524 partial differential equations

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1 Index adjoint operator, 114 airfoils, 334 analog signal, 386 analytic function, 307 angular frequency, 372 area element, 217 Argand diagram, 287 associated Laguerre polynomials, 210 associated Legendre functions, 143, 194 average of a function, 76, 184 backward difference, 445 beat frequency, 510 Bernoulli numbers, 398 Bernoulli, Daniel, 35, 147, 287 Bernoulli, Jacob, 287 Bernoulli, Jacob II, 287 Bernoulli, Johann, 287 Bernoulli, Johann II, 287 Bernoulli, Johann III, 287 Bernoulli, Nicolaus I, 287 Bernoulli, Nicolaus II, 287 Bessel functions, 147, 170 first kind, 148 Fourier-Bessel series, 150 generating function, 149 identities, 149 orthogonality, 149 recursion formula, 149 second kind, 148 spherical, 204 Bessel s inequality, 153 Bessel, Friedrich Wilhelm, 147 big-oh, 491 binomial coefficients, 487 binomial expansion, 486 Bohr radius, 210 Bose-Einstein integrals, 398 boundary conditions, 35, 109 nonhomogeneous, 241, 251 time dependent, 256 time independent, 252 boundary value problem, 1, 39 box function, 375 branch cut, 293 branch point, 329 breaking time, 14 Bromwich integral, 411 Bromwich, Thomas John I Anson, 411 Burgers equation, inviscid, 12 cakes baking, 184 cylindrical, 188 rectangular, 185 Cardano, Girolamo, 287 Cauchy Integral Formula, 309 Cauchy principal value integral, 323 Cauchy s Theorem, 303, 304 Cauchy, Augustin-Louis, 298 Cauchy-Euler equations, 514 nonhomogeneous, 518 Cauchy-Riemann equations, 298 central difference, 446 Chain Rule, 463 characteristic equation, 502, 515 characteristics fan, 18 Charpit equations, 26 circle of convergence, 308 circular membrane, 166 circulation, 336 Clairaut s Theorem, 2 Clairaut, Alexis Claude, 2 classical orthogonal polynomials, 131, 134 classification of PDEs, 50 complete basis, 154 complex differentiation, 297 complex functions, 290 multivalued, 293 natual logarithm, 293

2 524 partial differential equations real and imaginary parts, 291 complex numbers, 287 addition, 288 Cartesian form, 288 complex conjugate, 288 imaginary part, 287 modulus, 287 multiplication, 288 nth root, 289 nth roots of unity, 290 polar form, 288 quotient, 288 real part, 287 complex plane, 287 extended, 349 complex velocity, 336 conformal mapping, 346 conics, 54 connected set, 300 conservation law differential form, 11 integral form, 11 conservation laws., 10 conservation of mass, 335 constant coefficient equations, 502 complex roots, 504 repeated roots, 503 contour deformation, 304 contour integral, 323 convergence in the mean, 153 convolution Fourier transform, 379 Laplace transform, 407 convolution theorem Fourier transform, 379 Laplace transform, 407 cooking cakes, 184 turkey, 203 coordinates curvilinear, 213 cylindrical, 216 spherical, 213 cosine series, 88 cube roots of unity, 290 curvilinear coordinates, 213 cutoff frequency, 386 cylindrical coordinates, 216 d Alembert s formula, 55 d Alembert, Jean le Rond, 35 d Alembert, Jean-Baptiste, 55 de Moivre s Formula, 486 de Moivre, Abraham, 287, 486 de Vries, Gustav, 358 derivatives, 455 table, 463 difference equation, 394 differential equations, 399 autonomous, 498 first order, 497, 499 linear, 497, 501 nonhomogeneous, 225, 505 Runge-Kutta methods, 440 second order, 501 separable, 498 series, 147 series solutions, 209 Taylor methods, 435 differential operator, 501 Differentiation Under Integral, 470 dipole, 345 Dirac delta function, 226, 239, 368 Laplace transform, 404 sifting property, 369 three dimensional, 423 Dirac, Paul Adrien Maurice, 369 Dirichlet boundary conditions, 109 Dirichlet kernel, 100 Dirichlet, Gustav, 109 dispersion relation, 358 distribution, 369 domain, 300 domain coloring, 293 double factorial, 139 easy as π, 171 eigenfunction expansion, 119 Green s functions, 242 eigenfunctions complete, 111 orthogonal, 111 eigenvalues real, 111 Einstein, Albert, ix electric dipole, 345 electric field, 48, 333 electric potential, 48, 333 electrostatics, 48 elliptic equation, 51 entire function, 307 equilibrium solution, 48 equilibrium temperature, 178 essential singularity, 314 integrand, 320 residue, 318

