Asmar N.H. — Partial Differential Equations with fourier series and boundary value problems :: Электронная библиотека попечительского совета мехмата МГУ

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Asmar N.H. — Partial Differential Equations with fourier series and boundary value problems

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Название: Partial Differential Equations with fourier series and boundary value problems

Автор: Asmar N.H.

Аннотация:

This example-rich reference fosters a smooth transition from elementary ordinary differential equations to more advanced concepts. Asmar's relaxed style and emphasis on applications make the material accessible even to readers with limited exposure to topics beyond calculus. Encourages computer for illustrating results and applications, but is also suitable for use without computer access. Contains more engineering and physics applications, and more mathematical proofs and theory of partial differential equations, than the first edition. Offers a large number of exercises per section. Provides marginal comments and remarks throughout with insightful remarks, keys to following the material, and formulas recalled for the reader's convenience. Offers Mathematica files available for download from the author's website. A useful reference for engineers or anyone who needs to brush up on partial differential equations

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Second edition

Год издания: 2005

Количество страниц: 814

Добавлена в каталог: 28.03.2010

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Предметный указатель

Heat equation, solution with the Laplace transform      503
Heat equation, two dimensional      104 160
Heat equation, two dimensional with one radiating side      185
Heat equation, two dimensional, double Fourier series solution      161
Heat equation, two dimensional, nonhomogeneous      169
Heat equation, uniqueness of solution      189
Heat equation, varying the thermal diffusivity      124
Heat flux      143
Heat kernel      421
Heat problem in a bar insulated ends      146
Heat problem in a bar insulated lateral surface with non zero boundary conditions      140 142
Heat problem in a bar insulated lateral surface with zero boundary conditions      135
Heat problem in a bar with one radiating and one insulated end      152
Heat problem in a bar with one radiating end      147 450
Heat problem in a bar with two radiating ends      154
Heat problem in a thin circular ring      152
Heat problem with positive and negative separation constants      153
Heat source      143
Heat transfer constant      147 185 341 346
Heaviside unit step function      447 491
Heaviside unit step function, derivative of      448
Heaviside, Oliver      479
Heisenberg’s inequality      593
Heisenberg’s inequality for the Gaussian      595
Heisenberg’s uncertainty principle      573 574 590-592
Helmholtz equation      173 178 231
Helmholtz equation in a ball      291 293
Helmholtz equation in a disk      231 232
Helmholtz equation in rectangular coordinates      173
Helmholtz equation in spherical coordinates      291
Helmholtz equation, eigenvalues and eigenfunctions      293
Helmholtz equation, expansion in terms of the eigenfunctions of the      232 294
Helmholtz equation, three dimensional      178
Helmholtz equation, two dimensional      173
Hennite polynomial      332 597 606
Hennite polynomial, generating function for      608
Hennite polynomial, recurrence relation for      607
Hermite coefficient      599
Hermite function      410 580
Hermite series      581 599
Hermite’s differential equation      580 597
Hooke’s law      7
Hydrogen atom      573 582
Hydrogen atom and Schr $\ddot{o}$ dinger’s equation      582
Hydrogen atom and spherical harmonics      583
Hydrogen atom, bound state of the      585
Hydrogen atom, energy levels in the      585
Hydrogen atom, probability distribution lor the      586
Hydrogen atom, time-dependent wave function of the      586
Identity Principle      A33
IDFT      see "Inverse discrete Fourier transform"
Imaginary part      645
Impulse of a force      115
Incomplete gamma function      466 588 589
Indicial equation      A54 238
Indicial root      A54 238
