Nonlinear partial differential equations for scientists and engineers:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2005
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Publisher description Inhaltsverzeichnis |
| Physical Description: | XX, 737 S. graph. Darst. 25 cm |
| ISBN: | 0817643230 9780817643232 |
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MARC
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| 100 | 1 | |a Debnath, Lokenath |d 1935- |e Verfasser |0 (DE-588)115600663 |4 aut | |
| 245 | 1 | 0 | |a Nonlinear partial differential equations for scientists and engineers |c Lokenath Debnath |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2005 | |
| 300 | |a XX, 737 S. |b graph. Darst. |c 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Differentiaalvergelijkingen |2 gtt | |
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| 650 | 4 | |a Équations différentielles non linéaires | |
| 650 | 4 | |a Differential equations, Nonlinear | |
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Record in the Search Index
| _version_ | 1804136264051458048 |
|---|---|
| adam_text | Contents
Preface
to the Second Edition
.............................. xiii
Preface to the First Edition
................................. xv
1
Linear Partial Differential Equations
......................... 1
1.1
Introduction
............................................. 1
1.2
Basic Concepts and Definitions
............................ 2
1.3
The Linear Superposition Principle
.......................... 5
1.4
Some Important Classical Linear Model Equations
............. 8
1.5
Second-Order Linear Equations and Method of Characteristics
.. 11
1.6
The Method of Separation of Variables
....................... 22
1.7
Fourier Transforms and Initial Boundary-Value Problems
....... 35
1.8
Multiple Fourier Transforms and Partial Differential Equations
.. 48
1.9
Laplace Transforms and Initial Boundary-Value Problems
....... 53
1.10
Hankel Transforms and Initial Boundary-Value Problems
....... 63
1.11
Green s Functions and Boundary-Value Problems
.............. 71
1.12
Sturm-Liouville Systems and Some General Results
........... 84
1.13
Energy Integrals and Higher Dimensional Equations
........... 103
1.14
Fractional Partial Differential Equations
...................... 117
1.15
Exercises
............................................... 128
2
Nonlinear Model Equations andVanational Principles
.......... 149
2.1
Introduction
............................................. 149
2.2
Basic Concepts and Definitions
............................. 150
2.3
Some Nonlinear Model Equations
.......................... 150
2.4
Variational Principles and the Euler-Lagrange Equations
....... 156
2.5
The Variational Principle for Nonlinear Klein-Gordon Equations
. 169
2.6
The Variational Principle for Nonlinear Water Waves
........... 170
2.7
The
Euler
Equation of Motion and Water Wave Problems
....... 172
2.8
Exercises
............................................... 185
Contents
First-Order, Quasi-Linear Equations and Method
of Characteristics
.........................................
193
3.1
Introduction
............................................. 93
3.2
The Classification of First-Order Equations
................... 194
3.3
The Construction of a First-Order Equation
...................195
3.4
The Geometrical Interpretation of a First-Order Equation
.......198
3.5
The Method of Characteristics and General Solutions
..........201
3.6
Exercises
...............................................
215
First-Order Nonlinear Equations and Their Applications
........ 221
4.1
Introduction
.............................................221
4.2
The Generalized Method of Characteristics
...................222
4.3
Complete Integrals of Certain Special Nonlinear Equations
......225
4.4
The Hamilton-Jacobi Equation and Its Applications
............232
4.5
Applications to Nonlinear Optics
............................240
4.6
Exercises
...............................................248
Conservation Laws and Shock Waves
......................... 251
5.1
Introduction
.............................................251
5.2
Conservation Laws
.......................................252
5.3
Discontinuous Solutions and Shock Waves
...................264
5.4
Weak or Generalized Solutions
.............................266
5.5
Exercises
...............................................272
Kinematic Waves and Real-World Nonlinear Problems
.......... 277
6.1
Introduction
.............................................277
6.2
Kinematic Waves
.........................................278
6.3
Traffic Flow Problems
.....................................280
6.4
Flood Waves in Long Rivers
...............................292
6.5 Chromatographie
Models and Sediment Transport in Rivers
.....294
6.6
Glacier Flow
............................................300
6.7
Roll Waves and Their Stability Analysis
......................302
6.8
