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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/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1978

Количество страниц: 593

Добавлена в каталог: 26.03.2005

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Предметный указатель

Difference equations for exponential integrals, substitution for solving      53
Difference equations for exponential integrals, summing factor      39
Difference equations for exponential integrals, Taylor series for      210—212
Difference equations for exponential integrals, undetermined coefficients, method of, for      52—53 56p
Difference equations for exponential integrals, variation of parameters      49—51 56p
Difference equations for exponential integrals, Wronskian (Casoratian)      41—43 55p
Difference equations, asymptotic expansions of solutions of      225—231
Difference equations, Bernoulli      57p
Difference equations, boundary-value problems      39—40 43 47—49 55p 56p 231—233
Difference equations, classification of the point at $n = \infty$       206—207
Difference equations, constant-coefficient      40—41 54—55p
Difference equations, constants of summation      37 38
Difference equations, controlling factors from      209 214—218 223 229 232—233
Difference equations, definitions of      37—38 40 49
Difference equations, difference calculus for      36—37 207 211 214—216
Difference equations, differential equations vs.      36 49 205—210 214—215 237
Difference equations, eigenvalue      47—49 56p 231—233
Difference equations, Euler      44—46 55p
Differential equations in complex plane      29—30
Differential equations, asymptotic matching for, techniques of      336—341
Differential equations, boundary-layer theory for      419—483
Differential equations, boundary-value problems      see “Boundary-value problems”
Differential equations, constants of integration      3 5—6
Differential equations, controlling factors from      79—81 84 87 88 90—91 97 494
Differential equations, definitions of      3 4
Differential equations, eigenvalue problems      see “Eigenvalue problems”
Differential equations, existence and uniqueness theorems      6—7 30—3lp 33p 196p
Differential equations, general solutions of      3 8 152—154
Differential equations, global analysis of      417—568
Differential equations, initial-value problems      see “Initial-value problems”
Differential equations, integral representations for solutions to      see “Integral representations”
Differential equations, leading asymptotic behaviors from      see “Leading asymptotic behaviors”
Differential equations, local analysis of linear      61—145
Differential equations, local analysis of nonlinear      146—204
Differential equations, multiple-scale perturbation theory for      544—568
Differential equations, patching      335—336 505 527
Differential equations, periodic behavior      see “Oscillators; Periodic behaviors”
Differential equations, perturbation techniques (elementary) for      321—324 326—341
Differential equations, random behavior of      188—197
Differential equations, singular points of      see “Singular points”
Differential equations, systems of      5 66 171—197
Differential equations, WKB theory for      484—543 (See also “Linear differential equations; Nonlinear differential equations”)
Digamma function $\psi$       213—214 241p 242p 309p 575 576
Dirac delta function      see “Delta function”
Discrete derivative D      36 207 211 213 215 216
Dispersive phenomena      484—486 (See also “WKB theory”)
Dissipative phenomena      484—486 (See also “Boundary-layer theory”)
Distinguished limits      435—446
Distinguished limits in boundary-layer theory      435—438 441 443 447 450 453 454 456 465 473 479
Distinguished limits in WKB theory      486 488
Distinguished limits, absence of      452
Distinguished limits, explanation of      435—437
Dominance      115 (See also “Subdominance”)
Dominant balance, method of controlling factors of difference equations found by      208—209 214 218 229 232—233
Dominant balance, method of corrections to leading behaviors found by      84—87
Dominant balance, method of discussion of      83—84 208—209 235—237
Dominant balance, method of distinguished limits found by      see “Distinguished limits”
Dominant balance, method of failure of      208 235—237
Dominant balance, method of three-term      106—107 14
Dominant balance, method of, for inhomogeneous linear differential equations      104—107 141p
Dorodnicyn formula for period of Rayleigh oscillator      477—479
Duffing’s equation      545—551 547f 551.f; 566—567p
Dunham’s generalization of WKB eigenvalue condition      537
e, calculation of      411p
Eigenfunctions of difference equations      48 56p
Eigenfunctions, completeness      351
Eigenfunctions, definition of      27
Eigenfunctions, expansion in terms of      351—352
Eigenfunctions, normalization of      29 490
Eigenfunctions, orthogonality of      29 48 56p 490
Eigenfunctions, WKB approximation to      490—491 491f 492f 524—526f
Eigenvalue condition, WKB      467 519—523 537—539
Eigenvalue problems for differential equations, asymptotic matching for      338—341
