Bender C., Orszag S. — Advanced Mathematical Methods for Scientists and Engineers :: Электронная библиотека попечительского совета мехмата МГУ

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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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Название: Advanced Mathematical Methods for Scientists and Engineers

Авторы: Bender C., Orszag S.

Аннотация:

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.

Язык:

Рубрика: Математика/Анализ/Асимптотические методы, Теория возмущений/

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Год издания: 1978

Количество страниц: 593

Добавлена в каталог: 26.03.2005

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Предметный указатель

$\pi$ , calculation of      246p 411p
Abel equation      34p
Abel formula for Wronskian for difference equations      42 43
Abel formula for Wronskian for differential equations      9 32p 33p
Abelian theorems      127
Airy equation, differential equation      13 151 156 506 569
Airy equation, eigenvalue problem for      28 521—522
Airy equation, inhomogeneous      105 106f 141—142p
Airy equation, local analysis, $(x \rightarrow +\infty)$       100—102 494—495
Airy equation, local analysis, $(x \rightarrow -\infty)$       107—111
Airy equation, local analysis, $(x \rightarrow 0)$       67—68
Airy equation, Stokes lines for      116—118 130—131
Airy functions Ai and Bi as Bessel functions      569
Airy functions Ai and Bi in one-turning-point problems      506—510 515—518 529—530 535—536
Airy functions Ai and Bi, asymptotic series, $(x \rightarrow +\infty)$       100—102
Airy functions Ai and Bi, asymptotic series, $(x \rightarrow -\infty)$       109—111
Airy functions Ai and Bi, asymptotic series, $(|z| \rightarrow \infty)$       133—135 142p 570
Airy functions Ai and Bi, definition of      68 69f
Airy functions Ai and Bi, functional relations for      131 569
Airy functions Ai and Bi, integral representations of      313—314p 570
Airy functions Ai and Bi, integrals of, behavior      256—258 308p 309p 364p
Airy functions Ai and Bi, leading asymptotic behaviors, $(x \rightarrow +\infty)$       100 101f 102
Airy functions Ai and Bi, leading asymptotic behaviors, $(x \rightarrow -\infty)$       107—108 108f 109f
Airy functions Ai and Bi, leading asymptotic behaviors, $(|z| \rightarrow \infty)$       113—117 133—135
Airy functions Ai and Bi, sector of validity for Bi      115 115f
Airy functions Ai and Bi, Stokes behavior of      130—131 313—314p
Airy functions Ai and Bi, Taylor series for      67—68 569
Almost periodic orbits      186—187 188f 189f
Anharmonic oscillator, classical (Duffing’s equation)      545—551 547f 551f 566—567p
Anharmonic oscillator, quantum      334—335 337—338 353 353t 359—360 366—367p
Anti-Stokes lines      116 (See also “Stokes lines”)
Antiderivative, continuous      3
Antiderivative, discrete      37 53p 54p
Approximants      381 (See also “Pade approximants”)
Arnold — Moser theorem      189 191
Asymptotic expansions      see “Asymptotic series”
Asymptotic matching for boundary-layer problems      421—483
Asymptotic matching for integrals      341—349 365—366p 515—519
Asymptotic matching for nonlinear differential equations      421—423 463—479
Asymptotic matching for nonlinear differential equations for WKB problems      504—543
Asymptotic matching for nonlinear differential equations, patching vs.      335—336 499
Asymptotic matching for nonlinear differential equations, principle of      335—336
Asymptotic matching for nonlinear differential equations, techniques of, for differential equations      336—341
Asymptotic matching, existence of matching region      428—430
Asymptotic matching, high-order      337 344—349 428—435 434—437f 438—452 442—444f 449f 478—479 535—536
