Nonlinear partial differential equations for scientists and engineers:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Birkhäuser
2012
|
| Edition: | 3. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XXIII, 860 S. graph. Darst. |
| ISBN: | 9780817682644 |
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MARC
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| 245 | 1 | 0 | |a Nonlinear partial differential equations for scientists and engineers |c Lokenath Debnath |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Birkhäuser |c 2012 | |
| 300 | |a XXIII, 860 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804148578508079104 |
|---|---|
| adam_text | Contents
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
.... 10
1.6
The Method of Separation of Variables
......................... 21
1.7
Fourier Transforms and Initial Boundary-Value Problems
......... 34
1.8
Multiple Fourier Transforms and Partial Differential Equations
.... 46
1.9
Laplace Transforms and Initial Boundary-Value Problems
........ 51
1.10
Hankel Transforms and Initial Boundary-Value Problems
......... 61
1.11
Green s Functions and Boundary-Value Problems
................ 70
1.12
Sturm-Liouville Systems and Some General Results
............. 82
1.13
Energy Integrals and Higher Dimensional Equations
............. 101
1.14
Fractional Partial Differential Equations
........................ 114
1.15
Exercises
................................................. 125
2
Nonlinear Model Equations and Variational Principles
............. 149
2.1
Introduction
............................................... 149
2.2
Basic Concepts and Definitions
............................... 150
2.3
Some Nonlinear Model Equations
............................. 151
2.4
Variational Principles and the Euler-Lagrange Equations
......... 159
2.5
The Variational Principle for Nonlinear Klein-Gordon Equations
.. 171
2.6
The Variational Principle for Nonlinear Water Waves
............. 171
2.7
The
Euler
Equation of Motion and Water Wave Problems
......... 173
2.8
The Energy Equation and Energy Flux
......................... 192
2.9
Exercises
................................................. 193
3
First-Order, Quasi-linear Equations and Method of Characteristics
.. 201
3.1
Introduction
............................................... 201
3.2
The Classification of First-Order Equations
..................... 202
xx Contents
3.3
The Construction of a First-Order Equation
..................... 203
3.4
The Geometrical Interpretation of a First-Order Equation
......... 206
3.5
The Method of Characteristics and General Solutions
............ 208
3.6
Exercises
................................................. 221
4
First-Order Nonlinear Equations and Their Applications
........... 227
4.1
Introduction
............................................... 227
4.2
The Generalized Method of Characteristics
..................... 228
4.3
Complete Integrals of Certain Special Nonlinear Equations
........ 231
4.4
The Hamilton-Jacobi Equation and Its Applications
............. 238
4.5
Applications to Nonlinear Optics
............................. 246
4.6
Exercises
................................................. 254
5
Conservation Laws and Shock Waves
............................. 257
5.1
Introduction
............................................... 257
5.2
Conservation Laws
......................................... 258
5-3
Discontinuous Solutions and Shock Waves
..................... 270
5.4
Weak or Generalized Solutions
............................... 271
5.5
Exercises
................................................. 277
6
Kinematic Waves and Real-World Nonlinear Problems
............. 283
6.1
Introduction
............................................... 283
6.2
Kinematic Waves
........................................... 284
6.3
Traffic Flow Problems
....................................... 286
6.4
Flood Waves in Long Rivers
................................. 297
6.5 Chromatographie
Models and Sediment Transport in Rivers
....... 300
6.6
Glacier Flow
.............................................. 305
6.7
Roll Waves and Their Stability Analysis
....................... 307
6.8
Simple Waves and Riemann s Invariants
....................... 311
6.9
The Nonlinear Hyperbolic System and Riemann s Invariants
...... 329
6.10
Generalized Simple Waves and Generalized Riemann s Invariants
.. 338
6.11
The
Lorenz
System of Nonlinear Differential Equations
and Deterministic Chaos
..................................... 342
6.12
Exercises
................................................. 346
7
Nonlinear Dispersive Waves and Whitham s Equations
............. 355
7.1
Introduction
............................................... 355
7.2
Linear Dispersive Waves
.................................... 356
7.3
Initial-Value Problems and Asymptotic Solutions
................ 359
7.4
Nonlinear Dispersive Waves and Whitham s Equations
........... 361
7.5
Whitham s Theory of Nonlinear Dispersive Waves
............... 364
7.6
Whitham s Averaged Variational Principle
...................... 367
7.7
Whitham s Instability Analysis of Water Waves
................. 369
7.8
Whitham s Equation: Peaking and Breaking of Waves
............ 371
7.9
Exercises
................................................. 377
Contents xxi
8
Nonlinear Diffusion-Reaction Phenomena
........................ 381
8.1
Introduction
............................................... 381
8.2
Burgers Equation and the Plane Wave Solution
.................. 382
8.3
Traveling Wave Solutions and Shock-Wave Structure
............. 384
8.4
The Exact Solution of the Burgers Equation
.................... 386
8.5
The Asymptotic Behavior of the Burgers Solution
............... 391
8.6
The iV-Wave Solution
....................................... 392
8.7
