type Methods for Some Nonlinear Differential Problems	p. 1
1.1 The General Framework	p. 1
1.2 Nonlinear Boundary Value Problems	p. 6
1.3 Spectral Differential Problems	p. 9
1.4 Newton Method for the Matrix Eigenvalue Problem	p. 13
References	p. 14
2 Nodal and Laplace Transform Methods for Solving 2D Heat Conduction	p. 17
2.1 Introduction	p. 17
2.2 Nodal Method in Multi-layer Heat Conduction	p. 18
2.3 Numerical Results	p. 24
2.4 Final Remarks	p. 26
References	p. 27
3 The Cauchy Problem in the Bending of Thermoelastic Plates	p. 29
3.1 Introduction	p. 29
3.2 Prerequisites	p. 29
3.3 Homogeneous System	p. 32
3.4 Homogeneous Initial Data	p. 33
References	p. 35
4 Mixed Initial-boundary Value Problems for Thermoelastic Plates	p. 37
4.1 Introduction	p. 37
4.2 Prerequisites	p. 37
4.3 The Parameter-dependent Problems	p. 39
4.4 The Main Results	p. 43
References	p. 45
5 On the Structure of the Eigenfunctions of a Vibrating Plate with a Concentrated Mass and Very Small Thickness	p. 47
5.1 Introduction and Statement of the Problem	p. 47
5.2 Asymptotics in the Case r = 1	p. 50
5.3 Asymptotics in the Case r > 1	p. 56
References	p. 58
6 A Finite-dimensional Stabilized Variational Method for Unbounded Operators	p. 61
6.1 Introduction	p. 61
6.2 Background	p. 63
6.3 The Tikhonov-Morozov Method	p. 64
6.4 An Abstract Finite Element Method	p. 65
References	p. 70
7 A Converse Result for the Tikhonov-Morozov Method	p. 71
7.1 Introduction	p. 71
7.2 The Tikhonov-Morozov Method	p. 73
7.3 Operators with Compact Resolvent	p. 74
7.4 The General Case	p. 76
References	p. 77
8 A Weakly Singular Boundary Integral Formulation of the External Helmholtz Problem Valid for All Wavenumbers	p. 79
8.1 Introduction	p. 79
8.2 Boundary Integral Formulation	p. 79
8.3 Numerical Methods	p. 81
8.4 Numerical Results	p. 83
8.5 Conclusions	p. 86
References	p. 86
9 Cross-referencing for Determining Regularization Parameters in Ill-Posed Imaging Problems	p. 89
9.1 Introduction	p. 89
9.2 The Parameter Choice Problem	p. 90
9.3 Advantages of CREF	p. 91
9.4 Examples	p. 92
9.5 Summary	p. 95
References	p. 95
10 A Numerical Integration Method for Oscillatory Functions over an Infinite Interval by Substitution and Taylor Series	p. 99
10.1 Introduction	p. 99
10.2 Taylor Series	p. 100
10.3 Integrals of Oscillatory Type	p. 101
10.4 Numerical Examples	p. 103
10.5 Conclusion	p. 104
References	p. 104
11 On the Stability of Discrete Systems	p. 105
11.1 Introduction	p. 105
11.2 Main Definitions and Preliminaries	p. 105
11.3 Stability of Periodic Systems	p. 107
11.4 Stability of Almost Periodic Systems	p. 110
References	p. 115
12 Parallel Domain Decomposition Boundary Element Method for Large-scale Heat Transfer Problems	p. 117
12.1 Introduction	p. 117
12.2 Applications in Heat Transfer	p. 118
12.3 Explicit Domain Decomposition	p. 125
12.4 Iterative Solution Algorithm	p. 127
12.5 Parallel Implementation on a PC Cluster	p. 130
12.6 Numerical Validation and Examples	p. 130
12.7 Conclusions	p. 132
References	p. 133
13 The Poisson Problem for the Lame System on Low-dimensional Lipschitz Domains	p. 137
13.1 Introduction and Statement of the Main Results	p. 137
13.2 Estimates for Singular Integral Operators	p. 141
13.3 Traces and Conormal Derivatives	p. 146
13.4 Boundary Integral Operators and Proofs of the Main Results	p. 152
13.5 Regularity of Green Potentials in Lipschitz Domains	p. 153
13.6 The Two-dimensional Setting	p. 158
References	p. 159
14 Analysis of Boundary-domain Integral and Integro-differential Equations for a Dirichlet Problem with a Variable Coefficient	p. 161
14.1 Introduction	p. 161
14.2 Formulation of the Boundary Value Problem	p. 162
14.3 Parametrix and Potential-type Operators	p. 163
14.4 Green Identities and Integral Relations	p. 165