Asymptotic power series      see “Asymptotic series”
Asymptotic relations (asymptotic order relations) for functions with noncoincident zeros      108—109
Asymptotic relations (asymptotic order relations) for oscillatory functions      107—112
Asymptotic relations (asymptotic order relations) in complex plane      112—118
Asymptotic relations (asymptotic order relations), definition of, $\ll$       78—79
Asymptotic relations (asymptotic order relations), definition of, $\sim$       78—79
Asymptotic relations (asymptotic order relations), definition of, O      318
Asymptotic relations (asymptotic order relations), differentiation of      127—128 145p
Asymptotic relations (asymptotic order relations), integration of      81 139p 142p
Asymptotic relations (asymptotic order relations), sector of validity for      113—118
Asymptotic relations (asymptotic order relations), Stokes phenomenon and subdominance      115—118
Asymptotic series for Airy functions      100—102 109—111 133—135 142p 570
Asymptotic series for Bessel functions      111—112 112f 228—229 294 572—573
Asymptotic series for digamma function      309p 575
Asymptotic series for exponential integrals      576
Asymptotic series for Fourier integrals      276—277 281—287
Asymptotic series for logarithmic integral function      310p
Asymptotic series for modified Bessel functions      93—94 144p 265 571
Asymptotic series, arithmetical operations on      125—126
Asymptotic series, construction of analytic function asymptotic to      144p
Asymptotic series, construction of continuous function asymptotic to      119—120 119f
Asymptotic series, convergent and divergent      118—120
Asymptotic series, definition of      89 118
Asymptotic series, derived by integration by parts      252—261 276—277
Asymptotic series, derived by Laplace’s method and Watson’s lemma      262—265
Asymptotic series, derived by steepest-descents method      281—287 294
Asymptotic series, derived from difference equations      218—219 225—231
Asymptotic series, derived from linear differential equations      84—87 90—94 98—101 104—105 109—111 129 133—136
Asymptotic series, derived from nonlinear differential equations      150 154—155 157—158 165—167
Asymptotic series, differentiation of      127—128
Asymptotic series, equating coefficients in      125
Asymptotic series, integration of      126
Asymptotic series, nonuniqueness of (subdominance)      123—124
Asymptotic series, optimal asymptotic approximation for      94—102 122—124 222 223f 224t 242
Asymptotic series, pfor parabolic cylinder functions      98—100 131—133 265 574
Asymptotic series, proof of existence of      128—129 133—136 144
Asymptotic series, Stieltjes series as      78 120—123 260
Asymptotic series, Stirling series for gamma function      218—220
Asymptotic series, Taylor series vs.      90—91 92f 118—119 220 221t 222 254f 255
Asymptotic series, uniqueness of coefficients      89—90 124—125
Asymptotic summation of series      376 379 380t
Autonomous equations, definition of      24
Autonomous equations, reduction of order for      24—26 156
Autonomous systems      171—197
Autonomous systems with almost periodic orbits      186—187 188f 189f
Autonomous systems, C-systems      192—193 204p
Autonomous systems, critical points of      173—177
Autonomous systems, definition of      171
Autonomous systems, Duffing’s equation      546 547f
Autonomous systems, Hamiltonian      189—191 189f 190f 204p
Autonomous systems, Henon and Heiles, example of      188—190 189f 190f
Autonomous systems, higher-dimensional      185—197
Autonomous systems, linear      177—178
Autonomous systems, Lorenz model      194—195 194—197f 204p
Autonomous systems, one-dimensional      174—175
