Integral methods in science and engineering: Theoretical and practical aspects

310 Pages·2005·2.624 MB·English

by  Christian Constanda, M. Zuhair Nashed, D. Rollins

Integral Methods in
Science and Engineering
Theoretical and
Practical Aspects
C. Constanda
Z. Nashed
D. Rollins
Editors
Birkha¨user
Boston • Basel • Berlin
C.Constanda Z.Nashed
UniversityofTulsa UniversityofCentralFlorida
DepartmentofMathematical DepartmentofMathematics
andComputerSciences 4000CentralFloridaBlvd.
600SouthCollegeAvenue Orlando,FL32816
Tulsa,OK74104 USA
USA
D.Rollins
UniversityofCentralFlorida
DepartmentofMathematics
4000CentralFloridaBlvd.
Orlando,FL32816
USA
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AMSSubjectClassification:45-06,65-06,74-06,76-06
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Integralmethodsinscienceandengineering:theoreticalandpracticalaspects/C.
Constanda,Z.Nashed,D.Rollins(editors).
p.cm.
Includesbibliographicalreferencesandindex.
ISBN0-8176-4377-X(alk.paper)
1.Integralequations–Numericalsolutions–Congresses.2.Mathematical
analysis–Congresses.3.Science–Mathematics–Congresses.4.Engineering
mathematics–Congresses.I.Constanda,C.(Christian)II.Nashed,Z.(Zuhair)III.Rollins,
D.(David),1955-
QA431.I492005
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Contents
Preface xi
Contributors xiii
1 Newton-type Methods for Some Nonlinear Differential
Problems
Mario Ahues and Alain Largillier 1
1.1 The General Framework . . . . . . . . . . . . . . 1
1.2 Nonlinear Boundary Value Problems . . . . . . . . . 6
1.3 Spectral Differential Problems . . . . . . . . . . . . 9
1.4 Newton Method for the Matrix Eigenvalue Problem . . 13
References . . . . . . . . . . . . . . . . . . . . 14
2 Nodal and Laplace Transform Methods for Solving
2D Heat Conduction
Ivanilda B. Aseka, Marco T. Vilhena,
and Haroldo F. Campos Velho 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . 17
2.2 Nodal Method in Multi-layer Heat Conduction . . . . 18
2.3 Numerical Results . . . . . . . . . . . . . . . . . 24
2.4 Final Remarks . . . . . . . . . . . . . . . . . . . 26
References . . . . . . . . . . . . . . . . . . . . 27
3 The Cauchy Problem in the Bending of
Thermoelastic Plates
Igor Chudinovich and Christian Constanda 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . 29