Infinite square well      578
Initial condition      A5 3 105;
Initial condition, radially symmetric      199
Initial point      612
Initial value problem      A5
Initial value problem, existence and uniqueness of solutions of      A6
inner product of functions      327
Inner product of vectors      326
Inner product with respect to a weight      330
Integral representation of Bessel functions      261 264
Integral representation of Legendre polynomials      307
Integral representation of modified Bessel functions      388
Inverse cotangent      653
Inverse discrete Fourier transform      561
Inverse Fourier cosine transform      35
Inverse Fourier sine transform      135
Inverse Fourier transform      398
Inverse Laplace transform      485
Inverse tangent      652
Isolated ,singularity      682
Isotherm      220 130 657 668
Iteration methods      541
Jacobi iteration      511 512
Jordan’s Theorem      612
Korteweg-de Vries equation      419
Lagnerre’s differential equation      584 600
Laguerre coefficient      601
Laguerre polynomial      332 584
Laplace transform      480
Laplace transform and ordinary differential equations      488 193
Laplace transform and rational functions      488
Laplace transform inverse of      185
Laplace transform involving unit step functions      492
Laplace transform of ,       482
Laplace transform of $J_{0}$       500
Laplace transform of $t^{\alpha}$       181
Laplace transform of convolutions      194
Laplace transform of derivatives      183
Laplace transform of functions of exponential order      180
Laplace transform of partial derivatives      502
Laplace transform of periodic functions      499
Laplace transform of the error function      501
Laplace transform of the Gaussian function      101
Laplace transform, delined      480
Laplace transform, derivatives of      183
Laplace transform, existence of      480
Laplace transform, linearity      182
Laplace transform, operational properties      182
Laplace transform, shifting on the -axis      184
Laplace transform, shifting on the -axis      492
Laplace transform, table of      A70
Laplace’s equation      541
Laplace’s equation in a disk      218
Laplace’s equation in a wedge      222
Laplace’s equation in cylindrical coordinates      228
Laplace’s equation in cylindrical coordinates, solution      228 229
Laplace’s equation in polar coordinates      216
Laplace’s equation in rectangular coordinates      163
Laplace’s equation in spherical coordinates      270 274 281
Laplace’s equation in spherical coordinates with symmetry      274
Laplace’s equation invariance by conformal mapping      664
Laplace’s equation, boundary condition      164 216
Laplace’s equation, discretization of      535
Laplace’s equation, iteration methods for      541
Laplace’s equation, maximum principle for      189
Laplace’s equation, product solutions in spherical coordinates      271 273
Laplace’s equation, radially symmetric      270
Laplace’s equation, separating variables in spherical coordinates      270
Laplace’s equation, three dimensional      469
Laplace’s equation, two dimensional      104 163
Laplace’s integral formula for Legendre polynomials      307
Laplacian      104
Laplacian in cylindrical coordinates      196
Laplacian in polar coordinates      194 196
Laplacian in spherical coordinates      196 497
Laplacian, Green’s formula for its integral      648
Law of conservation of heat energv      443
Lebesgue, Henri      389
Legendre coefficient      312
Legendre function      300 303
Legendre scries      270 312
Legendre series with decreasing coefficients      318
Legendre series, polynomial of polynomials      318
Legendre series, test for pointwise convergence      318
Legendre series, test for uniform convergence      318
Legendre, polynomial      273 275 301 308 328
Legendre, polynomial formula for the      303
Legendre, polynomial integral formula      315
Legendre, polynomial, $P_{n}(0)$       305
Legendre, polynomial, $P_{n}(1)$       304 309
Legendre, polynomial, Bonnet’s resurrence relation      309
Legendre, polynomial, generating function      307
Legendre, polynomial, Laplace’s integral formula for      307
Legendre, polynomial, leading coefficient of      302

