Advanced mathematical methods for scientists and engineers: 1 Asymptotic methods and perturbation theory
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
New York [u.a.]
Springer
1999
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Originally published: New York : McGraw-Hill, c1978 |
| Beschreibung: | XIV, 593 S. graph. Darst. |
| ISBN: | 9780387989310 0387989315 |
Internformat
MARC
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| 100 | 1 | |a Bender, Carl M. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Advanced mathematical methods for scientists and engineers |n 1 |p Asymptotic methods and perturbation theory |c Carl M. Bender ; Steven A. Orszag |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1999 | |
| 300 | |a XIV, 593 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Originally published: New York : McGraw-Hill, c1978 | ||
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Orszag, Steven A. |e Verfasser |4 aut | |
| 773 | 0 | 8 | |w (DE-604)BV013032490 |g 1 |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-008878480 | ||
Datensatz im Suchindex
| _version_ | 1804127728859873280 |
|---|---|
| adam_text | CONTENTS!
Preface
xiii
PART I
FUNDAMENTALS
1
Ordinary Differential Equations
3
(E)
1.1
Ordinary Differential Equations
3
(définitions;
introductory examples)
(E)
1.2
Initial-Value and Boundary-Value Problems
5
(definitions; comparison of local and global analysis; examples of initial-
value problems)
(TE)
1.3
Theory of Homogeneous Linear Equations
7
(linear dependence and independence; Wronskians; well-posed and ill-posed
initial-value and boundary-value problems)
(E)
1.4
Solutions of Homogeneous Linear Equations
11
(how to solve constant-coefficient, equidimensional, and exact equations;
reduction of order)
(E)
1.5
Inhomogeneous Linear Equations
14
(first-order equations; variation of parameters; Green s functions; delta
function; reduction of order; method of undetermined coefficients)
(E)
1.6
First-Order Nonlinear Differential Equations
20
(methods for solving Bernoulli, Riccati, and exact equations; factoring;
integrating factors; substitutions)
(I)
1.7
Higher-Order Nonlinear Differential Equations
24
(methods to reduce the order of autonomous, equidimensional,
and scale-invariant equations)
t
Each section is labeled according to difficulty: (E)
=
easy, (I)
=
intermediate, (D)
=
difficult.
A section labeled (T) indicates that the material has a theoretical rather than an applied emphasis.
vii
viii CONTENTS
(E)
1.8
Eigenvalue Problems
27
(examples of eigenvalue problems on finite and infinite domains)
(TE)
1.9
Differential Equations in the Complex Plane
29
(comparison of real and complex differential equations)
Problems for Chapter
1 30
2
Difference Equations
36
(E)
2.1
The Calculus of Differences
36
(definitions; parallels between derivatives and differences, integrals, and sums)
(E)
2.2
Elementary Difference Equations
37
(examples of simple linear and nonlinear difference equations; gamma
function; general first-order linear homogeneous and inhomogeneous
equations)
(I)
2.3
Homogeneous Linear Difference Equations
40
(constant-coefficient equations; linear dependence and independence;
Wronskians; initial-value and boundary-value problems; reduction of order;
Buler equations; generating functions; eigenvalue problems)
(I)
2.4
Inhomogeneous Linear Difference Equations
49
(variation of parameters; reduction of order; method of undetermined
coefficients)
(E)
2.5
Nonlinear Difference Equations
53
(elementary examples)
Problems for Chapter
2 53
PART II
LOCAL ANALYSIS
3
Approximate Solution of Linear Differential Equations
61
,E)
3.1
Classification of Singular Points of Homogeneous Linear Equations
62
(ordinary, regular singular, and irreguiar singular points; survey of the
possible kinds of behaviors of solutions)
(E)
3.2
Local Behavior Near Ordinary Points of Homogeneous Linear
Equations
66
(Taylor series solution of first- and second-order equations; Airy equation)
(I)
3.3
Local Series Expansions About Regular Singular Points of
Homogeneous Linear Equations
68
(methods of
Fuchs
and Frobenius; modified Bessel equation)
(E)
3.4
Local Behavior at Irregular Singular Points of Homogeneous
Linear Equations
76
(failure of Taylor and Frobenius series; asymptotic relations; controlling
factor and leading behavior; method of dominant balance; asymptotic
series expansion of solutions at irregular singular points)
CONTENTS
ІХ
(E)
3.5
Irregular Singular
Point at Infinity
88
(theory of asymptotic power series; optimal asymptotic approximation;
behavior of modified Bessel, parabolic cylinder, and Airy functions for
large positive x)
(E)
3.6
Local Analysis of Inhomogeneous Linear Equations
103
(illustrative examples)
(TI)
3.7
Asymptotic Relations
107
(asymptotic relations for oscillatory functions; Airy functions and Bessel