3.2 Prerequisites . . . . . . . . . . . . . . . . . . . 29
3.3 Homogeneous System . . . . . . . . . . . . . . . . 32
3.4 Homogeneous Initial Data . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . 35
4 Mixed Initial-boundary Value Problems for
Thermoelastic Plates
Igor Chudinovich and Christian Constanda 37
4.1 Introduction . . . . . . . . . . . . . . . . . . . 37
4.2 Prerequisites . . . . . . . . . . . . . . . . . . . 37
4.3 The Parameter-dependent Problems . . . . . . . . . 39
vi Contents
4.4 The Main Results . . . . . . . . . . . . . . . . . 43
References . . . . . . . . . . . . . . . . . . . . 45
5 On the Structure of the Eigenfunctions of a
Vibrating Plate with a Concentrated Mass and Very
Small Thickness
Delfina Go´mez, Miguel Lobo, and Eugenia P´erez 47
5.1 Introduction and Statement of the Problem . . . . . . 47
5.2 Asymptotics in the Case r =1 . . . . . . . . . . . . 50
5.3 Asymptotics in the Case r >1 . . . . . . . . . . . . 56
References . . . . . . . . . . . . . . . . . . . . 58
6 A Finite-dimensional Stabilized Variational Method
for Unbounded Operators
Charles W. Groetsch 61
6.1 Introduction . . . . . . . . . . . . . . . . . . . 61
6.2 Background . . . . . . . . . . . . . . . . . . . . 63
6.3 The Tikhonov–Morozov Method . . . . . . . . . . . 64
6.4 An Abstract Finite Element Method . . . . . . . . . 65
References . . . . . . . . . . . . . . . . . . . . 70
7 A Converse Result for the Tikhonov–Morozov Method
Charles W. Groetsch 71
7.1 Introduction . . . . . . . . . . . . . . . . . . . 71
7.2 The Tikhonov–Morozov Method . . . . . . . . . . . 73
7.3 Operators with Compact Resolvent . . . . . . . . . 74
7.4 The General Case . . . . . . . . . . . . . . . . . 76
References . . . . . . . . . . . . . . . . . . . . 77
8 A Weakly Singular Boundary Integral Formulation
of the External Helmholtz Problem Valid for All
Wavenumbers
Paul J. Harris, Ke Chen, and Jin Cheng 79
8.1 Introduction . . . . . . . . . . . . . . . . . . . 79
8.2 Boundary Integral Formulation . . . . . . . . . . . 79
8.3 Numerical Methods . . . . . . . . . . . . . . . . 81
8.4 Numerical Results . . . . . . . . . . . . . . . . . 83
8.5 Conclusions . . . . . . . . . . . . . . . . . . . . 86
References . . . . . . . . . . . . . . . . . . . . 86
9 Cross-referencing for Determining Regularization
Parameters in Ill-Posed Imaging Problems
John W. Hilgers and Barbara S. Bertram 89
9.1 Introduction . . . . . . . . . . . . . . . . . . . 89
9.2 The Parameter Choice Problem . . . . . . . . . . . 90
9.3 Advantages of CREF . . . . . . . . . . . . . . . . 91
9.4 Examples . . . . . . . . . . . . . . . . . . . . . 92
9.5 Summary . . . . . . . . . . . . . . . . . . . . . 95
References . . . . . . . . . . . . . . . . . . . . 95
Contents vii
10 A Numerical Integration Method for Oscillatory
Functions over an Infinite Interval by Substitution
and Taylor Series
Hiroshi Hirayama 99
10.1 Introduction . . . . . . . . . . . . . . . . . . . 99
10.2 Taylor Series . . . . . . . . . . . . . . . . . . . 100
10.3 Integrals of Oscillatory Type . . . . . . . . . . . . 101
10.4 Numerical Examples . . . . . . . . . . . . . . . . 103
10.5 Conclusion . . . . . . . . . . . . . . . . . . . . 104
References . . . . . . . . . . . . . . . . . . . . 104
11 On the Stability of Discrete Systems
Alexander O. Ignatyev and Oleksiy A. Ignatyev 105
11.1 Introduction . . . . . . . . . . . . . . . . . . . 105
11.2 Main Definitions and Preliminaries . . . . . . . . . . 105
11.3 Stability of Periodic Systems . . . . . . . . . . . . 107
11.4 Stability of Almost Periodic Systems . . . . . . . . . 110
References . . . . . . . . . . . . . . . . . . . . 115
12 Parallel Domain Decomposition Boundary Element
