Introduction to the foundations of applied mathematics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Springer
2009
|
| Series: | Texts in applied mathematics
56 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 467 S. Ill., graph. Darst. |
| ISBN: | 9780387877495 9780387877655 |
Staff View
MARC
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| 100 | 1 | |a Holmes, Mark H. |e Verfasser |0 (DE-588)1107050499 |4 aut | |
| 245 | 1 | 0 | |a Introduction to the foundations of applied mathematics |c Mark H. Holmes |
| 264 | 1 | |a Dordrecht [u.a.] |b Springer |c 2009 | |
| 300 | |a XIV, 467 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Texts in applied mathematics |v 56 | |
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Record in the Search Index
| _version_ | 1804139318547054592 |
|---|---|
| adam_text | Contents
Preface
.......................................................
vii
1
Dimensional
Analysis
..................................... 1
1.1
Introduction
........................................... 1
1.2
Examples of Dimensional Reduction
...................... 3
1.2.1
Maximum Height of a Projectile
.................... 5
1.2.2
Drag on a Sphere
................................. 6
1.2.3
Toppling Dominoes
............................... 13
1.2.4 Endnotes........................................ 15
1.3
Theoretical Foundation
.................................. 16
1.3.1
Pattern Formation
................................ 19
1.4
Similarity Variables
..................................... 22
1.5
Nondimensionalization and Scaling
....................... 25
1.5.1
Projectile Problem
............................... 26
1.5.2
Weakly Nonlinear Diffusion
........................ 30
1.5.3 Endnotes........................................ 32
Exercises
.................................................. 33
2
Perturbation Methods
.................................... 43
2.1
Regular Expansions
..................................... 43
2.2
How to Find a Regular Expansion
........................ 48
2.2.1
Given a Specific Function
......................... 48
2.2.2
Given an Algebraic or Transcendental Equation
...... 51
2.2.3
Given an Initial Value Problem
.................... 53
2.3
Introduction to Singular Perturbations
,................... 58
2.4
Introduction to Boundary Layers
......................... 60
2.4.1 Endnotes........................................ 66
2.5
Multiple Boundary Layers
............................... 68
2.6
Multiple Scales and Two-Timing
......................... 72
Exercises
.................................................. 79
Contents
Kinetics
.................................................. 87
3.1
Introduction
........................................... 87
3.1.1
Radioactive Decay
................................ 87
3.1.2
Predator-Prey
................................... 88
3.1.3
Epidemic Model
.................................. 88
3.2
Kinetic Equations
...................................... 89
3.2.1
The Law of Mass Action
.......................... 91
3.2.2
Conservation Laws
............................... 92
3.2.3
Steady-States
.................................... 94
3.2.4
Examples
....................................... 94
3.2.5
End Notes
.............._,,....................... 96
3.3
General Mathematical Formulation
....................... 97
3.4
Michaelis-Menten Kinetics
............................... 100
3.4.1
Numerical Solution
...............................102
3.4.2
Quasi-Steady-State Approximation
.................103
3.4.3
Perturbation Approach
............................105
3.5
Assorted Applications
...................................
Ill
3.5.1
Elementary and Nonelementary Reactions
...........
Ill
3.5.2
Reverse Mass Action
..............................113
3.6
Steady-States and Stability
..............................114
3.6.1
Reaction Analysis
................................ 115
3.6.2
Geometric Analysis
............................... 115
3.6.3
Perturbation Analysis
............................. 118
3.7
Oscillators
............................................. 126
3.7.1
Stability
........................................ 128
Exercises
..................................................132
Diffusion
..................................................141
4.1
Introduction
...........................................141
4.2
Random Walks and Brownian Motion
.....................142
4.2.1
Calculating w{m, N)
..............................145
4.2.2
Large
N
Approximation
...........................148
4.3
Continuous Limit
.......................................149
4.3.1
What Does
D
Signify?
............................151
4.4
Solving the Diffusion Equation
...........................153
4.4.1
Point Source
.....................................154
4.4.2
Fourier Transform
................................157
4.5
Continuum Formulation of Diffusion
......................169
4.5.1
Balance Law
.....................................169
4.5.2
Fick s Law of Diffusion
............................171
4.5.3
Reaction-Diffusion Equations
......................177
4.6
Random Walks and Diffusion in Higher Dimensions
.........179
4.6.1
Diffusion Equation
...............................182
4.7
Langevin
Equation
.....................................185
4.7.1
Properties of the Forcing
..........................188
Contents
4.7.2 Endnotes........................................194
Exercises ..................................................