3 index 525 Euler s Formula, 485 Euler s Method, 431 Euler, Leonhard, 35, 147, 287 even functions, 82 exponential order., 410 fan characteristics, 18 Feynman s trick, 470 Feynman, Richard, 396, 460, 470 Fibonacci sequence, 487 filtering, 386 finite difference method, 445 finite wave train, 378, 387 flow rate, 335 fluid flow, 48, 333 forward difference, 445 Fourier analysis, 34, 74 Fourier coefficients, 75 generalized, 120, 130 Fourier series, 74 complex exponential, 364 generalized, 130 Maple code, 91 representation on [0, 2π], 75 Fourier sine transform, 419 Fourier transform, 366 convolution, 374 differential equations, 413 properties, 372 shifitng properties, 373 three dimensions, 422 Fourier, Joseph, 1, 35, 147, 497 Fourier-Bessel series, 147 Fourier-Legendre series, 134, 143 Fredholm Alternative, 248 Fredholm alternative, 122 frequency, 372 function average, 472 function spaces, 128 functions exponential, 456 hyperbolic, 461 inverse trigonometric functions, 460 logarithmic, 456 polynomial, 456 rational, 456 trigonometric, 457 Fundamental Theorem of Calculus, 464, 470 Gödel, Kurt, 33 Gamma function, 145, 395 gate function, 375, 386 Gaussian function, 374, 385 Gaussian integral, 375 Gegenbauer polynomials, 134 generalized function, 369 geometric series, 308, 311, 477 Gibbs phenomenon, 91, 98 Gibbs, Josiah Willard, 98 Gram-Schmidt Orthogonalization, 131 gravitational potential, 138 Green s first identity, 282 Green s function, modified, 249 Green s functions, 225, 501 boundary value, 231, 279 differential equation, 239 heat equation, 258, 419, 421 initial value, 227, 279 Laplace s equation, 278, 417 Poisson s equation, 280 properties, 234 series representation, 242 steady state, 280 time dependent equations, 279 wave equation, 259 Green s identity, 116, 239, 249, 271 Green s second identity, 226, 282 Green s Theorem in the Plane, 303, 335 Green, George, 225, 304 Gregory, James, 479 group velocity, 362 Gudermann, Christoph, 466 Gudermannian, 466 harmonic conjugate, 299 harmonic functions, 48, 192, 298, 333 harmonics, 35 meridional, 199 sectoral, 199 spherical, 199 surface, 199 tesseral, 199 zonal, 199 heat equation, 33 1D, 42, 93 derivation, 38 eigenfunction expansion, 262 fully nonhomogeneous, 270 Green s function, 419 Laplace transform, 418 nonhomogeneous, 251, 419 transform, 414 heat kernel, 415 Heaviside function, 144, 238, 402

