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Bender C., Orszag S. — Advanced Mathematical Methods for Scientists and Engineers
Bender C., Orszag S. — Advanced Mathematical Methods for Scientists and Engineers



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Íàçâàíèå: Advanced Mathematical Methods for Scientists and Engineers

Àâòîðû: Bender C., Orszag S.

Àííîòàöèÿ:

This book gives a clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory and explains how to use these methods to obtain approximate analytical solutions to differential and difference equations. These methods allow one to analyze physics and engineering problems that may not be solvable in closed form and for which brute-force numerical methods may not converge to useful solutions. The objective of this book is to teaching the insights and problem-solving skills that are most useful in solving mathematical problems arising in the course of modern research. Intended for graduate students and advanced undergraduates, the book assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations; develops local asymptotic methods for differential and difference equations; explains perturbation and summation theory; and concludes with a an exposition of global asymptotic methods, including boundary-layer theory, WKB theory, and multiple-scale analysis. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach the reader how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions; over 600 problems, of varying levels of difficulty; and an appendix summarizing the properties of special functions.


ßçûê: en

Ðóáðèêà: Ìàòåìàòèêà/Àíàëèç/Àñèìïòîòè÷åñêèå ìåòîäû, Òåîðèÿ âîçìóùåíèé/

Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö

ed2k: ed2k stats

Ãîä èçäàíèÿ: 1978

Êîëè÷åñòâî ñòðàíèö: 593

Äîáàâëåíà â êàòàëîã: 26.03.2005

Îïåðàöèè: Ïîëîæèòü íà ïîëêó | Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
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Ïðåäìåòíûé óêàçàòåëü
Laplace integrals, conversion of Fourier integrals into      281
Laplace integrals, definition of      258 261
Laplace integrals, integration by parts for      258—259
Laplace integrals, Watson’s lemma for      263—265 281 284
Laplace’s method      261—276
Laplace’s method for integrals with movable maxima      274—276 286 309—311p
Laplace’s method, applied to sums      304—305
Laplace’s method, approximation explained carefully      269—272 270f 271f
Laplace’s method, failure of      296—297 300
Laplace’s method, finding higher-order behaviors by      272—274 280
Laplace’s method, finding leading asymptotic behaviors by      261—263 266—272
Laplace’s method, Watson’s lemma      263—265 281 284
Leading asymptotic behaviors of Airy functions      100 101f 102f 107—108 108f 109f 113 115—117
Leading asymptotic behaviors of Bessel functions      111 280 291 293—294 298—299 309p 312p
Leading asymptotic behaviors of modified Bessel functions      91—92 92f 268 270—272 310p 313p
Leading asymptotic behaviors of Pade approximants      402—405
Leading asymptotic behaviors of parabolic cylinder functions      97 117 134—136 142—143p
Leading asymptotic behaviors of sums      303—305
Leading asymptotic behaviors of Taylor series      90—92 304—305
Leading asymptotic behaviors, comparison of, from difference and differential equations      207—210 214—216 237
Leading asymptotic behaviors, definition of      79—80
Leading asymptotic behaviors, derived by asymptotic matching      336—349
Leading asymptotic behaviors, derived by boundary-layer theory      422—423 425 437—440 455—477
Leading asymptotic behaviors, derived by elementary methods for integrals      249—250 253 258—259
Leading asymptotic behaviors, derived by Laplace’s method      262—263 266—272
Leading asymptotic behaviors, derived by multiple-scale analysis      548—560 563—564
Leading asymptotic behaviors, derived by stationary phase, method of      278—280 311—312p
Leading asymptotic behaviors, derived by steepest descents, method of      283 287 294—300
Leading asymptotic behaviors, derived by WKB theory (physical optics)      487—533
Leading asymptotic behaviors, derived from difference equations, effect of initial data on      248—249
Leading asymptotic behaviors, derived from difference equations, near irregular singular points      223—233
Leading asymptotic behaviors, derived from difference equations, near ordinary and regular singular points      207—210
Leading asymptotic behaviors, derived from difference equations, nonlinear      233—239
Leading asymptotic behaviors, derived from differential equations, effect of initial data on      247—248
Leading asymptotic behaviors, derived from differential equations, higher-order corrections to      84—87 92—94 96—102
Leading asymptotic behaviors, derived from differential equations, near irregular singular points      79—88 90—92 96—97 100 102—103