Eigenvalue problems for differential equations, elementary examples of      27—29 35p
Eigenvalue problems for differential equations, mathematical structure of perturbative      350—361
Eigenvalue problems for differential equations, nonlinear      338—341
Eigenvalue problems for differential equations, perturbative treatment of      330—335 338—341
Eigenvalue problems for differential equations, Schrodinger equation      see Schrodinger equation
Eigenvalue problems for differential equations, Sturm — Liouville      see “Sturm — Liouville problem”
Eigenvalue problems for differential equations, WKB analysis of      490—492 519—523 523t 524—526f 537—539 542p 543p
Eigenvalue problems, for difference equations      47—49 56p 231—233
Eikonal equation      487
Elliptic functions      158 161 246p
Emden equation (Prob. 1.37)      35p
Energy eigenvalue, defined      331 (See also “Schrodinger equation”)
Energy integrals      185 193 546 552—553
Equidimensional (Euler) differential equations, linear      12
Equidimensional (Euler) differential equations, nonlinear      4 25 27 156
Euler difference equations      44—46 55p
Euler differential equations      see “Equidimensional differential equations”
Euler summation      381 383 412p
Euler — Maclaurin sum formula      305—306 306t 315p 379 411p
Eulerian wobble      202—203p 203f 204f
Euler’s constant y      252 281 306 307p 316p 342 411p 575p
exact equations      13 23 33p
Existence and uniqueness theorems      6—7 30—31p 33p 196p
Exponential approximation      see “Exponential substitution”
Exponential integrals      252 483p 575—576
Exponential substitution, for difference equations      214 216—217
Exponential substitution, for differential equations as basis for approximation methods      493
Exponential substitution, for differential equations in WKB theory      485—486
Exponential substitution, for differential equations, finding controlling factors using      80—81 87 88 90—91 97
Factorial function      see “Gamma function”
Factoring, linear difference operators      55p
Factoring, linear differential operators      22 33p 34p
Fibonacci numbers      56p
Fixed singular points      146 158
Floquet theory      560—561 568p
Fourier integrals, asymptotic expansion of using asymptotic matching      347—349
Fourier integrals, asymptotic expansion of using integration by parts      276—278 311p
Fourier integrals, asymptotic expansion of using method-of steepest descents      281—287 291 293—294
Fourier integrals, conversion into Laplace integrals      281
Fourier integrals, definition of ordinary and generalized      276
Fourier integrals, Fresnel integral      308p
Fourier integrals, leading asymptotic behavior      278—280
Fourier integrals, Riemann — Lebesgue lemma for      277—278 285 31
Frequency shift      548 551 552f 553
Frequency, natural      544
Fresnel integral      308p
Frobenius series, for difference equations      212—214
Frobenius series, for differential equations at irregular singular points      77—78
Frobenius series, for differential equations at ordinary points      73
Frobenius series, for differential equations at regular singular points      68—76 452
Frobenius series, for differential equations for Bessel functions      572
Frobenius series, for differential equations for exponential integrals      252 576
Frobenius series, for differential equations for higher-order equations      76 138p
Frobenius series, for differential equations for incomplete gamma function      251—252 344—347 365p
Frobenius series, for differential equations for modified Bessel functions      71 74. 342 570—571
Frobenius series, for differential equations, controlling factor and leading behavior of      80
Frobenius series, for differential equations, definition of      69
Frobenius series, for differential equations, differentiation of, with respect to indicial exponents      73—76
Frobenius series, for differential equations, indicial exponents for      63 69—76 452
Frobenius series, for differential equations, indicial polynomial for      70 71 76
Fuchs, theorems of      62 63
Functional relations for Airy functions      131 569
Functional relations for Bessel functions      143p 572
Functional relations for gamma function      575
Functional relations for modified Bessel functions      143p 571
Functional relations for parabolic cylinder functions      132 574
Functional relations for probability density      238 245p
Functional relations, difference equations solved by      40 53 56p 57p
Galerkin method      352
Gamma function $\Gamma(x)$ , definition and properties of      38 39f 54p 574—575
Gamma function $\Gamma(x)$ , difference equation for      38 223 575
Gamma function $\Gamma(x)$ , digamma function      see “Digamma function”
Gamma function $\Gamma(x)$ , incomplete      251—252 344—346 365—366p
Gamma function $\Gamma(x)$ , integral representation for      38 54p 248 275 295 574
Gamma function $\Gamma(x)$ , Stirling series for      see “Stirling series”
Gamma function $\Gamma(x)$ , Taylor series for      220 221f 222t 254f 255 385 387t
Gamma function $\Gamma(x)$ , use in Taylor and Frobenius series for difference equations      211—214

Gaussian elimination      398
General solution of difference equations      38
