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Mickens R.E. Ч Mathematical Methods for the Natural and Engineering Sciences
Mickens R.E. Ч Mathematical Methods for the Natural and Engineering Sciences

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Ќазвание: Mathematical Methods for the Natural and Engineering Sciences

јвтор: Mickens R.E.


This book provides a variety of methods required for the analysis and solution of equations which arise in the modeling of phenomena from the natural and engineering sciences. It can be used productively by both undergraduate and graduate students, as well as others who need to learn and understand these techniques. A detailed discussion is also presented for several topics that are usually not included in standard textbooks at this level: qualitative methods for differential equations, dimensionalization and scaling, elements of asymptotics, difference equations, and various perturbation methods. Each chapter contains a large number of worked examples and provides references to the appropriate literature.

язык: en

–убрика: “ехнологи€/

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

ed2k: ed2k stats

√од издани€: 2004

 оличество страниц: 509

ƒобавлена в каталог: 05.12.2009

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
ѕредметный указатель
Airy equation      269 292Ч293
Asymptotics, asymptotic sequences      497
Asymptotics, elementary operations on asymptotic sequences      499Ч501
Asymptotics, expansions      497Ч498 410Ч412
Asymptotics, gauge functions      495
Asymptotics, generalized asymptotic series      503
Asymptotics, integration by parts      413Ч416
Asymptotics, Laplace's theorem for integrals      218
Asymptotics, order symbols, "o" and "O"      495Ч497
Asymptotics, Watson's lemma      417Ч418
Averaging method, perturbations, derivation      367Ч369
Averaging method, perturbations, procedure      369Ч370
Averaging method, perturbations, stability of limit-cycles      272
Averaging method, perturbations, two special cases      370Ч371
Averaging method, perturbations, worked examples      373Ч376
Averaging, difference equations      394Ч396
Averaging, worked examples      396Ч399
Bernoulli functions      423Ч425
Bernoulli functions, differential equation      426
Bernoulli numbers      425
Bessel equation      293Ч296
Bessel functions, general, asymptotic representations      324
Bessel functions, general, equations reducible to      348Ч350
Bessel functions, general, generating function      323
Bessel functions, general, integral representations      324
Bessel functions, general, Rodrique type formula      323
Bessel functions, general, series representation      322
Bessel functions, special types, Hankel functions      324
Bessel functions, special types, modified Bessel functions      325 326
Beta function, definition      71 72
Beta function, properties      88
Bifurcations, blue sky phenomena      181
Bifurcations, Hopf      184Ч186
Bifurcations, point      180
Boltzmann problem      464Ч467
Casorati (determinant)      221 223
Center      135 143 144
Center, second derivative test      178
Characteristic equation      231
Characteristic scales      4Ч6
Chebyshev polynomials      246Ч248
Clairaut's equation      241Ч242
Convolution theorems, Fourier transform      48Ч49
Convolution theorems, Laplace transform      55
Cosine integral      421Ч423
Delta function, Dirac      57 276Ч277 284 330Ч331
Delta Function, Dirac, definition      76Ч77
Delta Function, Dirac, derivative      80
Delta Function, Dirac, higher dimensions representations      80Ч81
Delta Function, Dirac, properties      77Ч80
DeMoivre's theorem      21
Determinants, special, Hessian      488
Determinants, special, Jacobian      487
Determinants, special, Wronskian      488
Detonation problem      464Ч467
Difference equations, $\Delta^{-1}$      206Ч209
Difference equations, definition      200
Difference equations, existence theorem      200
Difference equations, first-order (linear), form      215
Difference equations, first-order (linear), general solution      216
Difference equations, first-order (linear), homogeneous equation      215
Difference equations, first-order (linear), inhomogeneous equation      215
Difference equations, fundamental operators, E and A      202Ч203 204Ч205
Difference equations, general order (linear), existence and uniqueness theorem      221
Difference equations, general order (linear), fundamental theorems, homogeneous equations      222
Difference equations, general order (linear), homogeneous equation      220
Difference equations, general order (linear), inhomogeneous equation      220
Difference equations, general order (linear), inhomogeneous equation, solutions      223Ч224
Difference equations, genesis      197Ч200
Difference equations, linear, constant coefficients, characteristic equation      231
Difference equations, linear, constant coefficients, homogeneous equation      231
Difference equations, linear, constant coefficients, homogeneous equation, solutions      231Ч232