Leading asymptotic behaviors, derived from differential equations, nonlinear      149—152 154 157—158 166—167 171
Legendre polynomials $P_n(x)$, behavior of $(n\rightarrow\infty)$      229 231 232t 309p
Legendre polynomials $P_n(x)$, difference equation for      55p 229
Legendre polynomials $P_n(x)$, differential equation for      243p
Legendre polynomials $P_n(x)$, generating function for      55p
Legendre polynomials $P_n(x)$, integral representation for      309p
Legendre polynomials $P_n(x)$, zeros of      231 232t
Level crossing of eigenvalues for finite matrices      350
Level crossing of eigenvalues for Schrodinger equation eigenvalue problem      354—355 356—358f 360—361
Level crossing of eigenvalues, Simon, theorem by, on      363
Level curves      33p
Limit cycles in two dimensions      183 184f
Limit cycles of Rayleigh oscillator      468—479 476—478f 483p 554—556 556f 557f
Limit cycles of Van der Pol equation      483p 567p
Limit cycles, period of      476—479
Linear dependence      see “Linear independence”
Linear difference equations      see “Difference equations”
Linear differential equations, Airy      13 28 67—68 116—118 130—131 156 494—495 506 521—522 569
Linear differential equations, asymptotic matching for      336—341 (See also “Asymptotic matching”)
Linear differential equations, Bessel      14 111—112 143p 322 572 573
Linear differential equations, boundary-layer solutions of      419—463 479—482p
Linear differential equations, boundary-value problems for      see “Boundary-value problems”
Linear differential equations, constant-coefficient      11—12 32p 545
Linear differential equations, controlling factors from      see “Controlling factors”
Linear differential equations, definition of      3 4
Linear differential equations, eigenvalue-problems      see “Eigenvalue problems”
Linear differential equations, equidimensional (Euler)      12 32p
Linear differential equations, exact      13
Linear differential equations, factoring linear differential operators      22 33p 34p
Linear differential equations, Floquet theory      560—561 568p
Linear differential equations, for Stieltjes function      77—78
Linear differential equations, Frobenius series for      see “Frobenius series”
Linear differential equations, general solution of      8
Linear differential equations, Green’s functions      16—19 32p 498—504 540—541p
Linear differential equations, homogeneous      see “Homogeneous linear differential equations”
Linear differential equations, hyperairy equation      102—103 143—144p 315p 497
Linear differential equations, hypergeometric      35p
Linear differential equations, inhomogeneous      see “Inhomogeneous linear differential equations”
Linear differential equations, inhomogeneous Airy      105 106f 141—142p
Linear differential equations, initial-value problems for      see “Initial-value problems”
Linear differential equations, integrating factors for      13 14
Linear differential equations, Langer’s uniform approximation      510—511 541p
Linear differential equations, leading asymptotic behaviors from      see “Leading asymptotic behaviors”
Linear differential equations, Legendre      55p 229
Linear differential equations, local analysis of      61—145
Linear differential equations, Mathieu      339 560—566 568p
Linear differential equations, modified Bessel      92—96 570
Linear differential equations, multiple-scale perturbation theory for      556—566
Linear differential equations, ordinary, regular singular, and irregular singular points of      see “Ordinary points; Singular points”
Linear differential equations, parabolic cylinder (Weber — Hermite) equation      see “Parabolic cylinder equation”
Linear differential equations, patching      335—336 497—505 527
Linear differential equations, periodic behaviors of      see “Oscillators; Periodic behaviors”
Linear differential equations, quantum anharmonic oscillator      see “Anharmonic oscillator”
Linear differential equations, quantum harmonic oscillator      see “Harmonic Oscillator”
Linear differential equations, reduction of order for      13 19 32p 333
Linear differential equations, resonant behavior of      544—545
Linear differential equations, scattering      524—533 542—543p
Linear differential equations, Schrodinger      see “Schrodinger equation”
Linear differential equations, stability      see “Stability”
Linear differential equations, Stokes phenomenon      115—118 130—133
Linear differential equations, Sturm — Liouville problem      29 35p 351 490—492 540p
Linear differential equations, systems of      66 177—178
Linear differential equations, Taylor series for      see “Taylor series”
Linear differential equations, undetermined coefficients, method of, for      19—20
Linear differential equations, variation of parameters for      15 18—19 32p 103—104 201p 498
