Mathematical analysis and numerical methods for science and technology: 5 Evolution problems I
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| Main Authors: | , |
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| Format: | Book |
| Language: | English French |
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Berlin ; Heidelberg ; New York ; Paris ; Barcelona ; Hong Kong ; London
Springer-Verlag
2000
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| Edition: | [Nachdr.] |
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 739 S. |
| ISBN: | 3540661018 |
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| 100 | 1 | |a Dautray, Robert |d 1928- |0 (DE-588)133309347 |4 aut | |
| 240 | 1 | 0 | |a Analyse mathématique et calcul numérique pour les sciences et les techniques |
| 245 | 1 | 0 | |a Mathematical analysis and numerical methods for science and technology |n 5 |p Evolution problems I |c Robert Dautray ; Jacques-Louis Lions |
| 250 | |a [Nachdr.] | ||
| 264 | 1 | |a Berlin ; Heidelberg ; New York ; Paris ; Barcelona ; Hong Kong ; London |b Springer-Verlag |c 2000 | |
| 300 | |a XIV, 739 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Analyse mathématique |2 ram | |
| 650 | 7 | |a Analyse numérique |2 ram | |
| 650 | 7 | |a Equations d'évolution non linéaires - Solutions numériques |2 ram | |
| 650 | 7 | |a analyse mathématique |2 inriac | |
| 650 | 7 | |a calcul numérique |2 inriac | |
| 650 | 7 | |a diagonalisation |2 inriac | |
| 650 | 7 | |a méthode variationnelle |2 inriac | |
| 650 | 7 | |a problème Cauchy |2 inriac | |
| 650 | 7 | |a transformation Laplace |2 inriac | |
| 700 | 1 | |a Lions, Jacques-Louis |d 1928-2001 |0 (DE-588)124055397 |4 aut | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-008789678 | ||
Record in the Search Index
| _version_ | 1804127599921725440 |
|---|---|
| adam_text | Table
of Contents
Chapter
XIV.
Evolution Problems: Cauchy Problems in
Ж.
Introduction
.............................. 1
§1.
The Ordinary Cauchy Problems in Finite Dimensional Spaces
. ... 3
1.
Linear Systems with Constant Coefficients
............ 4
2.
Linear Systems with
Non
Constant Coefficients
.......... 6
§2.
Diffusion Equations
......................... 8
1.
Setting of Problem
........................ 9
2.
The Method of the Fourier Transform
.............. 10
3.
The Elementary Solution of the Heat Equation
.......... 15
4.
Mathematical Properties of the Elementary Solution and the
Semigroup Associated with the Heat Operator
.......... 16
§3.
Wave Equations
.......................... 21
1.
Model Problem: The Wave Equation in
R
............ 21
2.
The Euler-Poisson-Darboux Equation
.............. 44
3.
An Application of
§2
and
3:
Viscoelasticity
............ 48
§4.
The Cauchy Problem for the
Schrõdinger
Equation, Introduction
. . 53
1.
Model Problem
1.
The Case of Zero Potential
.......... 53
2.
Model Problem
2.
The Case of a Harmonic Oscillator
....... 57
§5.
The Cauchy Problem for Evolution Equations Related to Convolution
Products
.............................. 58
1.
Setting of Problem
........................ 58
2.
The Method of the Fourier Transform
.............. 59
3.
The Dirac Equation for a Free Particle
.............. 63
§6.
An Abstract Cauchy Problem. Ovsyannikov s Theorem
....... 66
Review of Chapter
XIV.......................... 72
Chapter XV. Evolution Problems: The Method of Diagonalisation
Introduction
.............................. 73
§1.
The Fourier Method or the Method of Diagonalisation
....... 74
1.
The Case of the Space
R
1 (n
= 1)................. 74
2.
The Case of Space Dimension
и
= 2............... 94
X Table of Contents
3.
The Case of Arbitrary Dimension
π
................ 99
Review
............................... 103
§2.
Variations. The Method of Diagonalisation for an Operator Having
Continuous Spectrum
........................ 104
1.
Review of Self-Adjoint Operators in Hubert Spaces
........ 104
2.
General Formulation of the Problem
............... 104
3.
A Simple Example of the Problem with Continuous Spectrum
... 108
§3.