Simple Waves and Riemann s Invariants
.....................308
6.9
The Nonlinear Hyperbolic System and Riemann s Invariants
___325
6.10
Generalized Simple Waves and Generalized Riemann s Invariants
335
6.11
Exercises
...............................................339
Nonlinear Dispersive Waves and Whitham s Equations
.......... 347
7.1
Introduction
.............................................347
7.2
Linear Dispersive Waves
..................................348
7.3
Initial-Value Problems and Asymptotic Solutions
..............351
7.4
Nonlinear Dispersive Waves and Whitham s Equations
.........354
7.5
Whitham s Theory of Nonlinear Dispersive Waves
.............356
7.6
Whitham s Averaged Variational Principle
....................359
7.7
Whitham s Instability Analysis of Water Waves
................361
7.8
Whitham s Equation: Peaking and Breaking of Waves
..........363
Contents ix
7.9
Exercises
...............................................369
8
Nonlinear Diffusion-Reaction Phenomena
..................... 373
8.1
Introduction
.............................................373
8.2
Burgers Equation and the Plane Wave Solution
...............374
8.3
Traveling Wave Solutions and Shock-Wave Structure
...........376
8.4
The Exact Solution of the Burgers Equation
..................378
8.5
The Asymptotic Behavior of the Burgers Solution
.............383
8.6
TheW-Wave Solution
.....................................385
8.7
Burgers Initial- and Boundary-Value Problem
................386
8.8
Fisher s Equation and Diffusion-Reaction Process
.............389
8.9
Traveling Wave Solutions and Stability Analysis
...............391
8.10
Perturbation Solutions of the Fisher Equation
.................395
8.11
Method of Similarity Solutions of Diffusion Equations
.........397
8.12
Nonlinear Reaction-Diffusion Equations
.....................405
8.13
Brief Summary of Recent Work
.............................409
8.14
Exercises
...............................................411
9 Solitons
and the Inverse Scattering Transform
................. 417
9.1
Introduction
.............................................417
9.2
The History of the
Solitons
and Soliton Interactions
............418
9.3
The Boussinesq and Korteweg-de
Vries
Equations
.............423
9.4
Solutions of the KdV Equation:
Solitons
and Cnoidal Waves
___446
9.5
The Lie Group Method and Similarity Analysis of the
KdV Equation
...........................................453
9.6
Conservation Laws and Nonlinear Transformations
............457
9.7
The Inverse Scattering Transform
(1ST)
Method
...............462
9.8
Bäcklund
Transformations and the Nonlinear Superposition
Principle
................................................483
9.9
The Lax Formulation and the Zakharov and Shabat Scheme
.....488
9.10
The AKNS Method
.......................................496
9.11
Asymptotic Behavior of the Solution of the KdV-Burgers
Equation
................................................498
9.12
Strongly Dispersive Nonlinear Equations and
Compactons
......499
9.13
Exercises
...............................................510
10
The Nonlinear
Schrödinger
Equation and Solitary Waves
........515
10.1
Introduction
.............................................515
10.2
The One-Dimensional Linear
Schrödinger
Equation
............516
10.3
The Nonlinear
Schrödinger
Equation and Solitary Waves
.......517
10.4
Properties of the Solutions of the Nonlinear
Schrödinger
Equation
522
10.5
Conservation Laws for the NLS Equation
....................528
10.6
The Inverse Scattering Method for the Nonlinear
Schrödinger
Equation
................................................531
χ
Contents
10.7
Examples of Physical Applications in Fluid Dynamics and
Plasma Physics
.......................................... 3
10.8
Applications to Nonlinear Optics
............................545
10.9
Exercises
...............................................
554
11
Nonlinear Klein-Gordon and Sine-Gordon Equations
........... 557
11.1
Introduction
.............................................557
11.2
The One-Dimensional Linear Klein-Gordon Equation
..........558
11.3
The Two-Dimensional Linear Klein-Gordon Equation
..........560
11.4
The Three-Dimensional Linear Klein-Gordon Equation
........562
11.5
The Nonlinear Klein-Gordon Equation and Averaging
Techniques
..............................................563
11.6
The Klein-Gordon Equation and the Whitham Averaged
Variational Principle
......................................576
11.7
The Sine-Gordon Equation: Soliton and Antisoliton Solutions
... 572
11.8
The Solution of the Sine-Gordon Equation by Separation of
Variables
................................................576
11.9
Bäcklund
Transformations for the Sine-Gordon Equation
.......584
1
l.lOThe Solution of the Sine-Gordon Equation by the Inverse
Scattering Method
........................................587
11.1
IThe Similarity Method for the Sine-Gordon Equation
..........591
11.