4 526 partial differential equations Heaviside, Oliver, 144 Heisenberg, Werner, 377 Helmholtz equation, 160 Helmholtz, Hermann Ludwig Ferdinand von, 160 Hermite polynomials, 134, 156 holomorphic function, 298, 307 Huen s method, 442 hydrogen atom, 208 hyperbolic cosine, 461 hyperbolic equation, 51 hyperbolic function identities, 462 substitution, 474 hyperbolic sine, 461 hyperbolic tangent, 461 identities double angle, 458 half angle, 458 product, 460 Pythagorean, 457 sum and difference, 458 tangent, 457 implicit solution, 498 impulse function, 387 unit impulse, 405 impulse response, 387 incompressible, 48, 333 infinite dimensional, 128 infinitesimal area element, 217 infinitesimal volume elements, 216 initial value problem, 1, 399, 497 inner product, 128 inner product spaces, 129 integral transforms, 389 integrals, 455 integration limits, 465 simple substitution, 464 table, 464 trigonometric, 471 integrating factor, 264, 499 integration by parts, 82, 466 integration by recursion, 393 interval of convergence, 483 inverse Fourier transform, 366 inverse Laplace transform, 400, 410 inversion mapping, 347 inviscid Burgers equation, 12 irrotational, 48, 333 Jacobi polynomials, 134 Jacobian determinant, 216 Jordan s lemma, 332 Julia set, 294 Kepler, Johannes, 35, 147 kernel, 227, 357 kettle drum, 166 Korteweg, Diederik, 358 Korteweg-de Vries equation, 358 Kronecker delta, 129 Lagrange s identity, 116 Lagrange, Joseph-Louis, 147 Lagrange-Charpit method, 26 Laguerre polynomials, 134, 210 Laguerre, Edmond, 210 Lambert, Johann Heinrich, 466 Laplace transform, 389 convolution, 407 differential equations, 399 inverse, 410 properties, 395 series summation, 396 transform pairs, 391 Laplace s equation Green s function, 417 half plane, 415 Polar coordinates, 182 Rectangular coordinates, 48, 176 rectangular plates, 178 spherical symmetry, 191 transform, 416 two dimensions, 48, 175 Laplace, Pierre-Simon, 192, 389 Laplacian, 33 cylindrical, 221 polar coordinates, 167 spherical, 221 Laurent series, 312 singular part, 314 Laurent, Pierre Alphonse, 312 least squares approximation, 152 Legendre polynomials, 134, 194 generating function, 138 leading coefficient, 137 normalization, 142 recurrence relation, 136 recursion formula, 136 Rodrigues formula, 135 Legendre, Adrien-Marie, 134, 145 line element, 215 linear fractional transformations, 348 linear operator, 501

5 index 527 linearity, 372, 501 logarithm multi-valued, 293 principal value, 293 Möbius transformation, 348 Möbius, August Ferdinand, 348 Maclaurin series, 481 Maclaurin, Colin, 479 Mandelbrot set, 294 MATLAB code, 296 mean square deviation, 152 mean value theorem, 184 membrane annular, 174 circlur, 166 rectangular, 161 meromorphic function, 307 Method of Characteristics, 5 Method of images, 261 Method of undetermined coefficients, 268, 501, 506 modified, 508 metric coefficients, 215 Midpoint method, 441 modified Euler method, 442 modified Green s function, 249 momentum, 361 Monge, Gaspard, 26 Morera s Theorem, 307 Morera, Giacinto, 307 multivalued functions, 289, 328 integration, 330 nested radicals, 459 Neumann boundary conditions, 109 Neumann function, 148, 170 Neumann, Carl, 109 Newton s Law of Cooling, 222 nodal curves, 163 nodal lines, 163 nonhomogeneous boundary conditions, 241 nonhomogeneous equations, 225 norm of functions, 131 normal, 303 normalization, 77, 130 numerical stability, 452 odd functions, 82 ode45, 443 open set, 300 ordinary differential equation, 1, 497 orthogonal families, 336 orthogonal functions, 77 orthogonal grid, 214 orthogonality eigenfunctions, 111 orthonormal, 77 oscillations forced, 508 Panofsky, Pief, 201 parabolic equation, 51 parametrization, 300 Parseval s equality, 154, 388 Parseval s identity, 103 Parseval, Marc-Antoine, 388 partial differential equation, 33 transforms, 413 partial fraction decomposition, 312, 400 particle wave function, 360 particular solution, 498 Pascal s triangle, 487 Pascal, Blaise, 487 path independence, 302 path integral, 300 period, 73 periodic boundary conditions, 109, 169 periodic extension, 75, 84 periodic function, 73, 460 phase, 74 phase shift, 74 phase velocity, 362 Plancherel s formula, 388 Plancherel, Michel, 388 Poisson Integral Formula, 184 Poisson kernel, 184, 276 Poisson s equation, 225 antisymmetric, 260 three dimensions, 422 two dimensions, 259 Poisson, Siméon, 147 polar coordinates grid, 215 poles, 314 potential function, 336 product solutions, 45 quadratic forms, 54 radicals nested, 459 radius of convergence, 308, 482 Rankine-Hugonoit jump condition, 17 Rayleigh quotient, 117 Rayleigh, Lord, 112 real eigenvalues, 111

