Bernatz R. Fourier Series and Numerical Methods for Partial Differential Equations
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INDEX
2D transport equation, 280
accumulation error, 227
accuracy, 208
adaptive methods, 210
algebraic equation, 208
alternating direction implicit (ADI), 236
analytic function, 6
backwards difference, 222
basis,
21, 216
BCs
linear homogeneous, 9
Bessel function
first kind of order n, 88
Bessel's equation, 85
Bessel's inequality, 45
bilinear form, 22
boundary condition, 7
second type, 100
boundary conditions
Dirichlet, 100
first
type,
100
Neumann, 101
third type, 101
boundary conditions of the first kind, 7
boundary conditions of the third kind, 7,
boundary function coefficients, 273
boundary functions, 282
boundary value problem, 1
Burgers' equation, 278
calculus of variations, 8, 216
Cauchy data, 5
Cauchy-Euler equation, 94
Cauchy-Kowalevski theorem, 5
centered difference, 222
Cholesky, 261
closed orthonormal set, 65
coefficient matrix, 226
composite grid, 212
conditionally stable, 231
conduction, 97
conservation of energy, 98
consistent, 218
control volume method, 217
convection, 97
converge
discrete method, 218
convergent, 218
convergent infinite series, 29
convolution, 128
Fourier Series and Numerical Methods for Partial Differential Equations,
First Edition. By Richard Bernatz
Copyright ©2010 John Wiley & Sons, Inc.
315
316 INDEX
coordinate systems, 208
corresponding Fourier series, 31
Crank-Nicolson (CN), 275
differentiable series, 29
direct methods, 261
Dirichlet condition, 7
Dirichlet kernel, 52
discretization error, 218
discretizing, 207
Duhamel's theorem
ID FA method, 299
eigenfunction expansion, 8
eigenfunctions, 11
eigenvalues, 11
elements, 207
equation
classification, 1, 2
partial differential, 2
error
accumulation, 227
roundoff,
227
truncation, 226
existence, 5
explicit finite difference technique, 275
FA method
consistency, 290
convergence, 290
stability, 290
FA solution
ID heat transport equation, 274
hybrid, 279
Burgers' equation, 278
FD method
explicit
stability, 276
semi-implicit
stability, 277
finite analytic coefficients
2D uniform grids, 284
up-winding, 286
finite analytic method
element, 217
outlined, 217
subproblems, 283
finite analytic solution
Laplace's equation, 288
Poisson equation, 287
finite difference
Forsythe and Wasow (FW), 275
finite difference (FD), 220
finite difference method
element, 214
outlined, 214
finite element
ID non-homogeneous heat equation, 264
finite element method
element, 216
general framework, 250
outlined, 216
weighted residual method, 216
forwards difference, 222
Fourier cosine series, 35
Fourier method, 9
Fourier Series
alternate form, 51
valid, 31
Fourier series, 1, 38
Fourier series representation, 31
Fourier Sine series, 30
Fourier's law, 99
Fourier, Joseph, 9
functional, 250
Galerkin method, 251
general Fourier series, 30
general principle of superposition, 28
geometric adaptation, 209, 210
Gibbs phenomena, 68
Gibbs phenomenon, 40
Gram-Schmidt process, 63
grid generation, 208
grid points, 207
heat equation, 3
ID,
97, 223
ID,
unsteady
explicit, 224
implicit, 224
2D,
unsteady, 234
explicit, 236
fully implicit, 236
heat exchange coefficient, 101
Hubert space, 30
IBVP
semihomogeneous, 116
implicit form
ID diffusion equation, 229
implicit-explicit parameter, 275
infinitely long string, 184
initial boundary value problem, 7
initial condition, 7
initial value problem, 1
inner product, 22
inner product space, 22
insulated boundary, 101
integral transforms, 8
intrinsic velocity, 183
intrinsic wave speed, 16
irregular coordinates, 210
jump discontinuity, 24
kernel, 26
Laplace equation, 3, 150
Laplace transform, 127
Laplacian, 105
left-hand limit, 25
Legendre function of the second kind, 92
Legendre polynomial, 92
Legendre's equation, 90
linear form, 22
linear function, 26
linearly independent vectors, 21
mean square error, 41
mixed BCs, 142, 163
mixed boundary conditions, 8
mixed partial, 2
moving boundary, 209
multigrids, 212
multilevel methods, 212
Neumann condition, 8
nodes
cell center, 207
cell vertex, 207
nonuniform grids, 210
normalized, 32
numerical diffusion, 210
numerical error, 226
optimal grid, 209
order of magnitude notation, 221
orthonormal, 31
partial derivative, 1
PDE
classification, 2
elliptic, 2, 3
general form, 2
homogeneous, 2
hyperbolic, 2
linear, 2
non-homogeneous, 2
nonlinear, 2
order, 2
parabolic, 2
quasilinear, 2
Peclet number, 272
piecewise continuous, 24
piecewise smooth, 47
Poisson equation, 150
principle of superposition, 26, 283
regional coordinates, 210
remainder, 221
removable discontinuity, 24
residual, 216
Reynolds number, 272
cell, 285
Riemann-Lebesgue Lemma, 51
right-hand limit, 25
Robin conditions, 7
roundoff error, 227
scalar multiplication, 19
scalar product, 22
Schwartz inequality, 63
second partial, 2
semi-implicit, 237
semihomogeneous, 113
separation of variables, 1, 8
shape, 216
singular
Sturm-Liouville problem, 94
solution adaptation, 209
solution adaptive, 210
sparse Matrix, 236
stability, 227
explicit case, 227
implicit case, 229
von Neumann method, 227
stiffness matrix, 260, 269
Sturm-Liouville, 10
eigenfunctions, 70
eigenvalues, 70
regular, 70
singular, 71
spectrum, 70
symmetric bilinear form, 22
Taylor series, 220
telephone equation, 183
thermal conductivity, 99
thermal diffusivity, 4, 100
triangle inequality, 63
tridiagonal matrix, 226
truncation error (TE), 221, 226
local, 218
order, 221
unconditionally stable, 231
uniform convergence, 67
uniqueness, 5
318
INDEX
variation of parameters, 16, 117
variational problem, 250
solution, 250
vector
distance between vectors, 22
norm, 22
orthogonal vectors, 23
vector addition, 19
vector space, 19
additive inverse, 20
axioms, 19
dimension, 21
infinite-dimensional, 21
inner product, 22
orthogonal basis, 23
orthonormal basis, 23
spanning set, 21
subspace, 21
zero vector space, 21
vector spaces
finite dimensional, 21
velocity
correction term, 281
von Neumann method, 227
von Neumann stability analysis, 231
wave equation, 3
ID
polar coordinates, 195
ID implicit FD, 231
d'Alembert's
solution, 184
one-dimensional, 183
weak solutions, 251
weighted residual methods, 216
well-posed problem, 7
Wronskian, 17
zero vector space, 21