functions; asymptotic relations in the complex plane; Stokes phenomenon;
subdominance)
(TD)
3.8
Asymptotic Series
118
(formal theory of asymptotic power series;
Stieltjes
series and integrals;
optimal asymptotic approximations; error estimates; outline of a rigorous
theory of the asymptotic behavior of solutions to differential equations)
Problems for Chapter
3 136
4
Approximate
Solution
of Nonlinear Differential Equations
146
(E)
4.1
Spontaneous Singularities
146
(comparison of the behaviors of solutions to linear and nonlinear equations)
(E)
4.2
Approximate Solutions of First-Order Nonlinear Differential
Equations
148
(several examples analyzed in depth)
(I)
4.3
Approximate Solutions to Higher-Order Nonlinear Differential
Equations
152
(Thomas-Fermi equatiorrffirst
Painlevé
transcendent; other examples)
(I)
4.4
Nonlinear Autonomous Systems
171
(phase-space interpretation; classification of critical points; one- and
two-dimensional phase space)
(I)
4.5
Higher-Order Nonlinear Autonomous Systems
185
(brief, nontechnical survey of properties of higher-order systems; periodic,
almost periodic, and random behavior;
Toda
lattice,
Lorenz
model, and
other systems)
Problems for Chapter
4 196
5
Approximate Solution of Difference Equations
205
(E)
5.1
Introductory Comments
205
(comparison of the behavior of differential and difference equations)
(I)
5.2
Ordinary and Regular Singular Points of Linear Difference
Equations
206
(classification of
π
=
oo as an ordinary, a regular singular, or an irregular
singular point; Taylor and Frobenius series at oo)
(E)
5.3
Local Behavior Near an Irregular Singular Point at Infinity:
Determination of Controlling Factors
214
(three general methods)
X
CONTENTS
(E)
5.4
Asymptotic Behavior of
η!
as
η -»οο:
The Stirling Series
218
[asymptotic behavior of the gamma function
Г(х)
as
χ
->
oc
obtained from
the difference equations
Г(.х
+ 1) =
хГ(х)]
(I)
5.5
Local Behavior Near an Irregular Singular Point at Infinity:
Full Asymptotic Series
227
(Bessel functions of large order; Legendre polynomials of large degree)
(E)
5.6
Local Behavior of Nonlinear Difference Equations
233
(Newton s method and other nonlinear difference equations; statistical
analysis of an unstable difference equation)
Problems for Chapter
5 240
6
Asymptotic Expansion of Integrals
247
(E)
6.1
Introduction
247
(integral representations of solutions to difference and differential equations)
(E)
6.2
Elementary Examples
249
(incomplete gamma function; exponential integral; other examples)
(E)
6.3
Integration by Parts
252
(many examples including some where the method fails)
(E)
6.4
Laplace s Method and Watson s Lemma
261
(modified
Besśel,
parabolic cylinder, and gamma functions
;
many other
illustrative examples)
(I)
6.5
Method of Stationary Phase
276
(leading behavior of integrals with rapidly oscillating integrands)
(I)
6.6
Method of Steepest Descents
280
(steepest ascent and descent paths in the complex plane; saddle points;
Stokes phenomenon)
(I)
6.7
Asymptotic Evaluation of Sums
302
(approximation of sums by integrals; Laplace s method for sums;
Euler-Maclaurin sum formula)
Problems for Chapter
6 306
PART III
PERTURBATION METHODS
7
Perturbation Series
319
(E)
7.1
Perturbation Theory
319
(elementary introduction; application to polynomial equations and initial-
value problems for differential equations)
(E)
7.2
Regular and Singular Perturbation Theory
324
(classification of perturbation problems as regular or singular;
introductory examples of boundary-layer, WKB, and multiple-scale problems)
(I)
7.3
Perturbation Methods for Linear Eigenvalue Problems
330
(Rayleigh-Schrodinger perturbation theory)
CONTENTS xi
(D)
7.4
Asymptotic Matching
335
(matched asymptotic expansions; applications to differential equations,
eigenvalue problems and integrals)
(TD)
7.5
Mathematical Structure of Perturbative Eigenvalue Problems
350
(singularity structure of eigenvalues as functions of complex perturbing
parameter; level crossing)
Problems for Chapter
7 361
8
Summation of Series
368
(E)
8.1
Improvement of Convergence
368
(Shanks transformation; Richardson extrapolation; Riemann
zeta
function)
(E)
8.2
Summation of Divergent Series
379
(Euler,
Borei,
and generalized
Borei
summation)
(I)
8.3
Padé
Summation
383
(one- and two-point
Padé
summation; generalized Shanks transformation;
many numerical examples)
(I)
8.4
Continued Fractions and
Padé
Approximants
395
(efficient methods for obtaining and evaluating
Padé
approximants)
(TD)
8.5
Convergence of
Padé
Approximants
400
(asymptotic analysis of the rate of convergence of
Padé
approximants)
(TD)
8.6
Padé
Sequences for
Stieltjes
Functions
405
(monotonicity;
convergence theory; moment problem; Carleman s condition)
Problems for Chapter
8 410
PART IV
GLOBAL ANALYSIS
9
Boundary Layer Theory