Linear differential equations, WKB theory for      484—543
Linear differential equations, Wronskian      8—11 15 31p 32p
Linear independence of continuous functions      8—9
Linear independence of discrete functions      41—42 55p
Linear independence, Wronskian test of      8 41—42 55p
Lipschitz condition      30 31
Local analysis      61—316
Local analysis of difference equations      205—246
Local analysis of integrals      247—316
Local analysis of linear differential equations      61—145
Local analysis of nonlinear differential equations      146—204
Local analysis of systems of differential equations      171—197
Local analysis, philosophy of      6—7 59 317—318 324 499
Local breakdown at boundary layers      484—485
Logarithmic integral function li (x)      310p
Logarithmic spirals      24 33p
Lorenz model      194—195 194—197f 204p
Matched asymptotic expansions      see “Asymptotic matching”
Matching regions for one-turning point problems      506—508 (see also “Asymptotic matching”)
Matching regions, existence of      428—430
Matching regions, extent of, in higherrorder      336—339 342—349 432—433
Mathieu equation      339 560—566 568p
Matrix algebra, elementary techniques      177—178 201p
Modified Bessel equation, arbitrary order      92—96 570
Modified Bessel equation, half-odd integer order      73 93
Modified Bessel equation, imaginary order      340
Modified Bessel equation, integer order      74—76 90—91 92f
Modified Bessel equation, local analysis near $x = \infty$      90—97
Modified Bessel equation, local analysis near x = 0      71 73—76
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$ as weight function in Stieltjes series      122
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$ in asymptotic matching      340 455
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, , leading asymptotic behaviors, $I_{\nu}(x \rightarrow +\infty)$      91 92f 270—272 310p
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, , leading asymptotic behaviors, $K_{\nu}(x \rightarrow +\infty)$      268
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, , leading asymptotic behaviors, $K_{\nu}(x \rightarrow +\infty, |\nu| \rightarrow +\infty)$      268 313p
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, , leading asymptotic behaviors, $K_{\nu}(\nu \rightarrow +\infty)$      310p
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, asymptotic series, $(x \rightarrow +\infty)$      93—94 144p 265
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, asymptotic series, $(|z| \rightarrow +\infty)$      571
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, definitions of      71 74
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, difference equation for      571
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, Frobenius series for $I_{\nu}$      71 570
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, Frobenius series for for $K_{n}$      74 76 342 571
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, Frobenius series for for $K_{\nu}$      570
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, functional relations for      143p 571
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, generating function for      572
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, integral representations for      268 270 313p 571
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, optimal asymptotic approximations to      95 95t 96f 97f 243p
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, proof of existence of asymptotic expansion for $K_{\nu}$      129 145p
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, Richardson extrapolation of recurrence relation for      376 378t
Modified Bessel functions $I_{\nu}$ and $K_{\nu}$, Stokes phenomenon for      143p
Moment integrals      121—122 405
Moment problem      410 414—415p
Movable maxima, Laplace’s method for      274—276 286 309—311p
Movable singular points      see “Spontaneous singular points”
Multiple-scale perturbation problems, boundary-layer problem solved as      559—560
Multiple-scale perturbation problems, damped harmonic oscillator      552—553 554f 555f
Multiple-scale perturbation problems, Duffing’s equation      545—551 547f 551f 552f
Multiple-scale perturbation problems, harmonic oscillator with aging spring      327—328 329f
Multiple-scale perturbation problems, Mathieu equation      560—566 568p
Multiple-scale perturbation problems, oscillator with slowly varying frequency      556 558—559
Multiple-scale perturbation problems, Rayleigh oscillator      554—556 556f 557f
Multiple-scale perturbation theory      544—568
Multiple-scale perturbation theory, averaging, method of      566p
Multiple-scale perturbation theory, boundary-layer approximation derived by      559—560
Multiple-scale perturbation theory, failure of (Cheng and Wu example)      568p
