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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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: Advanced Mathematical Methods for Scientists and Engineers

: Bender C., Orszag S.

:

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.


: en

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ed2k: ed2k stats

: 1978

: 593

: 26.03.2005

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$\pi$, 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 521522
Airy equation, inhomogeneous      105 106f 141142p
Airy equation, local analysis, $(x \rightarrow +\infty)$      100102 494495
Airy equation, local analysis, $(x \rightarrow -\infty)$      107111
Airy equation, local analysis, $(x \rightarrow 0)$      6768
Airy equation, Stokes lines for      116118 130131
Airy functions Ai and Bi as Bessel functions      569
Airy functions Ai and Bi in one-turning-point problems      506510 515518 529530 535536
Airy functions Ai and Bi, asymptotic series, $(x \rightarrow +\infty)$      100102
Airy functions Ai and Bi, asymptotic series, $(x \rightarrow -\infty)$      109111
Airy functions Ai and Bi, asymptotic series, $(|z| \rightarrow \infty)$      133135 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      313314p 570
Airy functions Ai and Bi, integrals of, behavior      256258 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, $(x \rightarrow -\infty)$      107108 108f 109f
Airy functions Ai and Bi, leading asymptotic behaviors, $(|z| \rightarrow \infty)$      113117 133135
Airy functions Ai and Bi, sector of validity for Bi      115 115f
Airy functions Ai and Bi, Stokes behavior of      130131 313314p
Airy functions Ai and Bi, Taylor series for      6768 569
Almost periodic orbits      186187 188f 189f
Anharmonic oscillator, classical (Duffings equation)      545551 547f 551f 566567p
Anharmonic oscillator, quantum      334335 337338 353 353t 359360 366367p
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      421483
Asymptotic matching for integrals      341349 365366p 515519
Asymptotic matching for nonlinear differential equations      421423 463479
Asymptotic matching for nonlinear differential equations for WKB problems      504543
Asymptotic matching for nonlinear differential equations, patching vs.      335336 499
Asymptotic matching for nonlinear differential equations, principle of      335336
Asymptotic matching for nonlinear differential equations, techniques of, for differential equations      336341
Asymptotic matching, existence of matching region      428430
Asymptotic matching, high-order      337 344349 428435 434437f 438452 442444f 449f 478479 535536
Asymptotic power series      see Asymptotic series
Asymptotic relations (asymptotic order relations) for functions with noncoincident zeros      108109
Asymptotic relations (asymptotic order relations) for oscillatory functions      107112
Asymptotic relations (asymptotic order relations) in complex plane      112118
Asymptotic relations (asymptotic order relations), definition of, $\ll$      7879
Asymptotic relations (asymptotic order relations), definition of, $\sim$      7879
Asymptotic relations (asymptotic order relations), definition of, O      318
Asymptotic relations (asymptotic order relations), differentiation of      127128 145p
Asymptotic relations (asymptotic order relations), integration of      81 139p 142p
Asymptotic relations (asymptotic order relations), sector of validity for      113118
Asymptotic relations (asymptotic order relations), Stokes phenomenon and subdominance      115118
Asymptotic series for Airy functions      100102 109111 133135 142p 570
Asymptotic series for Bessel functions      111112 112f 228229 294 572573
Asymptotic series for digamma function      309p 575
Asymptotic series for exponential integrals      576
Asymptotic series for Fourier integrals      276277 281287
Asymptotic series for logarithmic integral function      310p
Asymptotic series for modified Bessel functions      9394 144p 265 571
Asymptotic series, arithmetical operations on      125126
Asymptotic series, construction of analytic function asymptotic to      144p
Asymptotic series, construction of continuous function asymptotic to      119120 119f
Asymptotic series, convergent and divergent      118120
Asymptotic series, definition of      89 118
Asymptotic series, derived by integration by parts      252261 276277
Asymptotic series, derived by Laplaces method and Watsons lemma      262265
Asymptotic series, derived by steepest-descents method      281287 294
Asymptotic series, derived from difference equations      218219 225231
Asymptotic series, derived from linear differential equations      8487 9094 98101 104105 109111 129 133136
Asymptotic series, derived from nonlinear differential equations      150 154155 157158 165167
Asymptotic series, differentiation of      127128
Asymptotic series, equating coefficients in      125
Asymptotic series, integration of      126
Asymptotic series, nonuniqueness of (subdominance)      123124
Asymptotic series, optimal asymptotic approximation for      94102 122124 222 223f 224t 242
Asymptotic series, pfor parabolic cylinder functions      98100 131133 265 574
Asymptotic series, proof of existence of      128129 133136 144
Asymptotic series, Stieltjes series as      78 120123 260
Asymptotic series, Stirling series for gamma function      218220
Asymptotic series, Taylor series vs.      9091 92f 118119 220 221t 222 254f 255
Asymptotic series, uniqueness of coefficients      8990 124125
Asymptotic summation of series      376 379 380t
Autonomous equations, definition of      24
Autonomous equations, reduction of order for      2426 156
Autonomous systems      171197
