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Holmes M.H. Ч Introduction to Perturbation Methods
Holmes M.H. Ч Introduction to Perturbation Methods

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Ќазвание: Introduction to Perturbation Methods

јвтор: Holmes M.H.


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.

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1995

 оличество страниц: 356

ƒобавлена в каталог: 01.07.2008

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ѕредметный указатель
$\cong$ symbol      6
$\ll$ symbol      6
$\sim$ symbol      7 9
Acherberg Ч O'Malley resonance      81
Acoustic thermometer      207
Adiabatic explosion temperature      76
Air flow, supersonic      49
Airy functions      175 297Ч298
Airy functions, expansions of      298Ч299
Airy functions, graph of      298
Airy's equation      174 297
Arithmetic mean      232
Asymptotic approximations of Asymptotic approximations, algebraic equations      18Ч22
Asymptotic approximations, defined      7
Asymptotic approximations, differentiating      14
Asymptotic approximations, ideas underlying      13 47
Asymptotic approximations, of differential equations      26Ч31
Asymptotic approximations, of integrals      305Ч307
Asymptotic approximations, of transcendental equations      21Ч22
Asymptotic approximations, uniform      37Ч42
Asymptotic expansions      1
Asymptotic expansions, coefficients of      10
Asymptotic expansions, concept introduced      13Ч14
Asymptotic expansions, defined      9
Asymptotic expansions, differentiating      14Ч15
Asymptotic expansions, divergent      12
Asymptotic expansions, finding      9Ч11
Asymptotic expansions, integrating      15
Asymptotic expansions, manipulating      14Ч15
Asymptotic expansions, matched      see УMatched asymptotic expansionsФ
Asymptotic expansions, nonuniform      39
Asymptotic expansions, of eigenvalues      25Ч26 67 122 164 170 180 219Ч220
Asymptotic expansions, well ordered      19
Asymptotic sequences      9
Asymptotic series, accuracy versus convergence of      12Ч14
Asymptotic stability      257 258
Asymptotically equal functions      11
Asymptotically stable orbit      284
Asymptotically stable solutions      258
Attraction, basin of      250
Attractive solutions      258
Averaging in method of homogenization      243Ч244
Averaging procedure, properties of      230
Bachmann Ч Landau symbols      4
Balancing terms      19
Basin of attraction      250
Basis functions      9
Beam equation      171 189
Beam equation, damped      137
Beam equation, nonlinear      131 137 278
Bessel's equation      171
Bifurcating solutions, schematic of structure of      254
Bifurcation and stability, introduction to      249Ч291
Bifurcation and stability, introductory example      249Ч251
Bifurcation of periodic solutions      281Ч285
Bifurcation parameter      250
Bifurcation point      250 251
Bifurcation point, analysis of      251Ч255
Bifurcation, degenerate      284Ч285
Bifurcation, global      277
Bifurcation, imperfect      263
Bifurcation, pitchfork      250
Bifurcation, steady-state      249
Big oh      4
Big oh, two meanings of      20
Binomial expansion      3
Biological switching mechanism      270
Bistability      126 264
Bloch expansion      240
Boundary layers      50Ч51
Boundary layers, matching and      52 64Ч65 89
Boundary layers, multiple      62Ч65
Boundary layers, multiple scales and      132Ч133
Boundary layers, parabolic      90Ч91
Boundary-layer coordinate      50 63 87
Boundary-value problems, FORTRAN program for solving      309Ч312
Boundary-value problems, nonlinear, numerical solution of      309Ч312
Boussinesq equation      149
Breathers      158 281
Brusselator      293
Buckled states      278
Buckling      271Ч272
Buckling loads      260
Burgers' equation      91
Burgers' equation, solutions of      95
Caustics      200
cells      234Ч235
Cells, averages of functions over      236Ч237
Cells, in plane      235