6 528 partial differential equations rectangular membrane, 161 Reduction of Order, 503 rescaling, 202 residue, 316 Residue Theorem, 319 resonance, 268, 510 Riemann surface, 293, 328 Riemann zeta function, 397 Riemann, Georg Friedrich Bernhard, 225, 298 Riemann-Lebesgue Lemma, 154 RMS voltage, 472 root mean square, 472 Runge-Kutta, 440 fourth order, 442 second order, 441 Runge-Kutta-Fehlberg method, 443 Russell, Bertrand, 71, 455 Russell, J. Scott, 358 scalar product, 77 scale factors, 214 scheme algebraic system, 363 finite transform, 419 Fourier, 413 Fourier transform, 359 Laplace transform, 390 Schrödinger equation, 362 Schrödinger equation, 206, 360 Schwinger, Julian, 225 self-adjoint, 115 self-similarity transformation, 202 separation of variables, 42 sequence Fibonacci, 487 series binomial series, 489 Fourier series, 74 geometric series, 477 Laurent series, 311 Maclaurin series, 481 power series, 479 re-indexed, 311 summation by transforms, 396 Taylor series, 308, 481 telescoping, 479, 494 side conditions, 7 simple closed contour, 302 simple harmonic motion, 405 sinc function, 376 sine series, 88 double, 165 singularity, 314 double pole, 315 essential, 314 poles, 314 removable, 314 simple pole, 315 solitons, 461 source density, 11 space of square integrable functions, 154 Special Relativity, 490 spectral analysis, 34 spherical Bessel functions, 204 spherical harmonics, 193, 197 spherical symmetry, 191 spherical turkey, 201 square wave, 404, 412 stability numerical, 452 stagnation point, 343 steady state solution, 252 stencil, 446 step function, 238, 402 stereographic projection, 349 Stone-Weierstraß Theorem, 131 streamline function, 335 streamlines cylinder, 343 sink, 340 source, 340 uniform, 339 vortex, 341 Strutt, John William, 112 Sturm-Liouville operator, 108 Sturm-Liouville problem, 107, 150, 157, 158 substitution hyperbolic, 474 trigonometric, 472 Tabular Method, 468 Taylor polynomials, 441, 480 Taylor series, 308, 481 Taylor, Brook, 35, 479 Tchebychef polynomials, 134 Thanksgiving, 201 thermal equilibrium, 48 time dilation, 489 tones, 72 traffic flow, 12 transforms, 357 exponential Fourier, 365 finite Fourier sine, 419 Fourier, 366

7 index 529 Fourier sine, 419 inverse Laplace, 410 Laplace, 389 traveling wave, 7 trigonometric identities, 457 substitution, 472 trigonometric identities, 76 truncation error, 452 turkey, 201 uncertainty principle, 377 uniform flow, 338 variation of parameters, 501, 511 vector spaces, 501 function spaces, 128 inner product space, 129 vectors projection, 132 velocity complex, 336 vibrating sphere, 200 vibrating string, 34 Villecourt, Paul Charpit, 26 volume elements, 216 von Ettingshausen, Andreas Freiherr, 488 von Neumann, John, 431 vortices, 346 vorticity, 335 Wallis, John, 35 Watson, G.N., 147 wave equation, 34, 35 1D, 45, 96 circular, 166 derivation, 36 rectangular, 161 wave packet, 361 wave speed, 358 wavelength, 359 wavenumber, 359, 372 weak solutions, 16 Weierstraß substitution, 322 Weierstraß, Karl Theodor Wilhelm, 322 Wilbraham, Henry, 98 Wilkinson, Rebecca, 185 windowing, 387 Wronskian, 502

Index. c, 400 Ørstead, Christian, 397 matrix:symmetric, 160

Index. c, 400 Ørstead, Christian, 397 matrix:symmetric, 160 Index c, 400 Ørstead, Christian, 397 matrix:symmetric, 160 adjoint operator, 243 Ampère,André-Marie, 397 amplitude, 55 analog signal, 343 analytic function, 288 angular frequency, 55, 332 anticommutative,