Autonomous systems with almost periodic orbits      186187 188f 189f
Autonomous systems, C-systems      192193 204p
Autonomous systems, critical points of      173177
Autonomous systems, definition of      171
Autonomous systems, Duffings equation      546 547f
Autonomous systems, Hamiltonian      189191 189f 190f 204p
Autonomous systems, Henon and Heiles, example of      188190 189f 190f
Autonomous systems, higher-dimensional      185197
Autonomous systems, linear      177178
Autonomous systems, Lorenz model      194195 194197f 204p
Autonomous systems, one-dimensional      174175
Autonomous systems, predator-prey      179183 180182f
Autonomous systems, random behavior in      188189 190f 191f 192195 195197f
Autonomous systems, Rayleigh oscillator      468479 476478f 483p 554556 556f 557f
Autonomous systems, Toda lattice      187 188f 190 203p
Autonomous systems, two-dimensional      175185 468479 546547
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
Bernoullis method for finding roots of a polynomial      241p
Bessel equation, differential equation      14 111112 143p 572
Bessel equation, differential equations equivalent to      322 573
Bessel equation, local analysis $(x \rightarrow +\infty)$      111112
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, $(x \rightarrow +\infty)$      111112 112f294
Bessel functions $J_{\nu}$ and $Y_{\nu}$, asymptotic series, $(\nu \rightarrow +\infty)$      228229 230t
Bessel functions $J_{\nu}$ and $Y_{\nu}$, asymptotic series, $(|z| \rightarrow \infty)$      572573
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 293294 309p
Bessel functions $J_{\nu}$ and $Y_{\nu}$, leading asymptotic behaviors, ($(x \rightarrow +\infty)$ and $(\nu \rightarrow +\infty)$)      280 298299 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      381383 406 412p
Borel summation, generalized      382
Boundary layers, absence of      452
Boundary layers, definition of      326 327f 419420 420f
Boundary layers, internal      455463 466468 469475f 485 541542p
Boundary layers, local breakdown at      484485
Boundary layers, location of      425426
Boundary layers, mathematical structure of      426430
Boundary layers, multiple      437438 439f" 440f" 446455 449f" 458460 464475 465f 467478f 485
Boundary layers, nested      453455 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 $\delta = O(\varepsilon^{1/2})$      437439 442443 447 456 465 466
Boundary layers, thicknesses of $\delta = O(\varepsilon^{2/3})$      479 480p
Boundary layers, thicknesses of how to determine      435437
Boundary-layer problems with coordinate singularity      452453
Boundary-layer problems, absence of boundary layers      452
Boundary-layer problems, Carriers problem      464475
Boundary-layer problems, exactly-soluble      326 327f 419420 426430
Boundary-layer problems, internal      455463 466475 541542p
Boundary-layer problems, limit cycle of Rayleigh oscillator      468479 483p
Boundary-layer problems, linear, fourth-order      449452
Boundary-layer problems, linear, second-order      326 327f; 424f; 488490 559560
Boundary-layer problems, linear, third-order      446449
Boundary-layer problems, mathematical structure of      426430
Boundary-layer problems, nonlinear, first-order      421422 422f 423f
Boundary-layer problems, nonlinear, second-order      463479 483p
Boundary-layer theory      417483
Boundary-layer theory as singular perturbation theory      326 430
Boundary-layer theory for nonlinear equations      421422 422423f 463479 483p
Boundary-layer theory in higher order      428435 434437f 440452 442444f 449f 478479
Boundary-layer theory with underdetermined or overdetermined solutions      461467 465f 467475f 482p
Boundary-layer theory, failure of      461463 467 485 487
Boundary-layer theory, inner, outer, and intermediate limits in      426430
Boundary-layer theory, internal boundary layers      455463 466475
Boundary-layer theory, introductory discussion      326 327f 417426
Boundary-layer theory, multiple-scale theory vs.      559560
Boundary-layer theory, uniform approximation in      425 430 433
Boundary-layer theory, WKB theory vs.      484485 488490 498 503505
Boundary-value problems, for difference equations, eigenvalue      4749 56p 231233
Boundary-value problems, for difference equations, linear      3940
Boundary-value problems, for difference equations, local analysis of      231233
Boundary-value problems, for difference equations, well-posed      43 55p
Boundary-value problems, for differential equations, asymptotic matching for      336341
Boundary-value problems, for differential equations, boundary-layer structure in linear      326 327f 419420 422463 488490
Boundary-value problems, for differential equations, boundary-layer structure in nonlinear      463479
Boundary-value problems, for differential equations, definition of      67
Boundary-value problems, for differential equations, eigenvalue      see Eigenvalue problems
Boundary-value problems, for differential equations, existence and uniqueness of solutions to      7 1011
Boundary-value problems, for differential equations, Greens 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      167172
Boundary-value problems, for differential equations, multiple-scale analysis of      559560
Boundary-value problems, for differential equations, patching used to solve      335336
Boundary-value problems, for differential equations, well-posed      1011
Boundary-value problems, for differential equations, WKB analysis of      326327 328f 485 487492 497505 511 511514f 518t 519523 537539
C-systems      192193 204p
Carleman condition      410 415p 416p
Carlini Green Liouville approximation      see Exponential substitution
Casoratian      see Wronskian
Centers, almost periodic behavior near      186187 188f 189f