Characteristic coordinates, introducing      198
Coefficients, slowly varying      117Ч120
Cole Ч Hopf transformation      95
Composite expansion      54Ч55 56 61 63Ч64 72 80 89 101
Computing, symbolic      43Ч45
Confluent hypergeometric functions      81 299Ч302
Connection formulas      176
Convergent asymptotic expansions      12
Corner layers      77Ч81
Corner region      78
Corner-layer expansion      266Ч267
Cutoff frequency      196
Darcy's law      245
Debye Ч Huckel theory      30
Degenerate bifurcation      284Ч285
Difference equations, boundary layers and      98Ч102
Difference equations, multiple scales and      153Ч155
Difference equations, WKB and      212Ч218
Differential equations, approximate solutions of      1
Differential equations, asymptotic approximations of      26Ч31
Differential equations, homogenized      228
Differential equations, ordinary, systems of      287Ч291
Differential-difference equations      75
Diffusion, nonlinear      144Ч148
Diffusivity tensor      248
Dirichlet problem, inhomogeneous      234
Discrete WKB method      212Ч218
Disproportionate length scales      225
Dissipation function      136 186
Distinguished limit      51
Divergent asymptotic expansions      12
Divergent series, summability of      14
Dual-mode sorption model      246
Duffing's equation      35 249
Duffing's equation, forced      123 128
Effective coefficients      228
Eigenvalues of matrix      23
Eikonal equation      163
Elliptic equations      84Ч91
energy equation      136 186 190
Energy methods      186Ч188
Energy methods, wave propagation and      185Ч188
Energy, total      187
Euler buckling load      263
Exchange of stability      257
Expansion, composite      54Ч55
Exponential functions, transcendentally small      11
Exponentially small functions      9
Expression swell, intermediate      44
Extension theorem, Kaplun's      53
Fast dynamics      264Ч265
Fast length scale      225
Fisher Ч Wright Ч Haldane model      158
Fisher's equation      145 278
Flow, porous      241Ч245
Fokker Ч Planck equation      188
Forced motion near resonance      123Ч127
Fredholm alternative theorem      191 273
Fredholm integral equation      33
Functions, asymptotically equal      11 39
Functions, averages of, over cells      236Ч237
Functions, transcendentally small      5
Functions, well-ordered      9
Gauge Functions      9
Geometric mean      232
Ginzburg Ч Landau equation      278
Global asymptotic stability      257
Global bifurcation      277
Hamiltonian function      107 136
Harmonic mean      228 230
Harmonic resonance      127
Helmholtz equation      198
hessian      287
Hidden variables      239
Higher-order turning points      302Ч303
Homogenization      223
Homogenization procedure      236Ч237
Homogenization, method of      223Ч245
Homogenization, method of, averaging in      243Ч244
Homogenization, method of, introductory example      224Ч231
Homogenization, method of, multidimensional problem      234Ч239
Homogenization, method of, reduction using      242Ч243
Homogenization, method of, with nonlinear equations      231
Homogenized coefficients      228
Homogenized coefficients, substructure and      237
Homogenized differential equations      228
Hopf bifurcation      255 284
Hurwitz polynomial      291
Hypergeometric equation      299
Hypergeometric functions, confluent      81 299Ч302
Ignition period      76
Imperfect bifurcation      263
Implicit function theorem      251
Inhomogeneous Dirichlet problem      234
Initial-layer expansion      266
Inner solution      53
Inner-inner layer      65
Instability      257
Integrals, asymptotic approximations of      305Ч307
Interior layer      68Ч73
Interior layer, matching and      69Ч72
Interior-layer coordinate      70 92
Interior-layer expansion      267Ч268
Intermediate variables      53
Intrinsic phase average      245
Kaplun's extension theorem      53
Kaplun's hypothesis on domain of validity      53