194
Traffic Flow
..............................................205
5.1
Introduction
...........................................205
5.2
Continuum Variables
....................................206
5.2.1
Density
.........................................207
5.2.2
Flux
............................................208
5.3
Balance Law
...........................................209
5.3.1
Velocity Formulation
..............................210
5.4
Constitutive Laws
......................................211
5.4.1
Constant Velocity
................................212
5.4.2
Linear Velocity
...................................213
5.4.3
General Velocity Formulation
......................214
5.4.4
Flux and Velocity
................................216
5.4.5
Reality Check
....................................217
5.5
Constant Velocity
......................................218
5.5.1
Characteristics
...................................221
5.6
Nonconstant
Velocity
...................................225
5.6.1
Small Disturbance Approximation
..................226
5.6.2
Method of Characteristics
.........................229
5.6.3
Rankine-Hugoniot Condition
.......................233
5.6.4
Expansion Fan
...................................236
5.6.5
Shock Waves
.....................................241
5.6.6
Return of Phantom Traffic Jams
...................245
5.6.7
Summary
........................................247
5.7
Cellular Automata Modeling
.............................248
Exercises
..................................................254
Continuum Mechanics: One Spatial Dimension
...........265
6.1
Introduction
...........................................265
6.2
Coordinate Systems
.....................................265
6.2.1
Material Coordinates
.............................266
6.2.2
Spatial Coordinates
...............................267
6.2.3
Material Derivative
...............................270
6.2.4
End Notes
.......................................272
6.3
Mathematical Tools
.....................................273
6.4
Continuity Equation
....................................275
6.4.1
Material Coordinates
.............................276
6.5
Momentum Equation
...................................277
6.5.1
Material Coordinates
.............................279
6.6
Summary of the Equations of Motion
.....................279
6.7
Steady-State Solution
...................................280
6.8
Constitutive Law for an Elastic Material
..................282
6.8.1
Derivation of Strain
..............................284
Contents
6.8.2 Material
Linearity................................
286
6.8.3
End
Notes.......................................289
6.9
Morphological
Basis
for
Deformation......................290
6.9.1 Metals..........................................290
6.9.2 Elastomers......................................293
6.10
Restrictions on Constitutive Laws
........................294
6.10.1
Frame-Indifference
................................295
6.10.2
Entropy Inequality
...............................298
6.10.3
Hyperelasticity
...................................302
Exercises
..................................................304
Elastic and Viscoelastic Materials
.........................311
7.1
Linear Elasticity
.......................................311
7.1.1
Method of Characteristics
.........................313
7.1.2
Laplace Transform
................................316
7.1.3
Geometric Linearity
..............................327
7.2
Viscoelasticity
.........................................328
7.2.1
Mass, Spring, Dashpot Systems
....................329
7.2.2
Equations of Motion
..............................331
7.2.3
Integral Formulation
..............................335
7.2.4
Generalized Relaxation Functions
..................337
7.2.5
Solving Viscoelastic Problems
......................338
Exercises
..................................................342
Continuum Mechanics: Three Spatial Dimensions
.........351
8.1
Introduction
...........................................351
8.2
Material and Spatial Coordinates
.........................352
8.2.1
Deformation Gradient
.............................353
8.3
Material Derivative
.....................................356
8.4
Mathematical Tools
.....................................358
8.4.1
General Balance Law
.............................361
8.5
Continuity Equation
....................................362
8.5.1
Incompressibility
.................................362
8.6
Linear Momentum Equation
.............................363
8.6.1
Stress Tensor
.................................... 364
8.6.2
Differential Form of Equation
...................... 367
8.7
Angular Momentum
.................................... 367
8.8
Summary of the Equations of Motion
..................... 368
8.9
Constitutive Laws
...................................... 368
8.9.1
Representation Theorem and Invariants
.............372
8.10
Newtonian Fluid
....................................... 374
8.10.1
Pressure
.........................................374
8.10.2
Viscous Stress
...................................375
8.11
Equations of Motion for a Viscous Fluid
...................378
8.11.1
Incompressibility
.................................379
Contents xiii
8.11.2
Boundary and Initial Conditions