Difference equations, linear, constant coefficients, inhomogeneous equation      231
Difference equations, linear, constant coefficients, inhomogeneous equation, solutions      232Ч234
Difference equations, linear, nonlinear      201
Difference equations, nonlinear, Clairaut's equation      241Ч242
Difference equations, nonlinear, homogeneous      239
Difference equations, nonlinear, miscellaneous forms      242Ч243
Difference equations, nonlinear, Riccati equations      239Ч241
Difference equations, order      200
Difference equations, properties      205Ч206
Difference equations, uniqueness theorem      200
Differential equations, asymptotics of solutions, Airy equation      292Ч293
Differential equations, asymptotics of solutions, Bessel equation      293Ч296
Differential equations, asymptotics of solutions, elimination of first-derivative terms      287Ч288
Differential equations, asymptotics of solutions, general expansion procedure      296Ч298
Differential equations, asymptotics of solutions, Liouville Ч Green transformation      290Ч292
Differentiation of a definite integral      485
Diffusion equation      49Ч51 462Ч464
Dimensionless variables      4Ч6
Dirichlet integrals, definitions      82
Dirichlet integrals, expressed as gamma functions      83 84
Eigenfunctions      270Ч271
Eigenfunctions, completeness relation      273
Eigenfunctions, expansion of functions      272
Eigenfunctions, orthogonality      271
Eigenvalues      270Ч271
Eigenvalues of $2 \times 2$ matrix      485Ч486
Eigenvalues, reality of      277Ч278
Electron, magnetic moment      91
Elliptic functions, definitions      100
Elliptic functions, Fourier expansions      103Ч104
Elliptic functions, properties      100Ч103
Elliptic integrals, first kind      98
Elliptic integrals, second kind      98
Euler Ч Maclaurin sum formula      426Ч427
Euler Ч Maclaurin sum formula, worked examples      427Ч431
Euler's formula      21
Even function      26Ч28
Exponential order      53
Factorial functions      212Ч215
Family of a function      233
Fermi Ч Dirac integrals      89 90
First integral      132 133Ч134 177
Fisher equation, definition      448
Fisher equation, fixed-points      449
Fisher equation, initial conditions      448
Fisher equation, perturbation solution      453Ч455
Fisher equation, traveling wave solution      448Ч449
Fisher equation, velocity restrictions, traveling waves      449
Fixed points, classification      143
Fixed points, definition      116 132
Fixed points, stability      122Ч124
Fourier series      28Ч33
Fourier series, convergent properties      31
Fourier series, cosine      30
Fourier series, definition      29Ч30
Fourier series, differentiation      32Ч33
Fourier series, integration      32
Fourier series, Parseval's identity      33Ч34
Fourier series, sine      30
Fourier series, square integrable      32
Fourier transforms, convolution theorem      48Ч49
Fourier transforms, definition      44Ч45
Fourier transforms, properties      45Ч47
Functions, named cosine integral      94
Functions, named error function      95
Functions, named exponential integral      95
Functions, named Fresnel cosine and sine integrals      95
Functions, named incomplete gamma function      95
Functions, named sine integral      95
Gamma function, "derivation"      67Ч68
Gamma function, asymptotic expansion      70
Gamma function, definition      68
Gamma function, Euler expression      69
Gamma function, properties      84Ч85
Gamma function, Table, $1 \leq x \leq 2$      70
Gamma function, Weierstrass formula      69
Green's functions, construction of      282Ч283
Green's functions, definition      281
Green's functions, solutions of differential equations      281 284Ч285 285 286
Harmonic balance, conservative oscillators      386Ч387
Harmonic balance, direct method      386Ч388
Harmonic balance, non-conservative oscillators      387Ч388
Harmonic balance, worked examples      389Ч394
Harmonic oscillator, 1-dim oscillator      336Ч337
Harmonic oscillator, 3-dim oscillator      339Ч342
Harmonic oscillator, matrix elements      338
Homoclinic orbit      166
Hopf-bifurcation theorem      186
Index, fixed-point      186
Integrals, evaluation of by parameter differential      104Ч108
Korteweg Ч de Vries equation      455Ч457
Krylov and Bogoliubov, first approximation      369
l'Hopital's rule      487
Laplace transfer, definition      53
Laplace transfer, properties      54Ч56
Laplace transfer, theorem      53
Laplace's equation, spherical coordinates      331Ч333
Leibnitz's relation      487
Level curve      178
Limit-cycle      151
Lindstedt Ч Poincare method, formal procedure      379Ч381
Lindstedt Ч Poincare method, Logistic equation      125Ч126 249Ч250
Lindstedt Ч Poincare method, secular terms      376Ч378
Lindstedt Ч Poincare method, worked examples      382Ч385
Linearly independent functions      221Ч222
Lotka Ч Volterra equations      173Ч177 179Ч180
mathematical equations      3Ч4
Mathematical equations, budworm dynamics      11Ч12
Mathematical equations, decay      6Ч7