Examples of Application: The Diffusion Equation
.......... 112
1.
Example of Application
1:
The Monokinetic Diffusion Equation
for Neutrons
........................... 112
2.
Example of Application
2:
The Age Equation in Problems of Slowing
Down of Neutrons
........................ 118
3.
Example of Application
3:
Heat Conduction
........... 122
§4.
The Wave Equation: Mathematical Examples and Examples of
Application
............................. 126
1.
The Case of Dimension
η
= 1................... 126
2.
The Case of Arbitrary Dimension
π
................ 143
3.
Examples of Applications for
π
= 1................ 145
4.
Examples of Applications for
η
= 2.
Vibrating Membranes
..... 156
5.
Application to Elasticity; the Dynamics of Thin Homogeneous
Beams
.............................. 159
§5.
The
Schrödinger
Equation
..................... 169
1. The Cauchy Problem for the
Schrödinger
Equation in a Domain
O = ]0, l[cR
.......................... 170
2.
A Harmonic Oscillator
...................... 177
Review
............................... 183
§6.
Application with an Operator Having a Continuous Spectrum: Example
184
Review of Chapter XV
......................... 186
Appendix. Return to the Problem of Vibrating Strings
.......... 186
Chapter
XVI.
Evolution Problems: The Method of the Laplace
Transform
Introduction
.............................. 202
§1.
Laplace Transform of Distributions
................. 203
1.
Study of the Set
ƒ ƒ
and Definition of the Laplace Transform.
. . . 204
2.
Properties of the Laplace Transform
............... 210
3.
Characterisation of Laplace Transforms of Distributions of L+ (R)
. 212
§2.
Laplace Transform of Vector-valued Distributions
.......... 217
1.
Distributions with Vector-valued Values
............. 218
2.
Fourier and Laplace Transforms of Vector-valued Distributions
. . 222
Table
of
Contents
XI
§3.
Applications to First Order Evolution Problems
........... 225
1.
Vector-valued Distribution Solutions of an Evolution Equation
of First Order in
í
........................ 225
2.
The Method of Transposition
................... 231
3.
Application to First Order Evolution Equations. The Hubert Space
Case. L2 Solutions in Hubert Space
................ 233
4.
The Case where A is Defined by a Sesquilinear Form a(u, v)
. . . . 243
§4.
Evolution Problems of Second Order in
í
.............. 251
1.
Direct Method
.......................... 251
2.
Use of Symbolic Calculus
.................... 257
Review
............................... 261
§5.
Applications
............................ 261
1.
Hydrodynamical Problems
.................... 261
2.
A Problem of the Kinetics of Neutron Diffusion
.......... 265
3.
Problems of Diffusion of an Electromagnetic Wave
........ 267
4.
Problems of Wave Propagation
.................. 273
5.
Viscoelastic Problems
...................... 280
6.
A Problem Related to the
Schrödinger
Equation
......... 290
7.
A Problem Related to Causality, Analyticity and Dispersion Relations
292
8.
Remark
10............................ 295
Review of Chapter
XVI......................... 296
Chapter
XVII.
Evolution Problems: The Method of Semigroups
Introduction
.............................. 297
Part A. Study of Semigroups
....................... 301
§1.
Definitions and Properties of Semigroups Acting in a Banach Space
. 301
1.
Definition of a Semigroup of Class
<#
°
(Resp.
of a Group)
..... 301
2.
Basic Properties of Semigroups of
Classi0
............ 307
§2.
The Infinitesimal Generator of a Semigroup
............. 310
1.
Examples
............................ 310
2.
The Infinitesimal Generator of a Semigroup of Class 4>°
...... 315
§3.
The Hille-Yosida Theorem
..................... 321
1. A Necessary Condition
...................... 321
2.
The Hille-Yosida Theorem
.................... 323
3.
Examples of Application of the Hille-Yosida Theorem
...... 327
§4.
The Case of Groups of Class*0 and Stone s Theorem
........ 353
1.
The Characterisation of the Infinitesimal Generator of a Group
of Class*0
........................... 353
2.
Unitary Groups of Class «if °. Stone s Theorem
.......... 356
XII Table of
Contents
3.
Applications of Stone s Theorem.................
357
4.
Conservative
Operators and Isometric Semigroups in
Hubert
Space
362
Review
............................... 365
§5.
