Page 479 - Numerical Methods for Chemical Engineering

468 Index

Finite element method (FEM) (cont.) orthogonal functions 164
residual function 304 orthogonal polynomials 165
weight function 304 scalar product 164
weighted residual methods (see also Weighted singularities 166
residual methods) 304–305 square integrable functions 164
Finite volume method 297–299 weighted integrals 164
Floating Point Operation (FLOP) 18 Genetic algorithm 362–364
Fokker-Planck equation 347–351 Gershgorin’s theorem 112
in 1-D 350 Gibbs oscillations 437
corresponding SDE 351 GMRES method 287–288
in multiple dimensions 353 Gouy-Chapman theory 313
corresponding SDE 353 Gradient optimization methods 213–223
spurious drift 351 Gradient vector 212
Forward Kolmogorov equation 350 Gram-Schmidt orthogonalization 28
Fourier analysis 436–459
aliasing 446 Hamilton-Jacobi-Bellman (HJB) equation 249
convolution 447–449 numerical solution by finite differences 275
convolution theorem 448 Heaviside step function 232
correlation 449–450 Hermetian
Dirichlet’s theorem 436 conjugate 119
discrete Fourier transform 443 matrix 119
Fast Fourier Transform (FFT) 444 Hessian matrix
MATLAB fft, ifft 445–446 approximation by BFGS formula 224
MATLAB fft2, ifft2, fftn, ifftn 451–452 normal mode analysis 134
Fourier series 436–439 optimization 212, 223
Fourier Transform (FT) pair 439–446 Homotopy 88, 203
exponential-form Fourier series 438 Householder transformation (reflection) 129
Gibbs oscillations 437 Hyperbolic PDEs 278
in 1-D 436
discrete Fourier Transform 443 Identity matrix 37
Fourier Transform pair 439 Index of DAE system 198
MATLAB fft, ifft 445–446 Initial value problems (IVPs) 154–208
in multiple dimensions 450–452 arc length continuation 203
convolution 450 Differential Algebraic Equation (DAE) systems
correlation 450 195–202
discrete Fourier Transform 451 consistent initial conditions 198
Fourier transform pair 450 index 198
MATLAB fft2, ifft2, fftn, ifftn 451–452 mass matrix 195
periodic function 436 MATLAB ode15s 198
power spectrum 445 standard form 195
scattering theory (see also Scattering theory) Example. dynamic simulation of CSTR with two
452–458 reactions 181–183
Functional derivative 359 Example. dynamics on the 2-D circle 199
Example. heterogeneous catalysis in a packed bed
Galerkin method 304–305 reactor 199–202
Gauss-Markov conditions 384 Ordinary Differential Equation (ODE) systems
Gaussian elimination 10–23 standard form 155
basic algorithm 17 time-marching algorithms 176–184; A-stable
elementary row operation 12 methods 188; absolute stability 187;
fill-in 54, 284 Backward Difference Formula (BDF) methods
Gauss-Jordan elimination 19 179, 195–198; backward (implicit) Euler
MATLAB mldivide ‘/’ 53, 56 method (see also Euler integration method)
partial pivoting 20, 21 176; Crank-Nicholson method 176; error
solution of triangular systems by substitution 17, 18 analysis; local errors 186; global error 187;
Gaussian (normal) distribution 331–332 order of accuracy 187; rejection properties
MATLAB normrnd 334 190
MATLAB randn 334 forward (explicit) Euler method (see also Euler
multivariate distribution 337 integration method) 177; explicit methods
Gaussian quadrature 163–166 176; implicit methods 176; MATLAB ODE
accuracy 166 solvers 181–183; multi-step methods 178;
Legendre polynomials 166 MATLAB ode15s 182, 208
Lobatto quadrature 166 numerical stability 187, 188; predictor-corrector
MATLAB quadl 166 methods 180; Runge-Kutta method, 2nd order