A Primer on PDEs: models, methods, simulations
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| Main Authors: | , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Milan ; Heidelberg ; New York ; Dordrecht ; London
Springer
[2013]
|
| Series: | Unitext
volume 65 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 482 Seiten Illustrationen, Diagramme |
| ISBN: | 8847028612 9788847028616 |
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MARC
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| 020 | |a 9788847028616 |c pbk. |9 978-88-470-2861-6 | ||
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| 240 | 1 | 0 | |a Invito alle equazioni a derivate parziali |
| 245 | 1 | 0 | |a A Primer on PDEs |b models, methods, simulations |c Sandro Salsa, Federico M.G. Vegni, Anna Zaretti, Paolo Zunino |
| 264 | 1 | |a Milan ; Heidelberg ; New York ; Dordrecht ; London |b Springer |c [2013] | |
| 264 | 4 | |c © 2013 | |
| 300 | |a XIV, 482 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Unitext |v volume 65 | |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Vegni, Federico M. G. |e Verfasser |4 aut | |
| 700 | 1 | |a Zaretti, Anna |e Verfasser |4 aut | |
| 700 | 1 | |a Zunino, Paolo |e Verfasser |4 aut | |
| 830 | 0 | |a Unitext |v volume 65 |w (DE-604)BV047304938 |9 65 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025744949&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-025744949 | ||
Record in the Search Index
| _version_ | 1804150094990147584 |
|---|---|
| adam_text | Contents
Introduction
............................................. 1
1.1
Mathematical Modelling
.............................. 1
1.2
Partial Differential Equations
.......................... 2
1.3
Well Posed Problems
................................. 5
1.4
Basic Notations and Facts
............................. 7
1.5
Integration by Parts Formulas
......................... 10
1.6
Abstract Methods and Variational Formulation
.......... 11
1.7
Numerical approximation methods
..................... 12
Part I Differential Models
Scalar Conservation Laws
................................ 17
2.1
Introduction
......................................... 17
2.2
Linear transport equation
............................. 20
2.2.1
Distributed source
............................. 21
2.2.2
Extinction and localized source
.................. 22
2.2.3
Inflow and outflow characteristics. A stability
estimate
...................................... 24
2.3
Traffic Dynamics
..................................... 26
2.3.1
A macroscopic model
.......................... 26
2.3.2
The method of characteristics
................... 28
2.3.3
The green light problem. Rarefaction waves
....... 30
2.3.4
Traffic jam ahead. Shock waves. Rankine—Hugoniot
condition
..................................... 34
2.4
The method of characteristics revisited
.................. 37
2.5
Generalized solutions. Uniqueness and entropy condition
. . 40
2.6
The Vanishing Viscosity Method
....................... 44
VIII Contents
2.6.1
The viscous Burgers equation
.................. 48
2.7
Numerical methods
................................... 51
2.7.1
Finite difference approximation of scalar
conservation laws
.............................. 51
2.8
Exercises
............................................ 53
2.8.1
Numerical approximation of a constant coefficient
scalar conservation law
......................... 55
2.8.2
Numerical approximation of Burgers equation
..... 57
2.8.3
Numerical approximation of traffic dynamics
...... 57
3
Diffusion
................................................. 59
3.1
The Diffusion Equation
............................... 59
3.1.1
Introduction
.................................. 59
3.1.2
The conduction of heat
......................... 60
3.1.3
Well posed problems (n
= 1).................... 62
3.1.4
A solution by separation of variables
............. 65
3.1.5
Froblems in dimension
η
> 1 ................... 73
3.2
Uniqueness
.......................................... 76
3.2.1
Integral method
............................... 76
3.2.2
Maximum principles
........................... 78
3.3
The Fundamental Solution
............................ 81
3.3.1
Invariant transformations
....................... 81
3.3.2
Fundamental solution (n
=
l)
................... 83
3.3.3
The Dirac distribution
......................... 85
3.3.4
Pollution in a channel. Diffusion, drift and reaction
88
3.3.5
Fundamental solution (n
> 1)................... 89
3.4
The Global Cauchy Problem (n
= 1).................... 90
3.4.1
The homogeneous case
......................... 90
3.4.2
Existence of a solution
......................... 91
3.4.3
The
non
homogeneous case. Duhamel s method
... 92
3.5
An example of Nonlinear diffusion. The porous medium
equation
............................................ 95
3.6
Numerical methods
................................... 99
3.6.1
Finite difference approximation of the heat equation
99
3.6.2
Stability analysis for
Euler
methods
............. 101
3.6.3
The solution of the heat equation as a probability
density function
............................... 102
3.7
Exercises
............................................ 104
3.7.1
Application of
Euler
methods to the discretization
of the Cauchy-Dirichlet problem
................. 106
3.7.2
Application to the dynamics of chemicals
......... 107