General solution of differential equations      3 8 152—154
Generating functions for Bessel functions      55p 573
Generating functions for difference equations      46—47 53
Generating functions for Hermite polynomials      55p
Generating functions for Legendre polynomials      55p
Generating functions for modified Bessel functions      572
Geometrical optics approximation      494—495
Global analysis      317—367 417—568
Global analysis, asymptotic matching      335—349
Global analysis, boundary-layer theory      419—483
Global analysis, multiple-scale perturbation theory      544—568
Global analysis, perturbation series      319—367
Global analysis, philosophy of      6—7 59 317—318 417
Global analysis, WKB theory      484—543
Global breakdown      485
Green’s functions, boundary-value problems solved using      19 32p
Green’s functions, definition of      16—17
Green’s functions, variation of parameters vs.      18—19 498
Green’s functions, WKB approximation to      498—504 540—541
Hamiltonian systems      188—191 189f 190f 204p
Harmonic oscillator with aging spring      327—328 329f 568p
Harmonic oscillator, classical      544—545 548
Harmonic oscillator, damped      552—553 554f 555f
Harmonic oscillator, quantum      28 133 332—333 522—523 538
Heaviside step function      16—18
Herglotz function      357—359 366p 406 415p 416p
Hermite polynomials in eigenfunctions of quantum harmonic oscillator      28 133 333
Hermite polynomials , behavior of $(n\rightarrow\infty)$       244p
Hermite polynomials , difference equation for      55p 244p
Hill determinant      352
Homogeneous linear difference equations, classification of the point at $n = \infty$       206—207
Homogeneous linear difference equations, Frobenius series for      212—214.
Homogeneous linear difference equations, general form of      40
Homogeneous linear difference equations, general solution of      42—43
Homogeneous linear difference equations, leading asymptotic behaviors from      207—210 223 225 227—233 248—249
Homogeneous linear difference equations, linear independence of solutions to      41—42 55p
Homogeneous linear difference equations, local analysis of      205—233
Homogeneous linear difference equations, reduction of order for      43—44 46 56p
Homogeneous linear difference equations, solution of general first-order      38—39
Homogeneous linear difference equations, Taylor series for      210—212
Homogeneous linear difference equations, Wronskian (Casoratian) for      41—43 55p
Homogeneous linear differential equations, classification of singular points of      62—66
Homogeneous linear differential equations, definition of      4
Homogeneous linear differential equations, general solution of      8
Homogeneous linear differential equations, linear independence of solutions of      8—9
Homogeneous linear differential equations, local behavior near irregular singular points of      76—103
Homogeneous linear differential equations, local behavior near ordinary points of      66—69
Homogeneous linear differential equations, local behavior near regular singular points of      68—76
Homogeneous linear differential equations, reduction of order of      13 32p
Homogeneous linear differential equations, theory of      7—11
Homogeneous linear differential equations, Wronskian      8—11 15 31p 32p
Hyperairy equation      102—103 143—144p 315p 497
Hypergeometric functions, continued-fraction representation for      396 413p
Hypergeometric functions, differential equation for      35p
Ill-posed      see “Well-posed”
Incomplete gamma function $\Gamma(x, a)$ , $\gamma(x, a)$       251—252 344—346 365—366p
Indicial exponents for equidimensional equations      12
Indicial exponents for Frobenius series      63 69—76 452
Indicial polynomial $P(\alpha)$       70—71 73—76
Inhomogeneous Airy equation      105 106f 141—142p
Inhomogeneous linear difference equations, general form of      49
Inhomogeneous linear difference equations, reduction of order for      51—52 55p 56p
Inhomogeneous linear difference equations, solution of general first-order      39—40
Inhomogeneous linear difference equations, undetermined coefficients, method of, for      52—53 56p
Inhomogeneous linear difference equations, variation of parameters for      49—51 56p
Inhomogeneous linear differential equations, appearance of, in perturbation theory      321—322
Inhomogeneous linear differential equations, definition of      4
Inhomogeneous linear differential equations, form of solution to      14
Inhomogeneous linear differential equations, Green’s functions for      16—19 32p
Inhomogeneous linear differential equations, local analysis of      103—107 141p
Inhomogeneous linear differential equations, reduction of order for      19 32p 333
Inhomogeneous linear differential equations, resonant and secular behavior of      544—549
Inhomogeneous linear differential equations, solution of arbitrary first-order      14
Inhomogeneous linear differential equations, undetermined coefficients, method of, for      19—20
Inhomogeneous linear differential equations, variation of parameters for      15 18—19 32p 103—104 20lp 498
Inhomogeneous linear differential equations, WKB solutions of      497—504 500—505f