Differentiable Semigroups
...................... 365
§6.
Holomorphic Semigroups
...................... 367
§7.
Compact Semigroups
........................ 388
1.
Definition and Principal Properties
................ 388
2.
Characterisation of Compact Semigroups
............. 389
3.
Examples of Compact Semigroups
................ 394
Part B. Cauchy Problems and Semigroups
................ 397
§1.
Cauchy Problems
.......................... 397
§2.
Asymptotic Behaviour of Solutions as
t
-> 4-
co.
Conservation and
Dissipation in Evolution Equations
................. 406
§3.
Semigroups and Diffusion Problems
................. 412
§4.
Groups and Evolution Equations
.................. 420
1.
Wave Problems
......................... 420
2. Schrödinger
Type Problems
................... 424
3.
Weak Asymptotic Behaviour, for
í
-*
± oo, of Solutions of
Wave Type of
Schrödinger
Type Problems
............ 426
4.
The Cauchy Problem for Maxwell s Equations in an Open Set
ßcR3............................. 433
§5.
Evolution Operators in Quantum Physics. The Liouville-von Neumann
Equation
.............................. 439
1.
Existence and Uniqueness of the Solution of the Cauchy Problem for
the Liouville-von Neumann Equation in the Space of Trace Operators
439
2.
The Evolution Equation of (Bounded)
Observables
in the
Heisenberg
Representation
.......................... 446
3.
Spectrum and Resolvent of the Operator
h
............ 451
§6.
Trotter s Approximation Theorem
.................. 453
1.
Convergence of Semigroups
................... 453
2.
General Representation Theorem
................. 459
Summary of Chapter
XVII....................... 465
Chapter
XVIII.
Evolution Problems: Variation*! Methods
Introduction. Orientation
........................ 467
§1.
Some Elements of Functional Analysis
................ 469
1. Review of Vector-valued Distributions
.............. 469
2.
The Space
Ща,Ь;
V,V )
..................... 472
Table
of
Contents
XIII
3.
The Spaces W(a,b;X, Y)
..................... 479
4.
Extension to Banach Space Framework
.............. 482
5.
An Intermediate Derivatives Theorem
.............. 493
6.
Bidual.
Reflexivity.
Weak Convergence and Weak
*
Convergence
. 499
§2.
Galerkin Approximation of a Hubert Space
............. 503
1.
Definition
............................ 504
2.
Examples
............................ 504
3.
The Outline of a Galerkin Method
................ 507
§3.
Evolution Problems of First Order in
t
............... 509
1.
Formulation of Problem (P)
................... 509
2.
Uniqueness of the Solution of Problem (P)
............ 512
3.
Existence of a Solution of Problem (P)
.............. 513
4.
Continuity with Respect to the Data
............... 520
5.
Appendix: Various Extensions
-
Liftings
............. 521
§4.
Problems of First Order in
t
(Examples)
............... 523
1.
Mathematical Example
1.
Dirichlet Boundary Conditions
..... 524
2.
Mathematical Example
2.
Neumann Boundary Conditions
.... 524
3.
Mathematical Example
3.
Mixed Dirichlet-Neumann Boundary
Conditions
............... ............ 527
4.
Mathematical Example
4.
Bilinear Form Depending on Time
í
. . 528
5.
Evolution,
Positivity
and Maximum of Solutions of Diffusion
Equations in L (Q),
1
=ζ ρ =ζ οο
.................. 533
6.
Mathematical Example
5.
A Problem of Oblique Derivatives
. . . 539
7.
Example of Application. The Neutron Diffusion Equation
..... 542
8.
A Stability Result
........................ 548
§5.
Evolution Problems of Second Order in
í
.............. 552
1. General Formulation of Problem (P,)
.............. 552
2.
Uniqueness in Problem (PJ
................... 558
3.
Existence of a Solution of Problem (Pi)
.............. 561
4.
Continuity with Respect to the Data
............... 566
5.
Formulation of Problem (P2)
................... 570
§6.
Problems of Second Order in t. Examples
.............. 581
1.
Mathematical Example
1..................... 581
2.
Mathematical Example
2..................... 582
3.
Mathematical Example
3..................... 583
4.
Mathematical Example
4..................... 587
5.