Autonomous systems, predator-prey      179—183 180—182f
Autonomous systems, random behavior in      188—189 190f 191f 192—195 195—197f
Autonomous systems, Rayleigh oscillator      468—479 476—478f 483p 554—556 556f 557f
Autonomous systems, Toda lattice      187 188f 190 203p
Autonomous systems, two-dimensional      175—185 468—479 546—547
Autonomous systems, Van der Pol equation      202p 483p 567p
Averaging, method of      566p
Bernoulli equations, difference      57p
Bernoulli equations, differential      20 21 24
Bernoulli numbers      242p 305 306 314p 379
Bernoulli polynomials      305 315p
Bernoulli’s method for finding roots of a polynomial      241p
Bessel equation, differential equation      14 111—112 143p 572
Bessel equation, differential equations equivalent to      322 573
Bessel equation, local analysis $(x \rightarrow +\infty)$       111—112
Bessel equation, modified      see “Modified Bessel equation”
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , Airy functions as      569
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , asymptotic series, $(x \rightarrow +\infty)$       111—112 112f294
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , asymptotic series, $(\nu \rightarrow +\infty)$       228—229 230t
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , asymptotic series, $(|z| \rightarrow \infty)$       572—573
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , difference equation for      55p 228 243p 573
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , differential equations for      322 572 573
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , Frobenius series for      572
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , functional relations for      143p 572
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , generating function for      55p 573
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , integral representation for      280 291 293 298 309p 312p 573
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , leading asymptotic behaviors, $(x \rightarrow +\infty)$       111 291 293—294 309p
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , leading asymptotic behaviors, ( $(x \rightarrow +\infty)$ and $(\nu \rightarrow +\infty)$ )      280 298—299 312p
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , modified      see “Modified Bessel functions”
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , optimal asymptotic approximation to $J_{5}$       112 112f 113t
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , Taylor series for      143p
Bessel functions $J_{\nu}$ and $Y_{\nu}$ , uniform WKB approximation to      541p
Binomial theorem      204f 419
Blasius equation      34p
Borel summation      381—383 406 412p
Borel summation, generalized      382
Boundary layers, absence of      452
Boundary layers, definition of      326 327f 419—420 420f
Boundary layers, internal      455—463 466—468 469—475f 485 541—542p
Boundary layers, local breakdown at      484—485
Boundary layers, location of      425—426
Boundary layers, mathematical structure of      426—430
Boundary layers, multiple      437—438 439f" 440f" 446—455 449f" 458—460 464—475 465f 467—478f 485
Boundary layers, nested      453—455 481p
Boundary layers, thicknesses of $\delta = O(\varepsilon)$       420 426 431 450 454 463 473 477
Boundary layers, thicknesses of $\delta = O(\varepsilon\ln\varepsilon)$       421
Boundary layers, thicknesses of $\delta = O(\varepsilon^2)$       453 454
Boundary layers, thicknesses of $\delta = O(\varepsilon^{1/2})$       437—439 442—443 447 456 465 466
Boundary layers, thicknesses of $\delta = O(\varepsilon^{2/3})$       479 480p
Boundary layers, thicknesses of how to determine      435—437

