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. |
Contents
I | ix |
II | 1 |
III | 2 |
IV | 3 |
V | 4 |
VI | 5 |
VIII | 6 |
IX | 8 |
LII | 63 |
LIII | 65 |
LIV | 67 |
LV | 68 |
LVI | 69 |
LVIII | 70 |
LIX | 72 |
LX | 73 |
X | 9 |
XI | 11 |
XII | 12 |
XIII | 14 |
XIV | 15 |
XV | 16 |
XVI | 17 |
XVII | 18 |
XVIII | 19 |
XX | 20 |
XXI | 21 |
XXII | 23 |
XXIII | 24 |
XXIV | 25 |
XXV | 26 |
XXVI | 27 |
XXVIII | 28 |
XXIX | 29 |
XXX | 30 |
XXXII | 32 |
XXXIII | 33 |
XXXIV | 34 |
XXXV | 37 |
XXXVII | 39 |
XXXVIII | 41 |
XXXIX | 42 |
XL | 46 |
XLII | 48 |
XLIII | 50 |
XLIV | 52 |
XLV | 53 |
XLVII | 54 |
XLVIII | 55 |
XLIX | 57 |
L | 58 |
LI | 60 |
LXI | 74 |
LXII | 77 |
LXIII | 81 |
LXIV | 84 |
LXV | 87 |
LXVII | 90 |
LXVIII | 93 |
LXIX | 97 |
LXX | 102 |
LXXI | 103 |
LXXII | 104 |
LXXIII | 110 |
LXXIV | 116 |
LXXVI | 120 |
LXXVII | 123 |
LXXVIII | 126 |
LXXIX | 127 |
LXXX | 128 |
LXXXI | 130 |
LXXXII | 131 |
LXXXIII | 133 |
LXXXIV | 135 |
LXXXV | 136 |
LXXXVI | 137 |
LXXXVII | 140 |
LXXXVIII | 141 |
LXXXIX | 144 |
XCI | 147 |
XCII | 148 |
XCIII | 149 |
XCVI | 150 |
XCVII | 151 |
| 155 | |
| 158 | |
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Common terms and phrases
A₁ A₂ amplitude asymptotic approximation asymptotic expansions asymptotic sequence B₁ B₂ behaviour boundary condition boundary layer comparing coefficients complex plane condition at infinity correction term dashed curves derivative differential equations dispersion relation Dyke's rule eigenvalue eigenvector evaluated exact solution example Exercise expansion method f₁ fixed function governing equation group velocity Hence higher order integrand intermediate variable iterative method leading order term linear matched asymptotic expansions model problem nonlinear number of terms numerical O(e² obtain orbit ord(e ord(e² oscillation outer Padé approximants parameter perturbation pose an expansion potential quadratic r-approximation r-region radius of convergence relaxation oscillation rescaling right hand side root saddle saddle point satisfy the boundary scale sin² singular small region solve strained co-ordinates Substituting t₁ transform uniformly asymptotic wavenumber WKBJ x₁ θα მთ