Legendre, polynomial, maxima and minima of      304
Legendre, polynomial, normalization of      302
Legendre, polynomial, orthogonality of      275 310
Legendre, polynomial, properties of      303 304
Legendre, polynomial, recurrence relation      309
Legendre, polynomial, recurrence relation for the coefficients      301
Legendre, polynomial, Rodrigues’ formula for      308
Legendre, polynomial, zeros of      304
Legendre’s differential equation      A18 271 272 275 300 304
Legendre’s differential equation, solution by the method of power series      300
Legendre’s differential equation, used to compute areas      614
Leibniz rule      314
Liebmann’s iteration method      543
Line integral      643
Line of force      657
Line of heat flow      668
Linear correction term      82
Linear dependence for functions      A7
Linear fractional transformation      675
Linear fractional transformation inverse of      676
Linear fractional transformation, composition of      676
Linear fractional transformation, determined bv three points      678
Linear fractional transformation, image of circle or a line by      678
Linear fractional transformation, is analvtic      675
Linear fractional transformation, is one to-one      676
Linear fractional transformation, mapping the disk to the upper half-plane      678
Linear fractional transformation, mapping the upper half-plane to the disk      678
Linear independence      A7 326
Linear independence, Wronskian criterion for      A7
Longitudinal vibrations of elastic bars      1 12
M $\ddot{o}$ bius transformation      see "Linear fractional transformation"
mass density      1 43
Maximum principle      169 187
Maximum principle for Laplace’s, equation      169 189
Maximum principle for the heat equation      188
Maximum principle, failure with internal source      492
Maximum principle, proof of the      190
Maximum-minimum principle      625
Mean square error      53
Mean square error in terms of an integral      53
Mean square error in terms of the Fourier coefficients      55
Mean value property of harmonic functions      623
Method of characteristic curves      5 108
Method of eigenfunction expansions      170 473 231
Method of Frobenius      51 2 37
Method of images      638
Method of power series      300
Method of separation of variables      13 115
Method of separation of variables and conformal mapping      669
Method of separation of variables in cylindrical coordinates      228
Method of separation of variables in polar coordinates      199 208
Method of separation of variables in quantum mechanics      573
Method of separation of variables in spherical coordinates      270
Method of separation of variables, applied to Schr $\ddot{o}$ dinger’s equation      575
Method of separation of variables, failure with nonhomogeneous problems      170
Method of separation of variables, gist of the      116
Method of successive overrelaxetion      543
Method of the Fourier transform      411- 413
Method of the Laplace transform      502
Method of undetermined coefficients      A16
Method of undetermined coefficients, superposition rule for      A17
Modified Bessel function      229
Modified Bessel function integral representation      388
Modified Bessel function, asymptotic formula      388
Modified Bessel’s equation      230 231 387
Moment of inertia      360 368
Moments of the Gaussian      596
Momentum of an electron      575 591
Momentum of an electron, as a Fourier transform      591
Multiply connected      613
Natural frequencv      72
Neumann condition      181 221
Neumann condition in a Poisson problem      688
Neumann condition in polar coordinates      221
Neumann condition on a rectangle      181
Neumann function      681
Neumann function and conformal mappings      687
Neumann function for regions depicted by figures      689
Neumann function for the first quadrant      688
Neumann function for the unit disk      690
Neumann function for the upper half-plane      686
Neumann function for unbounded regions      688
Neumann function, defined      681
Neumann function, normal derivative of      685
Neumann problem      441 442 618 619 684
Neumann problem in a rectangle      180
Neumann problem in the upper half-plane      645
Neumann problem on a disk      225
Neumann problem on simply connected region      684
Neumann problem, compatibility condition in      186 618
Neumann problem, solution by Neumann function      685
Neumann problem, solution in the upper half-plane      687 689
Neumann problem, solution on a disk      690
Neumann problem, uniqueness of solution      619
Newton’s law of gravitation      308
Newton’s second law of motion      7 574
Nodal line      160
Nonhomogencous heat equation on a half-line      470
Nonhomogencous wave equation, solved using Duhamel’s principle      174
Nonhomogeneous heat equation      463 165 169 473
Nonhomogeneous heat equation, solved using Duhamel’s principle      173
Norm of a function      327
Norm of a vector      326
Norm with respect to a weight      330
Normal dcrivative      617
Normal mode      10 70 118;
Normal mode for a rectangular membrane      157
Normal mode, amplitude of      73
Nyquist sampling rate      519
Open set      612
Operations on Fourier series      33
Ordinary differential equation      A2
Ordinary differential equation in standard form      A2
Ordinary differential equation with constant coefficients, general solution      A15
Ordinary differential equation with constant coefficients, nth order      A10
Ordinary differential equation, existence of fundamental sets of solutions      A1
Ordinary differential equation, first order, linear      A2
Ordinary differential equation, first order, linear, solution of      A2
Ordinary differential equation, homogeneous      A2
Ordinary differential equation, linear      A2
Ordinary differential equation, nonhomogeneous      A2
Ordinary differential equation, nonhomogeneous, general solution of      A5
Ordinary differential equation, nonhomogeneous, particular solution of      A5
Ordinary differential equation, order of      A2
Ordinary differential equation, second order with constant coefficients      A16
Ordinary differential equation, second order, general solution      A16
Ordinary differential equation, solution of      A2
Ordinary point      A43 A51
Orthogonal functions      21 327
Orthogonal set of functions      327
Orthogonal trajectories      657
Orthogonal vectors      326
Orthogonality in Sturm - Liouville theory      336
Orthogonality of associated Legendre functions      322
Orthogonality of Bessel functions      250 252 338
Orthogonality of Chebyshev polynomials      315
Orthogonality of complex exponential system      65
Orthogonality of eigenfunctions in fourth order Sturm - Liouville problems      353
Orthogonality of eigenfunctions in Sturm - Liouville theory      336
Orthogonality of eigenfunctions of Helmholtz equation      232
Orthogonality of eigenfunctions of Helmholtz equation in a ball      293
Orthogonality of generalized Laguerre polynomials      603
Orthogonality of Hernite polynomials      598
Orthogonality of Laguerre polynomials      601
Orthogonality of Legendre polynomials      273 310 338
Orthogonality of spherical Bessel functions      261
Orthogonality of spherical harmonies      283
Orthogonality of trigonometric system      21
Orthogonality with respect to a weight      330 343
Orthonormal set of functions      327

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