Multiple-scale perturbation theory, formal procedures of      549—551
Multiple-scale perturbation theory, introductory discussion      327—328 329f 417 544
Multiple-scale perturbation theory, nonuniformity of regular perturbation expansions in      545—546
Multiple-scale perturbation theory, resonant and secular behavior      544—545
Multiple-scale perturbation theory, summation of leading secularities      547—549
Multiple-scale perturbation theory, time scales      549 551 556 558—559 564 567p
Multiple-scale perturbation theory, WKB approximation derived by      556 558—559
Natural frequency      544
Newton’s method      234—235 244p 245p
Nodes, one-dimensional      174—175 174f
Nodes, three-dimensional      186 187f
Nodes, two-dimensional      176—177 181 182f
Nonlinear difference equations      see “Difference equations”
Nonlinear differential equations, Abel      34p
Nonlinear differential equations, Arnold — Moser theorem      189 191
Nonlinear differential equations, asymptotic matching for      421—423 463—479
Nonlinear differential equations, autonomous      24—26 156
Nonlinear differential equations, autonomous systems of      171—197 (see also “Autonomous systems; Critical point analysis”)
Nonlinear differential equations, Bernoulli      20 21 24
Nonlinear differential equations, Blasius      34p
Nonlinear differential equations, boundary-layer solutions of      421—423 463—479 483p
Nonlinear differential equations, boundary-value problems for      see “Boundary-value problems”
Nonlinear differential equations, C-systems      192—193 204p
Nonlinear differential equations, critical points of      173—195
Nonlinear differential equations, Duffing’s      545—552 566—567p
Nonlinear differential equations, Emden (Prob. 1.37)      35p
Nonlinear differential equations, equidimensional (Euler)      4 25 27 156
Nonlinear differential equations, equivalence of, to a first-order system      5
Nonlinear differential equations, exact      23 33p
Nonlinear differential equations, general solution of      152—154
Nonlinear differential equations, Hamiltonian systems      189—191 204p
Nonlinear differential equations, Henon and Heiles, example of      188—190
Nonlinear differential equations, initial-value problems for      see “Initial-value problems”
Nonlinear differential equations, integrating factors for      23 34p
Nonlinear differential equations, leading asymptotic behaviors from      see “Leading asymptotic behaviors”
Nonlinear differential equations, local analysis of      146—204
Nonlinear differential equations, Lorenz model      194—197 204p
Nonlinear differential equations, multiple-scale perturbation theory for      545—557
Nonlinear differential equations, Painleve transcendents      158—165 198—199p
Nonlinear differential equations, periodic behavior of      see “Oscillators; Periodic behaviors”
Nonlinear differential equations, Poincare plot      188—190
Nonlinear differential equations, predator-prey systems      179—183
Nonlinear differential equations, random behavior of      188—197
Nonlinear differential equations, Rayleigh oscillator      468—479 483p 554—557
Nonlinear differential equations, reduction of order for      24—27 34p 35p 156
Nonlinear differential equations, Riccati      4 20—22 27 34p 149—152 150f 152f 153t 156 197—198p
Nonlinear differential equations, scale-invariant      25—26 156
Nonlinear differential equations, separable      3 23 24 30p
Nonlinear differential equations, singular points of      see “Singular points”
Nonlinear differential equations, stability and      see “Stability”
Nonlinear differential equations, substitutions for solving      23—24 151 156—157
Nonlinear differential equations, Taylor series for      see “Taylor series”
Nonlinear differential equations, Thomas-Fermi      25—26 167—170 168—169f 170f 199—200p
Nonlinear differential equations, Toda lattice      187—188 190 203p
Nonlinear differential equations, Van der Pol      202p 483p 567p
Nonuniformity of regular perturbation series      327—328 545—549
One-turning-point problems for first-order (simple) turning point      504—519 511—514f 529—530
One-turning-point problems for second-order turning point      532—533 541p
One-turning-point problems, connection formula      509 511—513 529—530
One-turning-point problems, higher-order WKB analysis of      534—536
One-turning-point problems, normalization of solution to      513—519
One-turning-point problems, two-turning-point problems derived from      519—520 530—531
One-turning-point problems, uniform asymptotic approximation for      510—511
One-turning-point problems, validity of WKB approximation for      506—508
Optimal asymptotic approximations for Bessel functions      112 112f 113
Optimal asymptotic approximations for modified Bessel functions      94—96 95f 96f 97f 243p