Initial-value problems for difference equations      42—43 248—249
Initial-value problems for differential equations, asymptotic matching for      336—337 421—423
Initial-value problems for differential equations, boundary-layer structure in      421—423 480p
Initial-value problems for differential equations, definition and examples of      5—7
Initial-value problems for differential equations, existence and uniqueness theorems for      6—7 30—31p
Initial-value problems for differential equations, local analysis of linear      67—69 247—248
Initial-value problems for differential equations, local analysis of nonlinear      147—153 186—188
Initial-value problems for differential equations, multiple-scale analysis of linear      327—328 329f 556—566 568p
Initial-value problems for differential equations, multiple-scale analysis of nonlinear      545—556 566—567p
Initial-value problems for differential equations, regular perturbation theory solutions of      321—322 323f 327 329f
Initial-value problems for differential equations, well-posed      9—10
Initial-value problems for differential equations, WKB structure in linear      488 489f
Inner, approximation in boundary-layer theory      421—483
Inner, limit, defined      427
Inner, region, defined      342—343 420 420f
Inner, solution, in boundary-layer theory      421—483
Inner, variables, defined      427 (See also “Boundary-layer theory”)
Integral representations for Airy functions      313—314f7 570
Integral representations for Bessel functions      280 291 293 298 309p 312p 573
Integral representations for difference equation solutions      248—249
Integral representations for digamma function      309p
Integral representations for Euler’s constant      252 281 307p
Integral representations for exponential integrals      252 483p 575
Integral representations for gamma function      38 54p 248 275 295 574
Integral representations for hyperairy function      315p 574
Integral representations for incomplete gamma function      251
Integral representations for Legendre polynomials      309p
Integral representations for logarithmic integral function      310p
Integral representations for modified Bessel functions      268 270 313p 571
Integral representations for moments      121—122 405
Integral representations for parabolic cylinder functions      315p
Integral representations for Riemann sums      303
Integral representations, Fourier integral      278
Integral representations, Fresnel integral      308p
Integral representations, general discussion      247—249
Integral representations, generalized Fourier integral      278
Integral representations, Laplace integrals      258—259
Integral representations, Stieltjes integral      120—122
Integrals, asymptotic expansion of      247—316
Integrals, asymptotic expansion of WKB approximation      490—491 500—504 513—519 540—54lp
Integrals, asymptotic expansion of, asymptotic matching for      341—349 365—366p 515—519
Integrals, asymptotic expansion of, elementary methods      249—252
Integrals, asymptotic expansion of, integration by parts      252—261 276—278 311p
Integrals, asymptotic expansion of, Laplace’s method      261—276 309—311p
Integrals, asymptotic expansion of, stationary phase, method of      276—280 311—312p
Integrals, asymptotic expansion of, steepest descents, method of      280—302
Integrating factors      13 14 23 34p
Integration by parts      252—261
Integration by parts for asymptotic expansion of integrals      252—261 348—349
Integration by parts for Fourier integrals      276—278 311p
Integration by parts for Laplace integrals      258—259
Integration by parts, failure of      259—261 281
Intermediate limits, definition of      428 (See also “Boundary-layer theory”)
Irregular singular points, of difference equations, asymptotic series about      255—231
Irregular singular points, of difference equations, controlling factors near      214—218 223
Irregular singular points, of difference equations, definition of      207
Irregular singular points, of difference equations, leading asymptotic behaviors near      223 225 227—233
Irregular singular points, of differential equations at $\infty$       64 88—103
Irregular singular points, of differential equations, controlling factors near      79—81 84 87 88 90—91 97
Irregular singular points, of differential equations, definition of      63—64
Irregular singular points, of differential equations, Frobenius series near, failure of      77—78
Irregular singular points, of differential equations, leading asymptotic behavior near      79—88 90—92 96—97 100 102—103
Irregular singular points, of differential equations, local behavior near      76—103
Irregular singular points, of differential equations, Taylor series near      65
Iteration methods, $\pi$ , calculation of      246p
Iteration methods, Bernoulli’s      241p
Iteration methods, continued exponentials      147
Iteration methods, continued square roots      245p
Iteration methods, Newton’s      234—235 244p 245p
Laguerre polynomials      244p
Langer’s uniform WKB approximation      510—511 541p
Laplace integrals, asymptotic expansion of      261—276

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