^Nonlinear Optics and the Sine-Gordon Equation
...............591
ll.BExercises
...............................................595
12
Asymptotic Methods and Nonlinear Evolution Equations
........ 599
12.1
Introduction
.............................................599
12.2
The Reductive Perturbation Method and Quasi-Linear
Hyperbolic Systems
......................................601
12.3
Quasi-Linear Dissipative Systems
...........................605
12.4
Weakly Nonlinear Dispersive Systems and the Korteweg-de
Vries
Equation
...........................................606
12.5
Strongly Nonlinear Dispersive Systems and the NLS Equation
... 618
12.6
The Perturbation Method of
Ostrovsky
and Pelinovsky
.........623
12.7
The Method of Multiple Scales
.............................627
12.8
Asymptotic Expansions and Method of Multiple Scales
.........633
12.9
Derivation of the NLS Equation and Davey-Stewartson
Evolution Equations
......................................641
13
Tables of Integral Transforms
............................... 653
13.1
Fourier Transforms
.......................................653
13.2
Fourier Sine Transforms
...................................655
13.3
Fourier Cosine Transforms
.................................657
13.4
Laplace Transforms
.......................................659
13.5
Hankel Transforms
.......................................663
13.6
Finite Hankel Transforms
..................................667
Contents xi
Answers and Hints to Selected Exercises
.......................... 669
1.15
Exercises
............................................... 669
2.8
Exercises
............................................... 681
3.6
Exercises
............................................... 682
4.6
Exercises
............................................... 687
5.5
Exercises
............................................... 689
6.11
Exercises
............................................... 692
7.9
Exercises
............................................... 695
8.14
Exercises
............................................... 696
1
l.BExercises
............................................... 697
Bibliography
................................................. 699
Index
........................................................ 727
|
| adam_txt |
Contents
Preface
to the Second Edition
. xiii
Preface to the First Edition
. xv
1
Linear Partial Differential Equations
. 1
1.1
Introduction
. 1
1.2
Basic Concepts and Definitions
. 2
1.3
The Linear Superposition Principle
. 5
1.4
Some Important Classical Linear Model Equations
. 8
1.5
Second-Order Linear Equations and Method of Characteristics
. 11
1.6
The Method of Separation of Variables
. 22
1.7
Fourier Transforms and Initial Boundary-Value Problems
. 35
1.8
Multiple Fourier Transforms and Partial Differential Equations
. 48
1.9
Laplace Transforms and Initial Boundary-Value Problems
. 53
1.10
Hankel Transforms and Initial Boundary-Value Problems
. 63
1.11
Green's Functions and Boundary-Value Problems
. 71
1.12
Sturm-Liouville Systems and Some General Results
. 84
1.13
Energy Integrals and Higher Dimensional Equations
. 103
1.14
Fractional Partial Differential Equations
. 117
1.15
Exercises
. 128
2
Nonlinear Model Equations andVanational Principles
. 149
2.1
Introduction
. 149
2.2
Basic Concepts and Definitions
. 150
2.3
Some Nonlinear Model Equations
. 150
2.4
Variational Principles and the Euler-Lagrange Equations
. 156
2.5
The Variational Principle for Nonlinear Klein-Gordon Equations
. 169
2.6
The Variational Principle for Nonlinear Water Waves
. 170
2.7
The
Euler
Equation of Motion and Water Wave Problems
. 172
2.8
Exercises
. 185
Contents
First-Order, Quasi-Linear Equations and Method
of Characteristics
.
193
3.1
Introduction
.'93
3.2
The Classification of First-Order Equations
. 194
3.3
The Construction of a First-Order Equation
.195
3.4
The Geometrical Interpretation of a First-Order Equation
.198
3.5
The Method of Characteristics and General Solutions
.201
3.6
Exercises
.