...................380
8.12
Material Equations of Motion
............................383
8.12.1
Frame-Indifference
................................385
8.12.2
Elastic Solid
.....................................387
8.12.3
Linear Elasticity
.................................389
8.13
Energy Equation
.......................................390
8.13.1
Incompressible Viscous Fluid
......................391
8.13.2
Elasticity
........................................391
Exercises
..................................................394
9
Fluids
....................................................403
9.1
Newtonian Fluids
.......................................403
9.2
Steady Flow
...........................................404
9.2.1
Plane
Couette
Flow
...............................405
9.2.2
Poiseuille Flow
...................................408
9.3
Vorticity
..............................................411
9.3.1
Vortex Motion
...................................412
9.4
Irrotational Flow
.......................................414
9.4.1
Potential Flow
...................................417
9.5
Ideal Fluid
............................................419
9.5.1
Circulation and Vorticity
..........................420
9.5.2
Potential Flow
...................................423
9.5.3
End Notes
.......................................426
9.6
Boundary Layers
.......................................427
9.6.1
Impulsive Plate
..................................427
9.6.2
Blasius Boundary Layer
...........................429
Exercises
..................................................434
A Taylor s Theorem
.........................................441
A.I Single Variable
.........................................441
A.2 Two Variables
.........................................441
A.3
Multivariable
Versions
..................................442
В
Fourier Analysis
..........................................445
B.I Fourier Series
..........................................445
B.2 Fourier Transform
......................................447
С
Stochastic Differential Equations
.........................449
D
Identities
.................................................451
D.I Trace
.................................................451
D.2 Determinant
...........................................451
D.3 Vector Calculus
........................................452
xiv Contents
E
Equations
for a Newtonian Fluid
.........................453
E.I Cartesian Coordinates
..................................453
E.2 Cylindrical Coordinates
.................................453
References
....................................................455
Index
.........................................................463
|
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| ctrlnum | (OCoLC)699336175 (DE-599)DNB99392929X |
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| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519 |
| dewey-search | 519 |
| dewey-sort | 3519 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| format | Book |
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| isbn | 9780387877495 9780387877655 |
| language | English |
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| spelling | Holmes, Mark H. Verfasser (DE-588)1107050499 aut Introduction to the foundations of applied mathematics Mark H. Holmes Dordrecht [u.a.] Springer 2009 XIV, 467 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 56 Engineering mathematics Angewandte Mathematik (DE-588)4142443-8 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Angewandte Mathematik (DE-588)4142443-8 s DE-604 Mathematisches Modell (DE-588)4114528-8 s Mathematische Physik (DE-588)4037952-8 s Texts in applied mathematics 56 (DE-604)BV002476038 56 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017693572&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Holmes, Mark H. Introduction to the foundations of applied mathematics Texts in applied mathematics Engineering mathematics Angewandte Mathematik (DE-588)4142443-8 gnd Mathematisches Modell (DE-588)4114528-8 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4142443-8 (DE-588)4114528-8 (DE-588)4037952-8 (DE-588)4123623-3 |
| title | Introduction to the foundations of applied mathematics |
| title_auth | Introduction to the foundations of applied mathematics |
| title_exact_search | Introduction to the foundations of applied mathematics |
| title_full | Introduction to the foundations of applied mathematics Mark H. Holmes |
| title_fullStr | Introduction to the foundations of applied mathematics Mark H. Holmes |
| title_full_unstemmed | Introduction to the foundations of applied mathematics Mark H. Holmes |
| title_short | Introduction to the foundations of applied mathematics |
| title_sort | introduction to the foundations of applied mathematics |
| topic | Engineering mathematics Angewandte Mathematik (DE-588)4142443-8 gnd Mathematisches Modell (DE-588)4114528-8 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Engineering mathematics Angewandte Mathematik Mathematisches Modell Mathematische Physik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017693572&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002476038 |
| work_keys_str_mv | AT holmesmarkh introductiontothefoundationsofappliedmathematics |