Contents
IX
4
The Laplace Equation
................................... 109
4.1
Introduction
......................................... 109
4.2
Well Posed Problems. Uniqueness
...................... 110
4.3
Harmonic Functions
.................................. 112
4.3.1
Mean value properties
......................... 112
4.3.2
Maximum principles
........................... 114
4.3.3
The Dirichlet problem in a circle. Poisson s formula
116
4.4
Fundamental Solution and Newtonian Potential
.......... 120
4.4.1
The fundamental solution
...................... 120
4.4.2
The Newtonian potential
....................... 122
4.5
The Green Function
.................................. 123
4.5.1
An integral identity
............................ 123
4.5.2
The Green function for the Dirichlet problem
..... 125
4.5.3
Green s representation formula
.................. 127
4.5.4
The Neumann function
.......................... 129
4.6
Numerical methods
................................... 130
4.6.1
The
5
point finite difference scheme for the
Poisson
problem
...................................... 130
4.7
Exercises
............................................ 134
4.7.1
Approximation of an elastic membrane using the
5
point scheme
................................ 136
4.7.2
Numerical simulations for testing maximum
principles
..................................... 137
5
Reaction-diffusion models
................................ 139
5.1
Reaction Models
..................................... 139
5.1.1
The mass action law
........................... 139
5.1.2
Inhibition, activation
.......................... 142
5.2
Diffusion and linear reaction
........................... 145
5.2.1
Pure diffusion. Asymptotic behavior
............. 145
5.2.2
Asymptotic behavior in general domains
......... 148
5.2.3
Linear reaction. Critical dimension
.............. 151
5.2.4
Linear reaction and diffusion in two dimensions
. . . 153
5.2.5
An Example in dimension n = 3
................. 155
5.3
Diffusion and nonlinear reaction
........................ 160
5.3.1
Monotone methods
............................ 160
5.3.2
The Fisher s equation
.......................... 163
5.3.3
Steady states, linearization and stability
.......... 165
5.3.4
Application to Fisher s equation (Dirichlet
conditions)
................................... 168
X
Contents
5.3.5 Application
to Fisher s equation (Neumann
conditions)
................................... 173
5.4
Turing instability
..................................... 175
5.5
Numerical methods
................................... 181
5.5.1
Numerical approximation of a nonlinear
reaction-diffusion problem
...................... 181
5.6
Exercises
............................................ 182
5.6.1
Numerical simulation of Fisher s equations
........ 183
5.6.2
Numerical approximation of travelling wave
solutions
..................................... 185
5.6.3
Numerical approximation of Turing instability and
pattern formation
............................. 186
6
Waves and vibrations
.................................... 189
6.1
General Concepts
.................................... 189
6.1.1
Types of waves
................................ 189
6.1.2
Group velocity and dispersion relation
............ 191
6.2
Transversal Waves in a String
.......................... 193
6.2.1
The model
.................................... 193
6.2.2
Energy
....................................... 195
6.3
The One-dimensional Wave Equation
................... 196
6.3.1
Initial and boundary conditions
................. 196
6.4
The d Alembert Formula
.............................. 198
6.4.1
The homogeneous equation
..................... 198
6.4.2
The nonhomogeneous equation. Duhamel s method
202
6.4.3
Dissipation and dispersion
...................... 203
6.5
Second Order Linear Equations
........................ 205
6.5.1
Classification
................................. 205
6.5.2
Characteristics and canonical form
.............. 208
6.6
The
Multi-
dimensional Wave Equation (n
> 1)........... 213
6.6.1
Special solutions
.............................. 213
6.6.2
Well posed problems. Uniqueness
................ 215
6.6.3
Small vibrations of an elastic membrane
.......... 217
6.6.4
Small amplitude sound waves
................... 221
6.7
The Cauchy Problem
................................. 225
6.7.1
Fundamental solution (n
= 3)
and strong Huygens
principle
..................................... 225
6.7.2
The
Kirchhoff
formula
......................... 228
6.7.3
Cauchy problem in dimension
2................. 229
6.7.4
Non
homogeneous equation. Retarded potentials
. . 231
Contents XI
6.8
Numerical methods
................................... 232
6.8.1
Numerical approximation of the one-dimensional
wave equation
................................. 232
6.9
Exercises
............................................ 234
6.9.1
Numerical simulation of a vibrating string
........ 236
6.9.2
Numerical simulation of a vibrating membrane
.... 238
Part II Functional Analysis Techniques for Differential Problems
Elements of Functional Analysis
......................... 243
7.1
Lebesgue Measure and Integral
......................... 244
7.1.1
A counting problem
........................... 244
7.1.2
Measures and measurable functions
.............. 245
7.1.3
The Lebesgue integral
.......................... 247
7.1.4
Some fundamental theorems