Boundary-layer problems with coordinate singularity      452—453
Boundary-layer problems, absence of boundary layers      452
Boundary-layer problems, Carrier’s problem      464—475
Boundary-layer problems, exactly-soluble      326 327f 419—420 426—430
Boundary-layer problems, internal      455—463 466—475 541—542p
Boundary-layer problems, limit cycle of Rayleigh oscillator      468—479 483p
Boundary-layer problems, linear, fourth-order      449—452
Boundary-layer problems, linear, second-order      326 327f; 424f; 488—490 559—560
Boundary-layer problems, linear, third-order      446—449
Boundary-layer problems, mathematical structure of      426—430
Boundary-layer problems, nonlinear, first-order      421—422 422f 423f
Boundary-layer problems, nonlinear, second-order      463—479 483p
Boundary-layer theory      417—483
Boundary-layer theory as singular perturbation theory      326 430
Boundary-layer theory for nonlinear equations      421—422 422—423f 463—479 483p
Boundary-layer theory in higher order      428—435 434—437f 440—452 442—444f 449f 478—479
Boundary-layer theory with underdetermined or overdetermined solutions      461—467 465f 467—475f 482p
Boundary-layer theory, failure of      461—463 467 485 487
Boundary-layer theory, inner, outer, and intermediate limits in      426—430
Boundary-layer theory, internal boundary layers      455—463 466—475
Boundary-layer theory, introductory discussion      326 327f 417—426
Boundary-layer theory, multiple-scale theory vs.      559—560
Boundary-layer theory, uniform approximation in      425 430 433
Boundary-layer theory, WKB theory vs.      484—485 488—490 498 503—505
Boundary-value problems, for difference equations, eigenvalue      47—49 56p 231—233
Boundary-value problems, for difference equations, linear      39—40
Boundary-value problems, for difference equations, local analysis of      231—233
Boundary-value problems, for difference equations, well-posed      43 55p
Boundary-value problems, for differential equations, asymptotic matching for      336—341
Boundary-value problems, for differential equations, boundary-layer structure in linear      326 327f 419—420 422—463 488—490
Boundary-value problems, for differential equations, boundary-layer structure in nonlinear      463—479
Boundary-value problems, for differential equations, definition of      6—7
Boundary-value problems, for differential equations, eigenvalue      see “Eigenvalue problems”
Boundary-value problems, for differential equations, existence and uniqueness of solutions to      7 10—11
Boundary-value problems, for differential equations, Green’s function solution of      19
Boundary-value problems, for differential equations, local analysis of linear      133 248
Boundary-value problems, for differential equations, local analysis of nonlinear      167—172
Boundary-value problems, for differential equations, multiple-scale analysis of      559—560
Boundary-value problems, for differential equations, patching used to solve      335—336
Boundary-value problems, for differential equations, well-posed      10—11
Boundary-value problems, for differential equations, WKB analysis of      326—327 328f 485 487—492 497—505 511 511—514f 518t 519—523 537—539
C-systems      192—193 204p
Carleman condition      410 415p 416p
Carlini — Green — Liouville approximation      see “Exponential substitution”
Casoratian      see “Wronskian”
Centers, almost periodic behavior near      186—187 188f 189f
Centers, higher-dimensional      186—188
Centers, random behavior near      188—189 190f 191f 192—195 195—197f
Centers, random behavior near spurious      184—185
Centers, two-dimensional      173f 177—179 185
Completeness      351
Complex plane, asymptotic relations in      112—118
Complex plane, asymptotic series in      130—136
Complex plane, differential equations in      29—30
Complex plane, Pade approximants in      383—410
Complex plane, physical optics approximation in      54lp 542p
Complex plane, spontaneous singular points in      149
Complex plane, steepest descents, method of, in      280—302
Complex plane, WKB theory in      54lp 542p
Connection formulas      509 511—513 529—530
Constant-coefficient equations, difference      40—41 54—55p
Constant-coefficient equations, differential      11—12 32p 545
Constant-phase paths      282—287 (See also “Steepest descents method
Constants of integration      3 5—6
Constants of summation      37 38
Continued fractions for Stieltjes function      396 406—407
Continued fractions, algorithm for computing coefficients of      397—398
Continued fractions, algorithm for evaluating      398—399
Continued fractions, definition of      395
Continued fractions, examples of      395—396 413p
Continued fractions, Pade approximants and      396—398
Continued function representations, algorithms for computing      399—400 413p
Continued function representations, exponentials      147 196p 399 413p 414p
Continued function representations, fractions      395—400 413p
Continued function representations, logarithms      399 413p 414p
Continued function representations, square roots      245p 399 413p
Controlling factors, from difference equations of gamma function      223
Controlling factors, from difference equations, controlling factors of differential equations vs.      214—215
Controlling factors, from difference equations, dominant-balance method used to find      209 214 218 229 232—233