Method for Large-scale Heat Transfer Problems
Alain J. Kassab and Eduardo A. Divo 117
12.1 Introduction . . . . . . . . . . . . . . . . . . . 117
12.2 Applications in Heat Transfer . . . . . . . . . . . . 118
12.3 Explicit Domain Decomposition . . . . . . . . . . . 125
12.4 Iterative Solution Algorithm . . . . . . . . . . . . . 127
12.5 Parallel Implementation on a PC Cluster . . . . . . . 130
12.6 Numerical Validation and Examples . . . . . . . . . 130
12.7 Conclusions . . . . . . . . . . . . . . . . . . . . 132
References . . . . . . . . . . . . . . . . . . . . 133
13 The Poisson Problem for the Lam´e System on
Low-dimensional Lipschitz Domains
Svitlana Mayboroda and Marius Mitrea 137
13.1 Introduction and Statement of the Main Results . . . . 137
13.2 Estimates for Singular Integral Operators . . . . . . . 141
13.3 Traces and Conormal Derivatives . . . . . . . . . . 146
13.4 Boundary Integral Operators and Proofs of the Main
Results . . . . . . . . . . . . . . . . . . . . . . 152
13.5 Regularity of Green Potentials in Lipschitz Domains . . 153
13.6 The Two-dimensional Setting . . . . . . . . . . . . 158
References . . . . . . . . . . . . . . . . . . . . 159
14 Analysis of Boundary-domain Integral and
Integro-differential Equations for a Dirichlet Problem
with a Variable Coefficient
Sergey E. Mikhailov 161
14.1 Introduction . . . . . . . . . . . . . . . . . . . 161
14.2 Formulation of the Boundary Value Problem . . . . . 162
viii Contents
14.3 Parametrix and Potential-type Operators . . . . . . . 163
14.4 Green Identities and Integral Relations . . . . . . . . 165
14.5 Segregated Boundary-domain Integral Equations . . . . 166
14.6 United Boundary-domain Integro-differential Equations
and Problem . . . . . . . . . . . . . . . . . . . 171
14.7 Concluding Remarks . . . . . . . . . . . . . . . . 174
References . . . . . . . . . . . . . . . . . . . . 175
15 On the Regularity of the Harmonic Green Potential
in Nonsmooth Domains
Dorina Mitrea 177
15.1 Introduction . . . . . . . . . . . . . . . . . . . 177
15.2 Statement of the Main Result . . . . . . . . . . . . 181
15.3 Prerequisites . . . . . . . . . . . . . . . . . . . 183
15.4 Proof of Theorem 1 . . . . . . . . . . . . . . . . 184
References . . . . . . . . . . . . . . . . . . . . 188
16 Applications of Wavelets and Kernel Methods in
Inverse Problems
Zuhair Nashed 189
16.1 Introduction and Perspectives . . . . . . . . . . . . 189
16.2 Sampling Solutions of Integral Equations of the First Kind 192
16.3 Wavelet Sampling Solutions of Integral Equations of the
First Kind . . . . . . . . . . . . . . . . . . . . 194
References . . . . . . . . . . . . . . . . . . . . 195
17 Zonal, Spectral Solutions for the Navier–Stokes Layer
and Their Aerodynamical Applications
Adriana Nastase 199
17.1 Introduction . . . . . . . . . . . . . . . . . . . 199
17.2 Qualitative Analysis of the Asymptotic Behavior of the
NSL’s PDE . . . . . . . . . . . . . . . . . . . . 201
17.3 Determination of the Spectral Coefficients of the Density
Function and Temperature . . . . . . . . . . . . . 204
17.4 Computation of the Friction Drag Coefficient of the
Wedged Delta Wing . . . . . . . . . . . . . . . . 205
17.5 Conclusions . . . . . . . . . . . . . . . . . . . . 207
References . . . . . . . . . . . . . . . . . . . . 207
18 Hybrid Laplace and Poisson Solvers. Part III:
Neumann BCs
Fred R. Payne 209
18.1 Introduction . . . . . . . . . . . . . . . . . . . 209
18.2 Solution Techniques . . . . . . . . . . . . . . . . 209
18.3 Results for Five of Each of Laplace and Poisson Neumann
BC Problems . . . . . . . . . . . . . . . . . . . 211
18.4 Discussion . . . . . . . . . . . . . . . . . . . . 212
18.5 Closure . . . . . . . . . . . . . . . . . . . . . . 214