.................... 248
7.2
Hubert Spaces
....................................... 249
7.2.1
Normed spaces
................................ 250
7.2.2
Inner product and Hubert Spaces
................ 254
7.2.3
Projections
................................... 258
7.2.4
Orthonormal
bases
............................ 263
7.3
Linear Operators and Duality
.......................... 267
7.3.1
Linear operators
.............................. 267
7.3.2
Punctionals and dual space
..................... 270
7.4
Abstract Variational Problems
......................... 273
7.4.1
Bilinear forms and the Lax-Milgram Theorem
..... 273
7.4.2
Minimization of quadratic functionals
............ 277
7.4.3
Approximation and Galerkin method
............ 278
7.5
Distributions and Functions
........................... 281
7.5.1
Preliminary concepts
.......................... 281
7.5.2
Test functions
................................. 283
7.5.3
Distributions
................................. 284
7.5.4
Calculus
..................................... 286
7.6
Sobolev Spaces
....................................... 293
7.6.1
The space
Η1 (Ω)
............................. 293
7.6.2
The spaces
Щ
(Ω)
and
Η}, (Ω)
................. 296
7.6.3
The dual of
Η^(Ω)
............................ 298
7.6.4
The spaces Hm
(Ω)
and
Яот
(Ω),
m
> 1.......... 300
7.6.5
Calculus rules
................................. 301
7.7
Exercises
............................................ 302
XII Contents
8
Variational
formulation
of elliptic problems
.............. 307
8.1
Elliptic Equations
.................................... 307
8.2
The
Poisson
Problem
................................. 309
8.3
Diffusion, Drift and Reaction (n
-- 1) ................... 311
8.3.1
The problem
.................................. 311
8.3.2
Dirichlet conditions
............................ 311
8.3.3
Neumann conditions
........................... 315
8.3.4
Robin and mixed conditions
.................... 318
8.4
Variational Formulation of Poisson s Problem
............ 320
8.4.1
The Dirichlet problem
......................... 320
8.4.2
Neumann, Robin and mixed problems
............ 322
8.5
Eigenvalues of the Laplace operator
.................... 325
8.5.1
Separation of variables revisited
................. 325
8.5.2
An asymptotic stability result
................... 328
8.6
Equations in Divergence Form
......................... 330
8.6.1
Basic assumptions
............................. 330
8.6.2
Dirichlet problem
.............................. 330
8.6.3
Neumann problem
............................. 333
8.6.4
Robin and mixed problems
..................... 335
8.7
A Control Problem
................................... 336
8.7.1
Structure of the problem
....................... 336
8.7.2
Existence and uniqueness of an optimal pair
...... 337
8.7.3 Lagrange
multipliers and optimality conditions
.... 339
8.8
Numerical methods
................................... 342
8.8.1
The finite element method in one space dimension
. 342
8.8.2
Error analysis of the
imite
element method
....... 344
8.8.3
The finite element method for the approximation of
advection, diffusion, reaction problems
........... 346
8.8.4
The finite element method in two space dimensions
348
8.9
Exercises
............................................ 350
8.9.1
Approximation of boundary conditions in the finite
element method
............................... 353
8.9.2
Approximation of Robin boundary conditions
..... 354
8.9.3
Approximation of a system of equations
.......... 355
8.9.4
Effect of problem regularity on the convergence of
the finite element method
...................... 356
9
Weak formulation of evolution problems
................. 359
9.1
Parabolic Equations
.................................. 359
9.2
The Heat Equation
................................... 360
9.2.1
The Cauchy-Dirichlet problem
.................. 360
Contents
XIII
9.2.2
Galer kin
approximations
....................... 363
9.2.3
Energy estimates
.............................. 366
9.2.4
Existence and stability
......................... 368
9.2.5
Neumann and mixed boundary conditions
........ 370
9.3
General Equations
.................................... 374
9.3.1
Weak formulation of initial-boundary value problems
374
9.4
Numerical methods
................................... 381
9.4.1
A Faedo-Galerkin/finite element method for the
heat equation
................................. 381
9.5
Exercises
............................................ 383
9.5.1
Verification of
Euler
methods stability properties
. . 385
9.5.2
Numerical simulation of mass transfer
............ 385
Part III Solutions
10
Solutions of selected exercises
........................... 389
10.1
Section
2.8 .......................................... 389
10.2
Section
3.7 .......................................... 398
10.3
Section
4.7 .......................................... 410
10.4
Section
5.6 .......................................... 418
10.5
Section
6.9 .......................................... 420
10.6
Section
7.7 .......................................... 426
10.7
Section
8.9 .......................................... 433
10.8
Section
9.5 .......................................... 441
Part IV Appendices
A Fourier Series
............................................ 449
A.I Fourier coefficients
................................... 449
A.
