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Elementary differential equations 7th edition - Boyce W.E

Boyce W.E Elementary differential equations 7th edition - Wiley publishing , 2001. - 1310 p.
ISBN 0-471-31999-6
Download (direct link): elementarydifferentialequat2001.pdf
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graphical construction of integral curves, 71 homogeneous, 24, 46-47 implicit solutions, 70-71 integrating factor for, 30-31, 35-36, 93-95, 96 interval of definition, 37, 43, 69-70 linear, 24, 29-39 nonlinear, 64-72 numerical solution of, see Numerical methods
740
Index
separable, 40-46 series solution of, 254 systems of, see Systems Fitzhugh-Nagumo equations,
531
Fourier, Joseph, 547, 573 Fourier coefficients, 671 Fourier series, 547-572
acceleration of convergence, 564
choice of series, 568-569 convergence of, 559, 564 convergence of partial sums, 554, 561 cosine series, 566, 635 error, 554-555, 561-562 Euler-Fourier formulas, 550-551 Gibbs phenomenon, 561, 567 integration of, 550 orthogonality of sines and cosines, 549-550 Parseval equation, 564, 571, 604, 676 periodicity of sines and cosines, 548-549 for sawtooth wave, 567-568, 571
sine series, 566-567, 635 specialized kinds, 571-572 for square wave, 560-562, 571 for triangular wave, 551-552, 553-555, 571 Fredholm, Erik Ivar, 644 Fredholm alternative theorem,
644
Frequency, natural, 190 of simple harmonic motion,
190
of vibrating string, 594 Frobenius, Ferdinand Georg, 268, 348, 459 Frobenius, method of, 268 Fuchs, Immanuel Lazarus, 251, 459
Fundamental matrix, 393-401, 405-406 Fundamental set of solutions,
142, 211, 370
Fundamental theorem of algebra, 215
G
Galois, Evariste, 216 Gamma function, 298-299 Gauss, Carl Friedrich, 215 Gaussian elimination, 352 Generalized function, 326 General solution of linear equations: first order, 11, 30, 36, 70, 71 nth order, 211
second order, 133, 142, 143, 155, 163, 170 systems of first order equations, 370 Gibbs, Josiah Willard, 561 Gibbs phenomenon, 561, 567 Global asymptotic stability, 487 Gompertz equation, 85 Graphical construction of integral curves, 71 Gravity, 2, 56 Green, George, 653 Green’s function, 653-656
H
Half-life, 16, 59
Harvesting a renewable resource, 86-87
Heat conduction equation, 17, 574, 615 bar with insulated ends, 584-587 boundary conditions, 574, 582, 584, 616 derivation of, 613-617 fundamental solutions of, 576, 586
nonhomogeneous boundary conditions, 582-584 nonhomogeneous source term, 617, 646-649 in polar coordinates, 581 smoothing of discontinuities in initial conditions, 582 solution of fundamental problem, 575-577
steady state solution, 582 transient solution, 583 Heaviside, Oliver, 294 Heaviside function, 310 Helmholtz equation, 668 Hereditary systems, 331 Hermite, Charles, 248, 348 Hermite equation, 248, 260, 627 Hermite polynomials, 248 Hermitian matrix, 248, 365-366, 379, 397, 398, 402, 637, 653
Heun formula, 431 History of differential equations, 23-26 Hodgkin, Alan L., 503 Hodgkin-Huxley equations, 503 Homogeneous algebraic equations, 357 Homogeneous boundary value problems, 542-544, 623-626 eigenfunctions, 545, 623 eigenvalues, 545, 623 singular Sturm-Liouville, 656-663 Sturm-Liouville, 629-641 Homogeneous differential
equations, with constant coefficients, 25, 129-137, 153-159, 160-169, 214-219 definition of, 130, 343 general theory of, 137-153, 210-211, 368-372 systems, 373-410 Homogeneous first order
differential equations, 24, 46-47 Hooke, Robert, 187 Hooke’s law, 187 Huxley, Andrew F., 503 Huygens, Christian, 336 Hypergeometric equation, 279
I
Identity matrix, 351 Implicit numerical method, 423, 441, 453
Index
74?
Implicit solution, 70-71 Improper integral, 294
comparison theorem for, 296 Improper node, 404, 462-464 Improved Euler method,
430-435 Impulse functions, 324-330 Impulse response, 335 Indicial equation, 269, 273, 274, 278
Inductance, 195, 196 Initial conditions, 11, 72, 116, 130,210, 342 propagation of discontinuities for wave equation, 598 smoothing of discontinuities for heat equation, 582 Initial value problem, 11, 65, 66, 105, 130, 210, 342 Inner product:
for functions, 549, 631 for vectors, 351 Instability:
of critical point, 79, 117, 119, 468, 471, 472, 475, 514 of numerical method, 448-453 of periodic orbit, 524 see also Diverging solutions Integral curves, 11, 71 Integral equation, 106 Volterra, 336 Integral transform, 293 Integrating factor, 25, 30-31, 35-36, 93-95, 96, 146 Interval of convergence, 233 Inverse Laplace transform, 303 Inverse matrix, 352-354 Invertible matrix, 352 Irregular singular point, 258, 279-280 Iteration, method of, 106 see also Successive approximations of difference equation, 116
J
Jordan, Camille, 348, 406 Jordan form of matrix, 406-407, 409, 410
Jump discontinuity, 295, 559, 598 K
Kernel, of convolution, 185 of integral transform, 293 Kirchhoff, Gustav R., 196 Kirchhoff’s laws, 196, 345 Kutta, M. Wilhelm, 436
L
Lagrange, Joseph-Louis, 25, 180, 547, 573, 591 Lagrange's identity, 630,
659-660 Laguerre, Edmond Nicolas, 272 Laguerre equation, 272, 627 Landau, Lev D., 89 Landau equation, 89 Laplace, Pierre Simon de, 25,
294, 604 Laplace’s equation, 25, 604-613, 666
boundary conditions, 605 in cylindrical coordinates, 613, 667
Dirichlet problem, 605 for circle, 609-611, 612 for rectangle, 606-608, 611-612 for sector, 612 for semicircle, 612 for semi-infinite strip, 612 mixed problem, 613 Neumann problem, 605 for circle, 613 for rectangle, 612 in polar coordinates, 609 in spherical coordinates, 668 Laplace transform, 293-338 of convolution, 330-333 definition of, 294 of derivatives, 300-301 of Dirac delta function, 326-327 existence of, 296 of integral equations, 336 inverse of, 303 of periodic functions, 315 of rectified sine wave, 316
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