Burgers Initial- and Boundary-Value Problem
................... 394
8.8
Fisher Equation and Diffusion-Reaction Process
................ 396
8.9
Traveling Wave Solutions and Stability Analysis
................ 399
8.10
Perturbation Solutions of the Fisher Equation
................... 402
8.11
Method of Similarity Solutions of Diffusion Equations
........... 404
8.12
Nonlinear Reaction-Diffusion Equations
....................... 412
8.13
Brief Summary of Recent Work
.............................. 416
8.14
Exercises
................................................. 417
9 Solitons
and the Inverse Scattering Transform
..................... 425
9.1
Introduction
............................................... 425
9.2
The History of the
Solitons
and Soliton Interactions
.............. 426
9.3
The Boussinesq and Korteweg-de
Vries
Equations
............... 431
9.4
Solutions of the KdV Equation:
Solitons
and Cnoidal Waves
...... 458
9.5
The Lie Group Method and Similarity Analysis of the KdV
Equation
.................................................. 466
9.6
Conservation Laws and Nonlinear Transformations
.............. 469
9.7
The Inverse Scattering Transform
(1ST)
Method
................. 474
9.8
Bäcklund
Transformations and the Nonlinear Superposition
Principle
.................................................. 494
9.9
The Lax Formulation and the Zakharov and Shabat Scheme
....... 499
9.10
The AKNS Method
......................................... 506
9.11
Asymptotic Behavior of the Solution of the Kd V-Burgers Equation
. 508
9.12
Strongly Dispersive Nonlinear Equations and
Compactons
........ 509
9.13
The Camassa-Holm (CH) and Degasperis-Procesi (DP) Nonlinear
Model Equations
........................................... 518
9.14
Exercises
................................................. 528
10
The Nonlinear
Schrödinger
Equation and Solitary Waves
........... 535
10.1
Introduction
............................................... 535
10.2
The One-Dimensional Linear
Schrödinger
Equation
.............. 536
10.3
The Nonlinear
Schrödinger
Equation and Solitary Waves
......... 537
10.4
Properties of the Solutions of the Nonlinear
Schrödinger
Equation
.. 541
10.5
Conservation Laws for the NLS Equation
...................... 547
10.6
The Inverse Scattering Method for the Nonlinear
Schrödinger
Equation
.................................................. 550
10.7
Examples of Physical Applications in Fluid Dynamics and Plasma
Physics
................................................... 552
xxii Contents
10.8
Applications to Nonlinear Optics
............................. 567
10.9
Exercises
................................................. 575
11
Nonlinear Klein-Gordon and Sine-Gordon Equations
.............. 579
11.1
Introduction
.............................................. 579
11.2
The One-Dimensional Linear Klein-Gordon Equation
.......... 580
11.3
The Two-Dimensional Linear Klein-Gordon Equation
.......... 582
11.4
The Three-Dimensional Linear Klein-Gordon Equation
......... 584
11.5
The Nonlinear Klein-Gordon Equation and Averaging Techniques
585
11.6
The Klein-Gordon Equation and the Whitham Averaged
Variational Principle
....................................... 592
11.7
The Sine-Gordon Equation: Soliton and Antisoliton Solutions
-----594
í
1.8
The Solution of the Sine-Gordon Equation by Separation
of Variables
.............................................. 598
11.9
Bäcklund
Transformations for the Sine-Gordon Equation
........ 605
11.10
The Solution of the Sine-Gordon Equation by the Inverse
Scattering Method
......................................... 608
11.11
The Similarity Method for the Sine-Gordon Equation
........... 611
11.12
Nonlinear Optics and the Sine-Gordon Equation
............... 612
11.13
Nonlinear Lattices and the Toda-Lattice Soliton
................ 615
11.14
Exercises
................................................ 619
12
Asymptotic Methods and Nonlinear Evolution Equations
........... 623
12.1
Introduction
.............................................. 623
12.2
The Reductive Perturbation Method and Quasi-linear Hyperbolic
Systems
................................................. 625
1
2.3
Quasi-linear Dissipative Systems
............................ 628
12.4
Weakly Nonlinear Dispersive Systems and the Korteweg-de
Vries
Equation
................................................. 630
12.5
Strongly Nonlinear Dispersive Systems and the NLS Equation
... 641
12.6
The Perturbation Method of
Ostrovsky
and Pelinovsky
..........646
12.7
The Method of Multiple Scales
.............................. 649
12.8
Asymptotic Expansions and Method of Multiple Scales
......... 656
12.9
Derivation of the NLS Equation and Davey-Stewartson Evolution
Equations
................................................ 663
12.10
Exercises
................................................ 674
13
Tables of Integral Transfonns
................................... 675
13Л
Fourier Transfonns
........................................ 675
13.2
Fourier Sine Transforms
.................................... 677
133
Fourier Cosine Transforms
................................. 679
13.4
Laplace Transforms
....................................... 680
13.5
Hankel Transforms
........................................ 683
13.6
Finite Hankel Transforms
.................................. 686
Contents xxiii
A Some Special Functions and Their Properties
...................... 689
A-
1
Gamma, Beta, and Error Functions
........................... 689
A-2 Bessel and Airy Functions
.................................. 697
A-3 Legendre and Associated Legendre Functions
................. 703
A-4 Jacobi and
Gegenbauer
Polynomials
......................... 706
A-5 Laguerre and Associated Laguerre Functions
.................. 710
A-6 Hermite Polynomials and Weber-Hermite Functions
............ 712
A-7 Mittag-Leffler Function