2
Expansion in Fourier series
............................ 452
В
Notes on ordinary differential equations
................. 455
B.I
Bidimensional
autonomous systems
..................... 455
B.2 Linear systems
....................................... 458
B.3 Non-linear systems
................................... 463
С
Finite difference approximation of time dependent
problems
................................................ 467
C.I Discrete scheme and the equivalence principle
............ 468
XIV Contents
D
Identities and Formulas
.................................. 471
D.I Gradient, Divergence, Curl, Laplacian
.................. 471
D.2 Formulas
............................................ 473
References
................................................... 475
Index
........................................................ 479
|
| any_adam_object | 1 |
| author | Salsa, Sandro 1950- Vegni, Federico M. G. Zaretti, Anna Zunino, Paolo |
| author_GND | (DE-588)1111815445 |
| author_facet | Salsa, Sandro 1950- Vegni, Federico M. G. Zaretti, Anna Zunino, Paolo |
| author_role | aut aut aut aut |
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| author_variant | s s ss f m g v fmg fmgv a z az p z pz |
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| bvnumber | BV040766472 |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV040766472 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:33:28Z |
| institution | BVB |
| isbn | 8847028612 9788847028616 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025744949 |
| oclc_num | 835300411 |
| open_access_boolean | |
| owner | DE-384 DE-355 DE-BY-UBR DE-29T DE-11 DE-188 |
| owner_facet | DE-384 DE-355 DE-BY-UBR DE-29T DE-11 DE-188 |
| physical | XIV, 482 Seiten Illustrationen, Diagramme |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series | Unitext |
| series2 | Unitext |
| spelling | Salsa, Sandro 1950- Verfasser (DE-588)1111815445 aut Invito alle equazioni a derivate parziali A Primer on PDEs models, methods, simulations Sandro Salsa, Federico M.G. Vegni, Anna Zaretti, Paolo Zunino Milan ; Heidelberg ; New York ; Dordrecht ; London Springer [2013] © 2013 XIV, 482 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Unitext volume 65 Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Vegni, Federico M. G. Verfasser aut Zaretti, Anna Verfasser aut Zunino, Paolo Verfasser aut Unitext volume 65 (DE-604)BV047304938 65 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025744949&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Salsa, Sandro 1950- Vegni, Federico M. G. Zaretti, Anna Zunino, Paolo A Primer on PDEs models, methods, simulations Unitext Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4123623-3 |
| title | A Primer on PDEs models, methods, simulations |
| title_alt | Invito alle equazioni a derivate parziali |
| title_auth | A Primer on PDEs models, methods, simulations |
| title_exact_search | A Primer on PDEs models, methods, simulations |
| title_full | A Primer on PDEs models, methods, simulations Sandro Salsa, Federico M.G. Vegni, Anna Zaretti, Paolo Zunino |
| title_fullStr | A Primer on PDEs models, methods, simulations Sandro Salsa, Federico M.G. Vegni, Anna Zaretti, Paolo Zunino |
| title_full_unstemmed | A Primer on PDEs models, methods, simulations Sandro Salsa, Federico M.G. Vegni, Anna Zaretti, Paolo Zunino |
| title_short | A Primer on PDEs |
| title_sort | a primer on pdes models methods simulations |
| title_sub | models, methods, simulations |
| topic | Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Partielle Differentialgleichung Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025744949&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV047304938 |
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