215
First-Order Nonlinear Equations and Their Applications
. 221
4.1
Introduction
.221
4.2
The Generalized Method of Characteristics
.222
4.3
Complete Integrals of Certain Special Nonlinear Equations
.225
4.4
The Hamilton-Jacobi Equation and Its Applications
.232
4.5
Applications to Nonlinear Optics
.240
4.6
Exercises
.248
Conservation Laws and Shock Waves
. 251
5.1
Introduction
.251
5.2
Conservation Laws
.252
5.3
Discontinuous Solutions and Shock Waves
.264
5.4
Weak or Generalized Solutions
.266
5.5
Exercises
.272
Kinematic Waves and Real-World Nonlinear Problems
. 277
6.1
Introduction
.277
6.2
Kinematic Waves
.278
6.3
Traffic Flow Problems
.280
6.4
Flood Waves in Long Rivers
.292
6.5 Chromatographie
Models and Sediment Transport in Rivers
.294
6.6
Glacier Flow
.300
6.7
Roll Waves and Their Stability Analysis
.302
6.8
Simple Waves and Riemann's Invariants
.308
6.9
The Nonlinear Hyperbolic System and Riemann's Invariants
_325
6.10
Generalized Simple Waves and Generalized Riemann's Invariants
335
6.11
Exercises
.339
Nonlinear Dispersive Waves and Whitham's Equations
. 347
7.1
Introduction
.347
7.2
Linear Dispersive Waves
.348
7.3
Initial-Value Problems and Asymptotic Solutions
.351
7.4
Nonlinear Dispersive Waves and Whitham's Equations
.354
7.5
Whitham's Theory of Nonlinear Dispersive Waves
.356
7.6
Whitham's Averaged Variational Principle
.359
7.7
Whitham's Instability Analysis of Water Waves
.361
7.8
Whitham's Equation: Peaking and Breaking of Waves
.363
Contents ix
7.9
Exercises
.369
8
Nonlinear Diffusion-Reaction Phenomena
. 373
8.1
Introduction
.373
8.2
Burgers' Equation and the Plane Wave Solution
.374
8.3
Traveling Wave Solutions and Shock-Wave Structure
.376
8.4
The Exact Solution of the Burgers Equation
.378
8.5
The Asymptotic Behavior of the Burgers Solution
.383
8.6
TheW-Wave Solution
.385
8.7
Burgers' Initial- and Boundary-Value Problem
.386
8.8
Fisher's Equation and Diffusion-Reaction Process
.389
8.9
Traveling Wave Solutions and Stability Analysis
.391
8.10
Perturbation Solutions of the Fisher Equation
.395
8.11
Method of Similarity Solutions of Diffusion Equations
.397
8.12
Nonlinear Reaction-Diffusion Equations
.405
8.13
Brief Summary of Recent Work
.409
8.14
Exercises
.411
9 Solitons
and the Inverse Scattering Transform
. 417
9.1
Introduction
.417
9.2
The History of the
Solitons
and Soliton Interactions
.418
9.3
The Boussinesq and Korteweg-de
Vries
Equations
.423
9.4
Solutions of the KdV Equation:
Solitons
and Cnoidal Waves
_446
9.5
The Lie Group Method and Similarity Analysis of the
KdV Equation
.453
9.6
Conservation Laws and Nonlinear Transformations
.457
9.7
The Inverse Scattering Transform
(1ST)
Method
.462
9.8
Bäcklund
Transformations and the Nonlinear Superposition
Principle
.483
9.9
The Lax Formulation and the Zakharov and Shabat Scheme
.488
9.10
The AKNS Method
.496
9.11
Asymptotic Behavior of the Solution of the KdV-Burgers
Equation
.498
9.12
Strongly Dispersive Nonlinear Equations and
Compactons
.499
9.13
Exercises
.510
10
The Nonlinear
Schrödinger
Equation and Solitary Waves
.515
10.1
Introduction
.515
10.2
The One-Dimensional Linear
Schrödinger
Equation
.516
10.3
The Nonlinear
Schrödinger
Equation and Solitary Waves
.517
10.4
Properties of the Solutions of the Nonlinear
Schrödinger
Equation
522
10.5
Conservation Laws for the NLS Equation
.528
10.6
The Inverse Scattering Method for the Nonlinear
Schrödinger
Equation
.531
χ
Contents
10.7
Examples of Physical Applications in Fluid Dynamics and
Plasma Physics
."3
10.8
Applications to Nonlinear Optics
.545
10.9
Exercises
.
554
11
Nonlinear Klein-Gordon and Sine-Gordon Equations
. 557
11.1
Introduction
.557
11.2
The One-Dimensional Linear Klein-Gordon Equation
.558
11.3
The Two-Dimensional Linear Klein-Gordon Equation
.560
11.4
The Three-Dimensional Linear Klein-Gordon Equation
.562
11.5
The Nonlinear Klein-Gordon Equation and Averaging
Techniques
.563
11.6
The Klein-Gordon Equation and the Whitham Averaged
Variational Principle
.576
11.7
The Sine-Gordon Equation: Soliton and Antisoliton Solutions
. 572
11.8
The Solution of the Sine-Gordon Equation by Separation of
Variables
.576
11.9
Bäcklund
Transformations for the Sine-Gordon Equation
.584
1
l.lOThe Solution of the Sine-Gordon Equation by the Inverse
Scattering Method
.587
11.1
IThe Similarity Method for the Sine-Gordon Equation
.591
11.