.................................... 713
A-8 The Jacobi Elliptic Integrals and Elliptic Functions
............. 715
В
Fourier Series, Generalized Functions, and Fourier and Laplace
Transforms
.................................................... 719
B-l Fourier Series and Its Basic Properties
........................ 719
B-2 Generalized Functions (Distributions)
........................ 740
B-3 Basic Properties of the Fourier Transforms
.................... 751
B-4 Basic Properties of Laplace Transforms
....................... 758
С
Answers and Hints to Selected Exercises
.......................... 767
1.15
Exercises
................................................ 767
2.9
Exercises
................................................ 785
3.6
Exercises
................................................ 788
4.6
Exercises
................................................ 794
5.5
Exercises
................................................ 796
6Л2
Exercises
................................................ 799
7.9
Exercises
................................................ 803
8.14
Exercises
................................................ 805
9.14
Exercises
................................................ 807
10.9
Exercises
................................................ 808
11.14
Exercises
................................................ 809
12.10
Exercises
................................................ 809
Bibliography
....................................................... 813
Index
............................................................. 847
Lokenath Debnath
Nonlinear Partial Differential Equations
for Scientists and Engineers, Third Edition
An exceptionatly complete overview of the latest developments in the field of PDEs. There
are numerous examples and the emphasis is on applications to almost all areas of science
and engineering. There is truly something for everyone here.
—
Applied Mechanics Review (Review of First Edition)
The revised and enlarged third edition of this successful book presents a comprehensive
and systematic treatment oflinear and nonlinear partial differential equations and their
various current applications. In on effort to make the book more useful for a diverse
readership, updated modern examples of applications have been chosen from areas of
fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics,
nonlinear optics, acoustics, and wave propagation.
The book gives thorough coverage of the derivation and solution methods for all
fundamental nonlinear
model
equations, such as Korteweg-de
Vries, Camassa-Holm,
Degaspcri
s
-Procesi,
Euler
Poincaré,Toda
lattice, Boussinesq, Burgers, Fisher, Whitham,
nonlinear Klein-Gordon,
sinet ¡ordon,
nonlinear
Schrödinger,
nonlinear reaction-
diffusion, and Euler-Lagrange equations. Other topics and key features include:
Improved presentations of results, solution methods, and proofs.
Solitons,
gravin
-capillary solitary waves, and the Inverse Scatterin;
•
Special emphasis on
compactons,
intrinsic localized modes, and nonlinear
instability of dispersive waves with applications to water waves and wave breaking
phenomena.
•
New section on the
Loren/
nonlinear system, the
Lorenz attractor,
and
deterministic chaos, and new examples of nonlinear quasi-harmonic waves,
modulational instability, nonlinear iattices, and the
Toda
lattice equation.
•
Over looo worked-out examples and end-of-chapter exercises with expanded hints
and answers to selected exercises.
•
Two new appendices on some special functions and their basic properties, Fourier
series, generalized functions,
ond
Fourier and Laplace transforms, with algebraic
and analytical properties of convolutions and applications.
•
Many aspects of modern theory that will put the reader at the forefront of current
research.
•
Completely updated list of references and enlarged index.
S
onlinear Partial Differential
t
(¡nations for Scientists and Engineers. Third Edition,
improves on an already complete and accessible resource
forsenior
undergraduate and
graduate students and professionals in mathematics, physics, science, and engineering.
It may be used to great effect as a course textbook, a research reference,
ora
self-study
guide.
ISBN
978-0-8176-8264-4
9 780817 682644
►
birkhauser-science.com
|
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| discipline | Mathematik |
| edition | 3. ed. |
| format | Book |
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| id | DE-604.BV039703922 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:09:22Z |
| institution | BVB |
| isbn | 9780817682644 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024552398 |
| oclc_num | 808254208 |
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| owner_facet | DE-384 DE-824 DE-20 DE-11 DE-355 DE-BY-UBR DE-703 DE-91G DE-BY-TUM |
| physical | XXIII, 860 S. graph. Darst. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Birkhäuser |
| record_format | marc |
| spelling | Debnath, Lokenath 1935- Verfasser (DE-588)115600663 aut Nonlinear partial differential equations for scientists and engineers Lokenath Debnath 3. ed. New York [u.a.] Birkhäuser 2012 XXIII, 860 S. graph. Darst. 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 Erscheint auch als Online-Ausgabe 978-0-8176-8265-1 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024552398&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024552398&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| 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_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://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024552398&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024552398&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT debnathlokenath nonlinearpartialdifferentialequationsforscientistsandengineers |