Introduction to Perturbation Methods

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Springer Science & Business Media, Jun 19, 1998 - Mathematics - 356 pages
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This book is an introductory graduate text dealing with many of the perturbation methods currently used by applied mathematicians, scientists, and engineers. The author has based his book on a graduate course he has taught several times over the last ten years to students in applied mathematics, engineering sciences, and physics. The only prerequisite for the course is a background in differential equations. Each chapter begins with an introductory development involving ordinary differential equations. The book covers traditional topics, such as boundary layers and multiple scales. However, it also contains material arising from current research interest. This includes homogenization, slender body theory, symbolic computing, and discrete equations. One of the more important features of this book is contained in the exercises. Many are derived from problems of up- to-date research and are from a wide range of application areas.
 

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Contents

Introduction to Asymptotic Approximations
1
12 Taylors Theorem and lHospitals Rule
3
13 Order Symbols
4
14 Asymptotic Approximations
7
141 Asymptotic Expansions
9
142 Accuracy versus Convergence of an Asymptotic Series
12
143 Manipulating Asymptotic Expansions
14
15 Asymptotic Solution of Algebraic and Transcendental Equations
18
43 Turning Points
173
44 Wave Propagation and Energy Methods
185
45 Wave Propagation and Slender Body Approximations
190
46 Ray Methods
197
47 Parabolic Approximations
207
48 Discrete WKB Method
212
The Method of Homogenization
223
52 Introductory Example
224

16 Introduction to the Asymptotic Solution of Differential Equations
26
17 Uniformity
37
18 Symbolic Computing
43
Matched Asymptotic Expansions
47
22 Introductory Example
48
23 Examples with Multiple Boundary Layers
62
24 Interior Layers
68
25 Corner Layers
77
26 Partial Differential Equations
84
27 Difference Equations
98
Multiple Scales
105
32 Introductory Example
106
33 Slowly Varying Coefficients
117
34 Forced Motion Near Resonance
123
35 Boundary Layers
132
36 Introduction to Partial Differential Equations
134
37 Linear Wave Propagation
139
38 Nonlinear Waves
142
39 Difference Equations
153
The WKB and Related Methods
161
42 Introductory Example
162
Periodic Substructure
234
54 Porous Flow
241
Introduction to Bifurcation and Stability
249
63 Analysis of a Bifurcation Point
251
64 Linearized Stability
255
65 Relaxation Dynamics
264
66 An Example Involving a Nonlinear Partial Differential Equation
271
67 Bifurcation of Periodic Solutions
281
68 Systems of Ordinary Differential Equations
287
Solution and Properties of Transition Layer Equations
297
A12 Confluent Hypergeometric Functions
299
A13 HigherOrder Turning Points
302
Asymptotic Approximations of Integrals
305
A22 Watsons Lemma
306
Numerical Solution of Nonlinear BoundaryValue Problems
309
A32 Examples
310
A33 Computer Code
311
References
313
Index
331
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Page 320 - Guedes, JM, and Kikuchi, N., 1990, "Preprocessing and Postprocessing for Materials Based on the Homogenization Method with Adaptive Finite Element Methods," Computer Methods in Applied Mechanics and Engineering, Vol.
Page 319 - ASYMPTOTIC SERIES By WB FORD Two VOLUMES IN ONE: Studies on Divergent Series and Summability and The Asymptotic Developments of Functions Defined by MacLaurin Series.
Page 327 - The initial development of the WKB solutions of linear second order ordinary differential equations and their use in the connection problem.
Page 327 - A dispersive effective medium for wave propagation in periodic composites.
Page 326 - Threedimensional acoustic waves in the ear canal and their interaction with the tympanic membrane,

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