References . . . . . . . . . . . . . . . . . . . . 216
Contents ix
19 Hybrid Laplace and Poisson Solvers. Part IV:
Extensions
Fred R. Payne 219
19.1 Introduction . . . . . . . . . . . . . . . . . . . 219
19.2 Solution Methodologies . . . . . . . . . . . . . . . 220
19.3 3D and 4D Laplace Dirichlet BVPs . . . . . . . . . 221
19.4 Linear and Nonlinear Helmholtz Dirichlet BVPs . . . . 223
19.5 Coding Considerations . . . . . . . . . . . . . . . 224
19.6 Some Remarks on DFI Methodology . . . . . . . . . 225
19.7 Discussion . . . . . . . . . . . . . . . . . . . . 226
19.8 Some DFI Advantages . . . . . . . . . . . . . . . 228
19.9 Closure . . . . . . . . . . . . . . . . . . . . . . 231
References . . . . . . . . . . . . . . . . . . . . 232
20 A Contact Problem for a Convection-diffusion
Equation
Shirley Pomeranz, Gilbert Lewis, and Christian Constanda 235
20.1 Introduction . . . . . . . . . . . . . . . . . . . 235
20.2 The Boundary Value Problem . . . . . . . . . . . . 235
20.3 Numerical Method . . . . . . . . . . . . . . . . . 237
20.4 Convergence . . . . . . . . . . . . . . . . . . . 239
20.5 Computational Results . . . . . . . . . . . . . . . 242
20.6 Conclusions . . . . . . . . . . . . . . . . . . . . 244
References . . . . . . . . . . . . . . . . . . . . 244
21 Integral Representation of the Solution of Torsion
of an Elliptic Beam with Microstructure
Stanislav Potapenko 245
21.1 Introduction . . . . . . . . . . . . . . . . . . . 245
21.2 Torsion of Micropolar Beams . . . . . . . . . . . . 245
21.3 Generalized Fourier Series . . . . . . . . . . . . . . 246
21.4 Example: Torsion of an Elliptic Beam . . . . . . . . 247
References . . . . . . . . . . . . . . . . . . . . 249
22 A Coupled Second-order Boundary Value Problem
at Resonance
Seppo Seikkala and Markku Hihnala 251
22.1 Introduction . . . . . . . . . . . . . . . . . . . 251
22.2 Results . . . . . . . . . . . . . . . . . . . . . . 253
References . . . . . . . . . . . . . . . . . . . . 256
23 Multiple Impact Dynamics of a Falling Rod and Its
Numerical Solution
Hua Shan, Jianzhong Su, Florin Badiu, Jiansen Zhu,
and Leon Xu 257
23.1 Introduction . . . . . . . . . . . . . . . . . . . 257
23.2 Rigid-Body Dynamics Model . . . . . . . . . . . . 258
23.3 Continuous Contact Model . . . . . . . . . . . . . 260
23.4 Discrete Contact Model for a Falling Rod . . . . . . . 261
x Contents
23.5 Numerical Simulation of a Falling Rigid Rod . . . . . 263
23.6 Discussion and Conclusion . . . . . . . . . . . . . 268
References . . . . . . . . . . . . . . . . . . . . 269
24 On the Monotone Solutions of Some ODEs.
I: Structure of the Solutions
Tadie 271
24.1 Introduction . . . . . . . . . . . . . . . . . . . 271
24.2 Some Comparison Results . . . . . . . . . . . . . . 273
24.3 Problem (E1). Blow-up Solutions . . . . . . . . . . 275
References . . . . . . . . . . . . . . . . . . . . 277
25 On the Monotone Solutions of Some ODEs.
II: Dead-core, Compact-support, and Blow-up
Solutions
Tadie 279
25.1 Introduction . . . . . . . . . . . . . . . . . . . 279
25.2 Compact-support Solutions . . . . . . . . . . . . . 280
25.3 Dead-core and Blow-up Solutions . . . . . . . . . . 284
References . . . . . . . . . . . . . . . . . . . . 288
26 A Spectral Method for the Fast Solution of Boundary
Integral Formulations of Elliptic Problems
Johannes Tausch 289
26.1 Introduction . . . . . . . . . . . . . . . . . . . 289
26.2 A Fast Algorithm for Smooth, Periodic Kernels . . . . 290
26.3 Extension to Singular Kernels . . . . . . . . . . . . 293
26.4 Numerical Example and Conclusions . . . . . . . . . 295
References . . . . . . . . . . . . . . . . . . . . 297
27 The GILTT Pollutant Simulation in a Stable