Kepler's equation      24Ч25
Klein Ч Gordon equation, nonlinear      138 142 281
Kolmogorov Ч Petrovsky Ч Piskunov (KPP) equation      145 278
Korteweg Ч de Vries (KdV) equation      149
Kummer's function      299Ч300
Kuramoto Ч Sivashinsky equation      279
l'Hospital's rule      4
Lagrangians, averaged      189
Landau symbols      4
Langmuir Ч Hinshelwood model      75
Laplace's approximation      306
Layer analysis, matching and      79Ч80
Least-squares problem      26
Length scales, disproportionate      225
Liebnitz's rule      197
Lindstedt's method      111
Linear stability theory      249
Linear wave propagation      139Ч141
Linearized stability      255Ч259
Linearized stability analysis      274Ч275
Liouville's equation      29
Little oh      4
Logistic equation with delay      286
Lyapunov Ч Schmidt method      252
Maclaurin series      3
Macroscale      225
Maintained vibrations      269Ч270
Maple commands for projectile problem      45
Matched asymptotic expansions      39 47Ч102
Matched asymptotic expansions, introductory example      48Ч57
Matching, boundary layers and      52 64Ч65
Matching, corner layer analysis and      79Ч80
Matching, importance of      56
Matching, interior layer and      69Ч72
Matching, overlap domain and      53
Matching, transition region and      175Ч176
Materials from multiple constituents      223
Mathieu's equation      121 261
Matrix perturbation problems      25Ч26
Matrix, eigenvalues of      23
Matrix, pseudo-inverse of      26
Maxwell's fisheye lens      206
Mercury's perihelion      128
Michaelis Ч Menten reaction scheme      59
Microscale      225
Microvoids      246
Mixture theory      245
Mixtures, rule of      233
Morse oscillator, forced      130
Multiple boundary layers      62Ч65
Multiple constituents, materials from      223
Multiple scales      105Ч155
Multiple scales, boundary layers and      132Ч133
Multiple scales, characteristic coordinates and      140 143
Multiple scales, difference equations and      153Ч155
Multiple scales, first-order systems and      116Ч117
Multiple scales, for traveling waves      139Ч141
Multiple scales, history of      105
Multiple scales, introductory example      106Ч113
Multiple scales, minimum error, principle of      112
Multiple scales, three time scales      110
Multiple solutions of nonlinear equations      70
Multiple-scale expansion      108Ч110
Negative damping      258
No-slip condition      241
Nonlinear boundary-value, problems, numerical solution of      309Ч312
Nonlinear diffusion      144Ч148
Nonlinear geometric optics method      167
Nonlinear wave equation      142Ч144
Nonlinear waves      142Ч148
Nonlinear WKB method      167
Nonuniform asymptotic expansions      39
Numerical solution of nonlinear boundary-value problems      309Ч312
Order symbols      4Ч6
Ordinary differential equations, systems of      287Ч291
Outer expansion      55
Outer scale      53
Outer solution      49Ч50 62 68 78 85 91 265
Parabolic boundary layers      90Ч91
Parabolic equations (PE)      91Ч94
Parabolic equations (PE), standard two-dimensional      209Ч211
Parabolic wave approximations      207Ч211
Paraxial approximation      209
Paul trap      115 142
Pe      see УParabolic equationsФ
Pendulum      114 121 262Ч263
Pendulum, elastic      294
Pendulum, forced      129 131
Pendulum, inverted      261Ч262
Periodic solutions, bifurcation of      281Ч285
Periodic substructure      231 234Ч239
Permeability tensor      245
Perturbation problems, matrix      25Ч26
Perturbation problems, regular      26
Perturbation problems, singular      48
Phase average      245
Phase fronts      198
Pitchfork bifurcation      250
Poincare Ч Andronov Ч Hopf bifurcation theorem      291
Poisson Ч Boltzmann equation      29
Porous flow      25 241Ч245
Power series functions, generalization of      9
Primary resonance      127
Projectile problem      1Ч3 27Ч28
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