## Perturbation MethodsIn this book the author presents the theory and techniques underlying perturbation methods in a manner that will make the book widely appealing to readers in a broad range of disciplines. Methods of algebraic equations, asymptotic expansions, integrals, PDEs, strained coordinates, and multiple scales are illustrated by copious use of examples drawn from many areas of mathematics and physics. The philosophy adopted is that there is no single or best method for such problems, but that one may exploit the small parameter given some experience and understanding of similar perturbation problems. The author does not look to perturbation methods to give quantitative answers but rather uses them to give a physical understanding of the subtle balances in a complex problem. |

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### Contents

I | ix |

II | 1 |

III | 2 |

IV | 3 |

V | 4 |

VI | 5 |

VIII | 6 |

IX | 8 |

LI | 63 |

LII | 65 |

LIII | 67 |

LIV | 68 |

LV | 69 |

LVII | 70 |

LVIII | 72 |

LIX | 73 |

X | 9 |

XI | 11 |

XII | 12 |

XIII | 14 |

XIV | 15 |

XV | 16 |

XVI | 17 |

XVII | 18 |

XVIII | 19 |

XIX | 20 |

XX | 21 |

XXI | 23 |

XXII | 24 |

XXIII | 25 |

XXIV | 26 |

XXV | 27 |

XXVII | 28 |

XXVIII | 29 |

XXIX | 30 |

XXXI | 32 |

XXXII | 33 |

XXXIII | 34 |

XXXIV | 37 |

XXXVI | 39 |

XXXVII | 41 |

XXXVIII | 42 |

XXXIX | 46 |

XLI | 48 |

XLII | 50 |

XLIII | 52 |

XLIV | 53 |

XLVI | 54 |

XLVII | 55 |

XLVIII | 57 |

XLIX | 58 |

L | 60 |

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### Common terms and phrases

amplitude applied approximation asymptotic approximation asymptotic expansions becomes behaviour body boundary condition boundary layer breaks called co-ordinates complex Consider constant continuous contribution convergence correction curve depend derivative determined differential equations direction drift error evaluated exact solution example Exercise exponential expression factor fixed flow forcing function further given gives governing equation Hence higher order infinity initial inner integral integrand intermediate iterative known leading order limit linear matching method moving necessary nonlinear Note numerical obtain ord(e order term original oscillation outer period perturbation phase pose positive possible potential powers problem produce radius of convergence range region regular relation rescaling respect result right hand side root rule saddle satisfies scale sequence singular slow slowly solution solve space starting Substituting transform variable varying wave yields zero