Controlling factors, from difference equations, methods for finding      214—218
Controlling factors, from differential equations      see “Leading asymptotic behaviors; Dominant balance method
Controlling factors, from differential equations, definition of      79—80
Controlling factors, from differential equations, exponential substitution used to find      80—81 84 87 88 90—91 97
Controlling factors, from differential equations, geometrical optics approximation and      494
Convergence, acceleration of      369—376
Convergence, acceleration of, asymptotic summation of series      376 379 380t
Convergence, acceleration of, generalized Shanks transformation      389—392
Convergence, acceleration of, Richardson extrapolation      375—376 377t 378t
Convergence, acceleration of, Shanks transformation      369—375
Cramer’s Rule      9 352
Critical points in phase space, centers      173f 177—179 180f 185—188
Critical points in phase space, definition of      173
Critical points in phase space, nodes      174—175 174f 176 177 181 182f 186 187f
Critical points in phase space, saddle points      174—175 174f 177 179—182 180—182f
Critical points in phase space, spiral points      176—179 181—184 181f 182f 184f 192—194f
Critical points in phase space, stability of      174—177 186 189 191
Critical-point analysis in phase space, one-dimensional      174—176
Critical-point analysis of C-systems      192—193 204p
Critical-point analysis of Hamiltonian systems      189—191 204p
Critical-point analysis of random systems      193—195 204p
Critical-point analysis of random systems, higher-dimensional      185—197 (See also “Autonomous systems; Critical points”)
Critical-point analysis of random systems, two-dimensional      175—183
Critical-point analysis, Arnold — Moser theorem      189 191
Critical-point analysis, energy integrals used in      185
Critical-point analysis, failure of linear      183—185
Critical-point analysis, linear      174—183
Crossing of eigenvalues      see “Level crossing”
Crossing of roots of polynomials      328—329 330f 331f 332f 363p
Cycles for difference equations      238 246p
Cycles in phase space      173f 175
Delta function $\delta(x)$       16 32p 310p 498 526—527 540—541p
Delta function potential      526—527
Determinants for Pade approximants      384 391
Determinants of N x N matrices      352—355 366p
Determinants of solutions to differential and difference equations      8—9 11 32p 41—42
Determinants, difference equations used to compute      55—56p 366p
Determinants, Hill      352
Determinants, Wronskian      see “Wronskian”
Difference calculus      36—37 207 211 214—216
Difference equations for Bessel functions      55p 228 243p 573
Difference equations for computing determinants      55—56p 366p
Difference equations for digamma function      214 575
Difference equations for exponential integrals      576
Difference equations for exponential integrals for gamma function      38 223 575
Difference equations for exponential integrals, factor linear difference operators      55p
Difference equations for exponential integrals, Fibonacci numbers      56p
Difference equations for exponential integrals, first-order linear      38—40
Difference equations for exponential integrals, for Hermite polynomials      55p 244p
Difference equations for exponential integrals, for Laguerre polynomials      244p
Difference equations for exponential integrals, for Legendre polynomials      55p 229
Difference equations for exponential integrals, for modified Bessel functions      571
Difference equations for exponential integrals, for parabolic cylinder functions      574
Difference equations for exponential integrals, Frobenius series for      212—214
Difference equations for exponential integrals, functional relations used to solve      40 53 56p 57p
Difference equations for exponential integrals, general solution of      38
Difference equations for exponential integrals, generating functions for      46—47 53 55p 572 573
Difference equations for exponential integrals, homogeneous linear      see “Homogeneous linear difference equations”
Difference equations for exponential integrals, inhomogeneous linear      see “Inhomogeneous linear difference equations”
Difference equations for exponential integrals, initial-value problems      43 248—249
Difference equations for exponential integrals, integral representations for solutions to      248—249
Difference equations for exponential integrals, leading asymptotic behaviors from      207—210 223 225 227—239 248—249
Difference equations for exponential integrals, local analysis of linear      205—233 248—249
Difference equations for exponential integrals, local analysis of nonlinear      233—239
Difference equations for exponential integrals, Newton’s method      234—235 244p 245p
Difference equations for exponential integrals, nonlinear local analysis of      233—239
Difference equations for exponential integrals, nonlinear methods for solving      40 53 56—57p
Difference equations for exponential integrals, ordinary, regular singular, and irregular singular points of      206—207
Difference equations for exponential integrals, periodic behavior of      238 246p
Difference equations for exponential integrals, random behavior of      237—239 239f
Difference equations for exponential integrals, reduction of order of      43—46 51—52 55p 56p

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