Centers, higher-dimensional      186188
Centers, random behavior near      188189 190f 191f 192195 195197f
Centers, random behavior near spurious      184185
Centers, two-dimensional      173f 177179 185
Completeness      351
Complex plane, asymptotic relations in      112118
Complex plane, asymptotic series in      130136
Complex plane, differential equations in      2930
Complex plane, Pade approximants in      383410
Complex plane, physical optics approximation in      54lp 542p
Complex plane, spontaneous singular points in      149
Complex plane, steepest descents, method of, in      280302
Complex plane, WKB theory in      54lp 542p
Connection formulas      509 511513 529530
Constant-coefficient equations, difference      4041 5455p
Constant-coefficient equations, differential      1112 32p 545
Constant-phase paths      282287 (See also Steepest descents method
Constants of integration      3 56
Constants of summation      37 38
Continued fractions for Stieltjes function      396 406407
Continued fractions, algorithm for computing coefficients of      397398
Continued fractions, algorithm for evaluating      398399
Continued fractions, definition of      395
Continued fractions, examples of      395396 413p
Continued fractions, Pade approximants and      396398
Continued function representations, algorithms for computing      399400 413p
Continued function representations, exponentials      147 196p 399 413p 414p
Continued function representations, fractions      395400 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.      214215
Controlling factors, from difference equations, dominant-balance method used to find      209 214 218 229 232233
Controlling factors, from difference equations, methods for finding      214218
Controlling factors, from differential equations      see Leading asymptotic behaviors; Dominant balance method
Controlling factors, from differential equations, definition of      7980
Controlling factors, from differential equations, exponential substitution used to find      8081 84 87 88 9091 97
Controlling factors, from differential equations, geometrical optics approximation and      494
Convergence, acceleration of      369376
Convergence, acceleration of, asymptotic summation of series      376 379 380t
Convergence, acceleration of, generalized Shanks transformation      389392
Convergence, acceleration of, Richardson extrapolation      375376 377t 378t
Convergence, acceleration of, Shanks transformation      369375
Cramers Rule      9 352
Critical points in phase space, centers      173f 177179 180f 185188
Critical points in phase space, definition of      173
Critical points in phase space, nodes      174175 174f 176 177 181 182f 186 187f
Critical points in phase space, saddle points      174175 174f 177 179182 180182f
Critical points in phase space, spiral points      176179 181184 181f 182f 184f 192194f
Critical points in phase space, stability of      174177 186 189 191
Critical-point analysis in phase space, one-dimensional      174176
Critical-point analysis of C-systems      192193 204p
Critical-point analysis of Hamiltonian systems      189191 204p
Critical-point analysis of random systems      193195 204p
Critical-point analysis of random systems, higher-dimensional      185197 (See also Autonomous systems; Critical points)
Critical-point analysis of random systems, two-dimensional      175183
Critical-point analysis, Arnold Moser theorem      189 191
Critical-point analysis, energy integrals used in      185
Critical-point analysis, failure of linear      183185
Critical-point analysis, linear      174183
Crossing of eigenvalues      see Level crossing
Crossing of roots of polynomials      328329 330f 331f 332f 363p
Cycles for difference equations      238 246p
Cycles in phase space      173f 175
Delta function $\delta(x)$      16 32p 310p 498 526527 540541p
Delta function potential      526527
Determinants for Pade approximants      384 391
Determinants of N x N matrices      352355 366p
Determinants of solutions to differential and difference equations      89 11 32p 4142
Determinants, difference equations used to compute      5556p 366p
Determinants, Hill      352
Determinants, Wronskian      see Wronskian
Difference calculus      3637 207 211 214216
Difference equations for Bessel functions      55p 228 243p 573
Difference equations for computing determinants      5556p 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      3840
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      212214
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      4647 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 248249
Difference equations for exponential integrals, integral representations for solutions to      248249
Difference equations for exponential integrals, leading asymptotic behaviors from      207210 223 225 227239 248249
Difference equations for exponential integrals, local analysis of linear      205233 248249
Difference equations for exponential integrals, local analysis of nonlinear      233239
Difference equations for exponential integrals, Newtons method      234235 244p 245p
Difference equations for exponential integrals, nonlinear local analysis of      233239
Difference equations for exponential integrals, nonlinear methods for solving      40 53 5657p
Difference equations for exponential integrals, ordinary, regular singular, and irregular singular points of      206207
Difference equations for exponential integrals, periodic behavior of      238 246p
Difference equations for exponential integrals, random behavior of      237239 239f
Difference equations for exponential integrals, reduction of order of      4346 5152 55p 56p
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