417
(E)
9.1
Introduction to Boundary-Layer Theory
419
(linear and nonlinear examples)
(E)
9.2
Mathematical Structure of Boundary Layers:
Inner, Outer, and Intermediate Limits
426
(formal boundary-layer theory)
(E)
9.3
Higher-Order Boundary Layer Theory
431
(uniformly valid global approximants to a simple boundary-value problem)
(I)
9.4
Distinguished Limits and Boundary Layers of Thickness
φ ε
435
(three illustrative examples)
(I)
9.5
Miscellaneous Examples of Linear Boundary-Layer Problems
446
(third- and fourth-order differential equations; nested boundary layers)
(D)
9.6
Internal Boundary Layers
455
(four cases including some for which boundary-layer theory fails)
XU CONTENTS
(I)
9.7
Nonlinear Boundary-Layer Problems
463
(a problem of Carrier; limit cycle of the Rayleigh oscillator)
Problems for Chapter
9 479
10
WKB Theory
484
(E)
10.1
The Exponential Approximation for Dissipative and Dispersive
Phenomena
484
(formal WKB expansion; relation to boundary-layer theory)
(E)
10.2
Conditions for Validity of the WKB Approximation
493
(geometrical and physical optics)
(E)
10.3
Patched Asymptotic Approximations: WKB Solution of
Inhomogeneous Linear Equations
497
(WKB approximations to Green s functions)
(I)
10.4
Matched Asymptotic Approximations: Solution of the
One-Turning-Point Problem
504
(connection formula; Langer s solution; normalization methods)
(I)
10.5
Two-Turning-Point Problems: Eigenvalue Condition
519
(approximate eigenvalues of
Schrödinger
equations)
(D)
10.6
Tunneling
524
(reflection and transmission of waves through potential barriers)
(D)
10.7
Brief Discussion of Higher-Order WKB Approximations
534
(second-order solution of one-turning-point problems; quantization condition
to all orders)
Problems for Chapter
10 539
11
Multiple-Scale Analysis
544
(E)
11.1
Resonance and Secular Behavior
544
(nonuniform
convergence of regular perturbation expansions)
(E)
11.2
Multiple-Scale Analysis
549
(formal theory; Duffing equation)
(I)
11.3
Examples of Multiple-Scale Analysis
551
(damped oscillator; approach to a limit cycle; recovery of WKB and
boundary layer approximations)
(I)
11.4
The
Mathieu
Equation and Stability
560
(Floquet theory; stability boundaries of the
Mathieu
equation)
Problems for Chapter
11 566
Appendix
—
Useful Formulas
569
References
577
Index
581
|
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| id | DE-604.BV013032491 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:37:58Z |
| institution | BVB |
| isbn | 9780387989310 0387989315 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008878480 |
| oclc_num | 313784779 |
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| owner_facet | DE-703 DE-355 DE-BY-UBR DE-20 DE-91G DE-BY-TUM DE-29T DE-634 DE-11 |
| physical | XIV, 593 S. graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Springer |
| record_format | marc |
| spelling | Bender, Carl M. Verfasser aut Advanced mathematical methods for scientists and engineers 1 Asymptotic methods and perturbation theory Carl M. Bender ; Steven A. Orszag New York [u.a.] Springer 1999 XIV, 593 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Originally published: New York : McGraw-Hill, c1978 Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Orszag, Steven A. Verfasser aut (DE-604)BV013032490 1 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008878480&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bender, Carl M. Orszag, Steven A. Advanced mathematical methods for scientists and engineers Numerisches Verfahren (DE-588)4128130-5 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4012249-9 |
| title | Advanced mathematical methods for scientists and engineers |
| title_auth | Advanced mathematical methods for scientists and engineers |
| title_exact_search | Advanced mathematical methods for scientists and engineers |
| title_full | Advanced mathematical methods for scientists and engineers 1 Asymptotic methods and perturbation theory Carl M. Bender ; Steven A. Orszag |
| title_fullStr | Advanced mathematical methods for scientists and engineers 1 Asymptotic methods and perturbation theory Carl M. Bender ; Steven A. Orszag |
| title_full_unstemmed | Advanced mathematical methods for scientists and engineers 1 Asymptotic methods and perturbation theory Carl M. Bender ; Steven A. Orszag |
| title_short | Advanced mathematical methods for scientists and engineers |
| title_sort | advanced mathematical methods for scientists and engineers asymptotic methods and perturbation theory |
| topic | Numerisches Verfahren (DE-588)4128130-5 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Numerisches Verfahren Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008878480&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013032490 |
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