Optimal asymptotic approximations for parabolic cylinder functions      98f 99—100 99f 100 243p
Optimal asymptotic approximations for Riemann zeta function      376 379 380t
Optimal asymptotic approximations for Stirling series      221t 222 223f 224t 242p
Optimal asymptotic approximations, definition of      94
Optimal asymptotic approximations, error estimates for      123 124f
Optimal asymptotic approximations, Pade approximants vs.      388—392
Optimal asymptotic approximations, use of      94—100 122—124 124f
Ordinary points of difference equations, definition of      207
Ordinary points of difference equations, leading asymptotic behavior near      207—210
Ordinary points of difference equations, Taylor series near      210—212
Ordinary points of differential equations at $\infty$      64
Ordinary points of differential equations, definition of      62
Ordinary points of differential equations, Frobenius series near      73
Ordinary points of differential equations, Taylor series near      62 65—70
Orthogonal families of curves      33p
Oscillators with slowly varying frequency      556 558—559 568p
Oscillators, anharmonic      see “Anharmonic oscillator”
Oscillators, harmonic      see “Harmonic oscillator”
Oscillators, Rayleigh      468—479 476—478f 483p 554—556 556f 557f
Oscillators, Van der Pol      202p 483p 567p
Oscillatory behaviors      see “Periodic behaviors”
Outer, approximation, in boundary-layer theory      421—484 488
Outer, limit, defined      427
Outer, region, characteristics of      420 420f 423—424
Outer, solution, in boundary-layer theory      421—484 (see also “Boundary-layer theory”)
Pade approximants      383—410
Pade approximants for $1/\Gamma$      385 387t
Pade approximants for $e^z$      384—385 385 386t 414p 416p
Pade approximants for $z^{-1}ln (1 + z)$      385 388 409 412p
Pade approximants for parabolic cylinder functions      389 392
Pade approximants for Stieltjes series      387—388 390 404—410 415—416p
Pade approximants for Stirling series      388—389 391
Pade approximants, continued fractions and      396—398
Pade approximants, convergence of      400—410
Pade approximants, definition of      383—384
Pade approximants, diagonal sequence of      383
Pade approximants, distribution of zeros and poles      384—385 386t 388t 412p 416p
Pade approximants, generalized (two-point)      393—395 394f 414p
Pade approximants, generalized Shanks transformation and      389—392 412p
Pade approximants, monotonicity of      407—409
Pade approximants, normal sequence of      396—398
Pade approximants, Taylor series vs.      384—387 385—388 400
Pade summation      383—410 (see also “Pade approximants”)
Painleve transcendents, first, asymptotic behavior $(x \rightarrow +\infty)$      158—161 161f 162
Painleve transcendents, first, asymptotic behavior $(x \rightarrow -\infty)$      161 163—164 164f 165f
Painleve transcendents, first, numerical solutions to      198—199p
Painleve transcendents, second      199p 200p
Parabolic cylinder equation, differential equation      14 337 573
Parabolic cylinder equation, eigenvalue problem for (quantum harmonic oscillator)      28 133 332—333 522—523 538
Parabolic cylinder equation, internal boundary layers described by      456—463 482p
Parabolic cylinder equation, local analysis $(x \rightarrow +\infty)$      96—100 495
Parabolic cylinder equation, regular perturbation of      334 335 354—355 355 356—358f
Parabolic cylinder equation, second-order one-turning-point problem described by      532
Parabolic cylinder equation, singular perturbation of      334—335 337—338 353—354
Parabolic cylinder functions, asymptotic series, $(x \rightarrow +\infty)$      98—100 265
Parabolic cylinder functions, asymptotic series, $(x \rightarrow -\infty)$      131—133
Parabolic cylinder functions, asymptotic series, $(|z| \rightarrow \infty)$      131—136 574
Parabolic cylinder functions, definition of      97
Parabolic cylinder functions, difference equation for      574
Parabolic cylinder functions, functional relations for      132 574
Parabolic cylinder functions, integral representations for      315p 574
Parabolic cylinder functions, leading asymptotic behaviors, $(x \rightarrow +\infty)$      97
Parabolic cylinder functions, leading asymptotic behaviors, $(\nu \rightarrow +\infty)$      315p
Parabolic cylinder functions, leading asymptotic behaviors, $(|z| \rightarrow \infty)$      117 134—136 142—143p
Parabolic cylinder functions, optimal asymptotic approximations for      98f 99—100 99f 100t 243p
Parabolic cylinder functions, Pade approximants for      389 392t
Parabolic cylinder functions, relation to Hermite polynomials      99 133 574
Parabolic cylinder functions, Stokes behavior of      117 131—133
Parabolic cylinder functions, Taylor series for      574
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