Application Examples
...................... 589
§7.
Other Types of Equation
...................... 620
1.
Schrödinger
Type Equations
.............. ..... 620
2.
Evolution Equations with Delay
................. 643
3.
Some Integro-Differential Equations
............... 651
4.
Optimal Control and Problems where the Unknowns are
Operators
............................ 662
XIV Table of
Contents
5.
The Problem of Coupled Parabolic-Hyperbolic Transmission
. . . 670
6.
The Method of Extension with Respect to a Parameter
...... 676
Review of Chapter
XVIII........................ 679
Bibliography
.............................. 680
Table of Notations
........................... 686
Index
.................................. 702
Contents of Volumes
1-4,6........................ 735
|
| any_adam_object | 1 |
| author | Dautray, Robert 1928- Lions, Jacques-Louis 1928-2001 |
| author_GND | (DE-588)133309347 (DE-588)124055397 |
| author_facet | Dautray, Robert 1928- Lions, Jacques-Louis 1928-2001 |
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| ctrlnum | (OCoLC)44551966 (DE-599)BVBBV012912861 |
| dewey-full | 519.4 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.4 |
| dewey-search | 519.4 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | [Nachdr.] |
| format | Book |
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| id | DE-604.BV012912861 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:35:55Z |
| institution | BVB |
| isbn | 3540661018 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008789678 |
| oclc_num | 44551966 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-703 DE-355 DE-BY-UBR DE-20 |
| owner_facet | DE-91G DE-BY-TUM DE-703 DE-355 DE-BY-UBR DE-20 |
| physical | XIV, 739 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Springer-Verlag |
| record_format | marc |
| spelling | Dautray, Robert 1928- (DE-588)133309347 aut Analyse mathématique et calcul numérique pour les sciences et les techniques Mathematical analysis and numerical methods for science and technology 5 Evolution problems I Robert Dautray ; Jacques-Louis Lions [Nachdr.] Berlin ; Heidelberg ; New York ; Paris ; Barcelona ; Hong Kong ; London Springer-Verlag 2000 XIV, 739 S. txt rdacontent n rdamedia nc rdacarrier Analyse mathématique ram Analyse numérique ram Equations d'évolution non linéaires - Solutions numériques ram analyse mathématique inriac calcul numérique inriac diagonalisation inriac méthode variationnelle inriac problème Cauchy inriac transformation Laplace inriac Lions, Jacques-Louis 1928-2001 (DE-588)124055397 aut (DE-604)BV013031479 5 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008789678&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dautray, Robert 1928- Lions, Jacques-Louis 1928-2001 Mathematical analysis and numerical methods for science and technology Analyse mathématique ram Analyse numérique ram Equations d'évolution non linéaires - Solutions numériques ram analyse mathématique inriac calcul numérique inriac diagonalisation inriac méthode variationnelle inriac problème Cauchy inriac transformation Laplace inriac |
| title | Mathematical analysis and numerical methods for science and technology |
| title_alt | Analyse mathématique et calcul numérique pour les sciences et les techniques |
| title_auth | Mathematical analysis and numerical methods for science and technology |
| title_exact_search | Mathematical analysis and numerical methods for science and technology |
| title_full | Mathematical analysis and numerical methods for science and technology 5 Evolution problems I Robert Dautray ; Jacques-Louis Lions |
| title_fullStr | Mathematical analysis and numerical methods for science and technology 5 Evolution problems I Robert Dautray ; Jacques-Louis Lions |
| title_full_unstemmed | Mathematical analysis and numerical methods for science and technology 5 Evolution problems I Robert Dautray ; Jacques-Louis Lions |
| title_short | Mathematical analysis and numerical methods for science and technology |
| title_sort | mathematical analysis and numerical methods for science and technology evolution problems i |
| topic | Analyse mathématique ram Analyse numérique ram Equations d'évolution non linéaires - Solutions numériques ram analyse mathématique inriac calcul numérique inriac diagonalisation inriac méthode variationnelle inriac problème Cauchy inriac transformation Laplace inriac |
| topic_facet | Analyse mathématique Analyse numérique Equations d'évolution non linéaires - Solutions numériques analyse mathématique calcul numérique diagonalisation méthode variationnelle problème Cauchy transformation Laplace |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008789678&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013031479 |
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