Mathematical equations, Duffings      9Ч11
Mathematical equations, Fisher      8Ч9
Mathematical equations, logistic      7Ч8
Mathematical modeling      1Ч3
Node      118 122 135 143
Nonlinearity      12Ч13
Nullclines      133 138
Odd function      26Ч28
Orthogonal polynomials, general, differential equations      309
Orthogonal polynomials, general, generating functions      310Ч311
Orthogonal polynomials, general, interval of definition      309Ч310
Orthogonal polynomials, general, orthogonality relation      310
Orthogonal polynomials, general, recurrence relations      311
Orthogonal polynomials, general, Rodrique's relations      311
Orthogonal polynomials, general, weight functions      310
Orthogonal polynomials, general, zeros      312
Orthogonal polynomials, particular, functions, Chebyshev      315Ч317
Orthogonal polynomials, particular, functions, Hermite      314Ч315
Orthogonal polynomials, particular, functions, Laguerre      317Ч318
Orthogonal polynomials, particular, functions, Legendre      312Ч314
Orthogonal polynomials, particular, functions, Legendre, second kind and associated functions      319Ч321
Parseval's identity      33Ч34
Partial differential equations, general, linear      438Ч441 462Ч464
Partial differential equations, general, pulse solutions      443
Partial differential equations, general, similarity methods      459Ч464
Partial differential equations, general, soliton solutions      442
Partial differential equations, general, traveling waves      441 442
Partial differential equations, types of, advective      442Ч444 444Ч446
Partial differential equations, types of, Boltzmann problem      464Ч467
Partial differential equations, types of, Burgers' equation      444Ч446 461Ч462
Partial differential equations, types of, diffusion equations      462Ч464 467Ч469
Partial differential equations, types of, Fisher equation      448Ч455
Partial differential equations, types of, Korteweg Ч de Vries equation      455Ч457
Partial differential equations, types of, Schroedinger equation, nonlinear      457Ч459
Partial fractions      491Ч492
Periodic functions      25
Perturbation methods, general, general procedure      358 360Ч361
Perturbation methods, general, related issues      362
Perturbation methods, general, worked examples      362Ч367
Perturbation methods, particular techniques, averaging, first order      367Ч371 394Ч396
Perturbation methods, particular techniques, harmonic balance      385Ч388
Perturbation methods, particular techniques, Lindstedt Ч Poincare method      376Ч382
Phase space, one-dimension      118
Phase space, two-dimension      134Ч137 138Ч139
Physical equations      3Ч4
Piecewise continuous functions      31
Piecewise smooth function      31
Pulse solutions, PDE's      442
Riccati equation      239Ч241
Riemann Ц Lebesgue theorem      40
Routh Ч Hurwitz theorem      486
Saddlepoint      118 122 135 143
Schroedinger equation, nonlinear      457Ч459
Separation of variables, application      308
Separation of variables, constants of separation      309
Sign function      92
Similarity methods, applied to first-order PDE's      459
Similarity methods, applied to second-order PDE's      460
Similarity methods, invariance under stretching transformations      460
Similarity methods, stretching transformations      459
Similarity methods, worked examples      461Ч464
Since integral      421Ч423
Soliton solution      442
Special functions      273Ч275
Special functions, associated Laguerre      274
Special functions, associated Legendre      274
Special functions, Chebyshev      246Ч248 274
Special functions, Hermite      274
Special functions, Laguerre      274
Sphere, in uniform flow      334Ч335
Square wave      35Ч36 56Ч57
Stability, asymptotically stable      141
Stability, linear      122Ч124 139Ч140
Stirling numbers      213
Stirling's formula      70 419
Stretching transformations      459Ч460
Sturm Ч Liouville problems, comparison theorem      266
Sturm Ч Liouville problems, definition      270
Sturm Ч Liouville problems, eigenfunctions and eigenvalues      270
Sturm Ч Liouville problems, orthogonality of eigenfunctions      271Ч272
Sturm Ч Liouville problems, separation theorem      266
Symmetries      134 163
Theta function      91
traveling wave      441
Traveling wavefront      442
van der Pol equation      374Ч376 383Ч385 393Ч394 398Ч399
Vibrating string, boundary conditions      260
Vibrating string, differential equation      260
Vibrating string, fixed and free ends      262Ч263
Vibrating string, fixed ends      260Ч261
Vibrating string, free ends      263Ч264
Watson's lemma      417
Wave equations, linear      51Ч52 438Ч441
Wave equations, linear, d'Alembert solution      441
Weight function      272
Zeta function, Riemann, definition      72
Zeta function, Riemann, generalized      74Ч75
Zeta function, Riemann, prime numbers      75Ч76
Zeta function, Riemann, properties      73
Zeta function, Riemann, related functions      73Ч74
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