Atmosphere
Sergio Wortmann, Marco T. Vilhena,
Haroldo F. Campos Velho, and Cynthia F. Segatto 299
27.1 Introduction . . . . . . . . . . . . . . . . . . . 299
27.2 GILTT Formulation . . . . . . . . . . . . . . . . 300
27.3 GILTT in Atmospheric Pollutant Dispersion . . . . . . 303
27.4 Final Remarks . . . . . . . . . . . . . . . . . . . 308
References . . . . . . . . . . . . . . . . . . . . 308
Index 309
Preface
ThepurposeoftheinternationalconferencesonIntegralMethodsinScience
and Engineering (IMSE) is to bring together researchers who make use of
analytic or numerical integration methods as a major tool in their work.
The first two such conferences, IMSE1985 and IMSE1990, were held
at the University of Texas at Arlington under the chairmanship of Fred
Payne. At the 1990 meeting, the IMSE consortium was created, charged
with organizing these conferences under the guidance of an International
Steering Committee. Thus, IMSE1993 took place at Tohoku University,
Sendai, Japan, IMSE1996 at the University of Oulu, Finland, IMSE1998
at Michigan Technological University, Houghton, MI, USA, IMSE2000 in
Banff, AB, Canada, IMSE2002 at the University of Saint-E´tienne, France,
andIMSE2004attheUniversityofCentralFlorida,Orlando,FL,USA.The
IMSEconferenceshavenowbecomeestablishedasaforumwherescientists
andengineersworkingwithintegralmethodsdiscussanddisseminatetheir
latest results concerning the development and applications of a powerful
class of mathematical procedures.
An additional, and quite rare, characteristic of all IMSE conferences
is their very friendly and socially enjoyable professional atmosphere. As
expected, IMSE2004, organized at the University of Central Florida in
Orlando, FL, continued that tradition, for which the participants wish to
express their thanks to the Local Organizing Committee:
David Rollins, Chairman;
Zuhair Nashed, Chairman of the Program Committee;
Ziad Musslimani;
Alain Kassab;
Jamal Nayfeh.
The organizers and the participants also wish to acknowledge the sup-
port received from
The Department of Mathematics, UCF,
The College of Engineering, UCF,
andtheUniversityofCentralFloridaitselffortheexcellentfacilitiesplaced
at our disposal.
The next IMSE conference will be held in July 2006 in Niagara Falls,
Canada. Details concerning this event are posted on the conference web
page, http://www.civil.uwaterloo.ca/imse2006.
xii Preface
This volume contains eight invited papers and nineteen contributed pa-
pers accepted after peer review. The papers are arranged in alphabetical
order by (first) author’s name.
The editors would like to record their thanks to the referees for their
willingness to review the papers, and to the staff at Birkha¨user Boston,
who have handled the publication process with their customary patience
and efficiency.
Tulsa, Oklahoma, USA Christian Constanda, IMSE Chairman
The International Steering Committee of IMSE:
C. Constanda (University of Tulsa), Chairman
M. Ahues (University of Saint-E´tienne)
B. Bertram (Michigan Technological University)
I. Chudinovich (University of Guanajuato)
C. Corduneanu (University of Texas at Arlington)
P. Harris (University of Brighton)
A. Largillier (University of Saint-E´tienne)
S. Mikhailov (Glasgow Caledonian University)
A. Mioduchowski (University of Alberta, Edmonton)
D. Mitrea (University of Missouri-Columbia)
Z. Nashed (University of Central Florida)
A. Nastase (Rhein.-Westf. Technische Hochschule, Aachen)
F.R. Payne (University of Texas at Arlington)
M.E. P´erez (University of Cantabria, Santander)
S. Potapenko (University of Waterloo)
K. Ruotsalainen (University of Oulu)
P. Schiavone (University of Alberta, Edmonton)
S. Seikkala (University of Oulu)