^Nonlinear Optics and the Sine-Gordon Equation
.591
ll.BExercises
.595
12
Asymptotic Methods and Nonlinear Evolution Equations
. 599
12.1
Introduction
.599
12.2
The Reductive Perturbation Method and Quasi-Linear
Hyperbolic Systems
.601
12.3
Quasi-Linear Dissipative Systems
.605
12.4
Weakly Nonlinear Dispersive Systems and the Korteweg-de
Vries
Equation
.606
12.5
Strongly Nonlinear Dispersive Systems and the NLS Equation
. 618
12.6
The Perturbation Method of
Ostrovsky
and Pelinovsky
.623
12.7
The Method of Multiple Scales
.627
12.8
Asymptotic Expansions and Method of Multiple Scales
.633
12.9
Derivation of the NLS Equation and Davey-Stewartson
Evolution Equations
.641
13
Tables of Integral Transforms
. 653
13.1
Fourier Transforms
.653
13.2
Fourier Sine Transforms
.655
13.3
Fourier Cosine Transforms
.657
13.4
Laplace Transforms
.659
13.5
Hankel Transforms
.663
13.6
Finite Hankel Transforms
.667
Contents xi
Answers and Hints to Selected Exercises
. 669
1.15
Exercises
. 669
2.8
Exercises
. 681
3.6
Exercises
. 682
4.6
Exercises
. 687
5.5
Exercises
. 689
6.11
Exercises
. 692
7.9
Exercises
. 695
8.14
Exercises
. 696
1
l.BExercises
. 697
Bibliography
. 699
Index
. 727 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Debnath, Lokenath 1935- |
| author_GND | (DE-588)115600663 |
| author_facet | Debnath, Lokenath 1935- |
| author_role | aut |
| author_sort | Debnath, Lokenath 1935- |
| author_variant | l d ld |
| building | Verbundindex |
| bvnumber | BV022263140 |
| callnumber-first | Q - Science |
| callnumber-label | QA377 |
| callnumber-raw | QA377 |
| callnumber-search | QA377 |
| callnumber-sort | QA 3377 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 540 |
| classification_tum | MAT 354f |
| ctrlnum | (OCoLC)55510752 (DE-599)BVBBV022263140 |
| dewey-full | 515/.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 |
| dewey-search | 515/.353 |
| dewey-sort | 3515 3353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV022263140 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:43:27Z |
| indexdate | 2024-07-09T20:53:38Z |
| institution | BVB |
| isbn | 0817643230 9780817643232 |
| language | English |
| lccn | 2004052899 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015473753 |
| oclc_num | 55510752 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-706 DE-83 DE-11 |
| owner_facet | DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-706 DE-83 DE-11 |
| physical | XX, 737 S. graph. Darst. 25 cm |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Birkhäuser |
| record_format | marc |
| spelling | Debnath, Lokenath 1935- Verfasser (DE-588)115600663 aut Nonlinear partial differential equations for scientists and engineers Lokenath Debnath 2. ed. Boston [u.a.] Birkhäuser 2005 XX, 737 S. graph. Darst. 25 cm txt rdacontent n rdamedia nc rdacarrier Differentiaalvergelijkingen gtt Niet-lineaire vergelijkingen gtt Équations différentielles non linéaires Differential equations, Nonlinear Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s DE-604 http://www.loc.gov/catdir/enhancements/fy0662/2004052899-d.html Publisher description Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015473753&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Debnath, Lokenath 1935- Nonlinear partial differential equations for scientists and engineers Differentiaalvergelijkingen gtt Niet-lineaire vergelijkingen gtt Équations différentielles non linéaires Differential equations, Nonlinear Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| subject_GND | (DE-588)4128900-6 |
| title | Nonlinear partial differential equations for scientists and engineers |
| title_auth | Nonlinear partial differential equations for scientists and engineers |
| title_exact_search | Nonlinear partial differential equations for scientists and engineers |
| title_exact_search_txtP | Nonlinear partial differential equations for scientists and engineers |
| title_full | Nonlinear partial differential equations for scientists and engineers Lokenath Debnath |
| title_fullStr | Nonlinear partial differential equations for scientists and engineers Lokenath Debnath |
| title_full_unstemmed | Nonlinear partial differential equations for scientists and engineers Lokenath Debnath |
| title_short | Nonlinear partial differential equations for scientists and engineers |
| title_sort | nonlinear partial differential equations for scientists and engineers |
| topic | Differentiaalvergelijkingen gtt Niet-lineaire vergelijkingen gtt Équations différentielles non linéaires Differential equations, Nonlinear Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| topic_facet | Differentiaalvergelijkingen Niet-lineaire vergelijkingen Équations différentielles non linéaires Differential equations, Nonlinear Nichtlineare partielle Differentialgleichung |
| url | http://www.loc.gov/catdir/enhancements/fy0662/2004052899-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015473753&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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