14.5 Segregated Boundary-domain Integral Equations	p. 166
14.6 United Boundary-domain Integro-differential Equations and Problem	p. 171
14.7 Concluding Remarks	p. 174
References	p. 175
15 On the Regularity of the Harmonic Green Potential in Nonsmooth Domains	p. 177
15.1 Introduction	p. 177
15.2 Statement of the Main Result	p. 181
15.3 Prerequisites	p. 183
15.4 Proof of Theorem 1	p. 184
References	p. 188
16 Applications of Wavelets and Kernel Methods in Inverse Problems	p. 189
16.1 Introduction and Perspectives	p. 189
16.2 Sampling Solutions of Integral Equations of the First Kind	p. 192
16.3 Wavelet Sampling Solutions of Integral Equations of the First Kind	p. 194
References	p. 195
17 Zonal, Spectral Solutions for the Navier-Stokes Layer and Their Aerodynamical Applications	p. 199
17.1 Introduction	p. 199
17.2 Qualitative Analysis of the Asymptotic Behavior of the NSL's PDE	p. 201
17.3 Determination of the Spectral Coefficients of the Density Function and Temperature	p. 204
17.4 Computation of the Friction Drag Coefficient of the Wedged Delta Wing	p. 205
17.5 Conclusions	p. 207
References	p. 207
18 Hybrid Laplace and Poisson Solvers. Part III: Neumann BCs	p. 209
18.1 Introduction	p. 209
18.2 Solution Techniques	p. 209
18.3 Results for Five of Each of Laplace and Poisson Neumann BC Problems	p. 211
18.4 Discussion	p. 212
18.5 Closure	p. 214
References	p. 216
19 Hybrid Laplace and Poisson Solvers. Part IV: Extensions	p. 219
19.1 Introduction	p. 219
19.2 Solution Methodologies	p. 220
19.3 3D and 4D Laplace Dirichlet BVPs	p. 221
19.4 Linear and Nonlinear Helmholtz Dirichlet BVPs	p. 223
19.5 Coding Considerations	p. 224
19.6 Some Remarks on DFI Methodology	p. 225
19.7 Discussion	p. 226
19.8 Some DFI Advantages	p. 228
19.9 Closure	p. 231
References	p. 232
20 A Contact Problem for a Convection-diffusion Equation	p. 235
20.1 Introduction	p. 235
20.2 The Boundary Value Problem	p. 235
20.3 Numerical Method	p. 237
20.4 Convergence	p. 239
20.5 Computational Results	p. 242
20.6 Conclusions	p. 244
References	p. 244
21 Integral Representation of the Solution of Torsion of an Elliptic Beam with Microstructure	p. 245
21.1 Introduction	p. 245
21.2 Torsion of Micropolar Beams	p. 245
21.3 Generalized Fourier Series	p. 246
21.4 Example: Torsion of an Elliptic Beam	p. 247
References	p. 249
22 A Coupled Second-order Boundary Value Problem at Resonance	p. 251
22.1 Introduction	p. 251
22.2 Results	p. 253
References	p. 256
23 Multiple Impact Dynamics of a Falling Rod and Its Numerical Solution	p. 257
23.1 Introduction	p. 257
23.2 Rigid-Body Dynamics Model	p. 258
23.3 Continuous Contact Model	p. 260
23.4 Discrete Contact Model for a Falling Rod	p. 261
23.5 Numerical Simulation of a Falling Rigid Rod	p. 263
23.6 Discussion and Conclusion	p. 268
References	p. 269
24 On the Monotone Solutions of Some ODEs. I: Structure of the Solutions	p. 271
24.1 Introduction	p. 271
24.2 Some Comparison Results	p. 273
24.3 Problem (E1). Blow-up Solutions	p. 275
References	p. 277
25 On the Monotone Solutions of Some ODEs. II: Dead-core, Compact-support, and Blow-up Solutions	p. 279
25.1 Introduction	p. 279
25.2 Compact-support Solutions	p. 280
25.3 Dead-core and Blow-up Solutions	p. 284
References	p. 288
26 A Spectral Method for the Fast Solution of Boundary Integral Formulations of Elliptic Problems	p. 289
26.1 Introduction	p. 289
26.2 A Fast Algorithm for Smooth, Periodic Kernels	p. 290
26.3 Extension to Singular Kernels	p. 293
26.4 Numerical Example and Conclusions	p. 295
References	p. 297
27 The GILTT Pollutant Simulation in a Stable Atmosphere	p. 299
27.1 Introduction	p. 299
27.2 GILTT Formulation	p. 300
27.3 GILTT in Atmospheric Pollutant Dispersion	p. 303
27.4 Final Remarks	p. 308
References	p. 308
Index	p. 309

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