## Introduction to Perturbation MethodsThis 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 |

313 | |

331 | |

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### Common terms and phrases

amplitude appropriate expansion assumed assumption asymptotic approximation asymptotic expansion asymptotic solution asymptotically stable bifurcation diagram bifurcation point boundary conditions boundary layer coefficients composite expansion Consider the problem coordinate curve damping depends derive determine difference equation eikonal equation exact solution example Exercise explain Find a composite Find a first-term Find a two-term find an asymptotic first-order first-term approximation first-term expansion following problem frequency given Hopf bifurcation initial conditions integral interior layer interval introduce linear matched asymptotic expansions multiple scales multiple-scale nonlinear nonzero numerical solution obtain oscillator outer expansion parameter partial differential equations periodic perturbation positive constant possible procedure region result satisfies second term Section secular terms shown in Fig situation smooth solve steady steady-state solution Substituting substructure Suppose Taylor's theorem tion transition layer two-term expansion uniformly valid valid for large values variable velocity WKB approximation WKB method yields zero

### Popular passages

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,