Page 480 - Numerical Methods for Chemical Engineering

Index 469

(RK 2) 178; Runge-Kutta method, 4th order Jordan form 118
(RK 4) 177; Runge-Kutta-Fehlberg method of a normal matrix 121
(RKF 45) 178; MATLAB ode45 182, 206
single-step methods 176; stiff system algorithms Karush-Kuhn-Tucker (KKT) conditions 238
180, 182, 192; stiff decay 192; symplectic Kroenecker delta 5
methods 194; time step restrictions 190–191; Krylov subspace 287
velocity Verlet method 195
Partial Differential Equation (PDE) systems Lagrange multiplier 234
Example. dynamic simulation of a tubular Lagrange’s equation of motion 136
chemical reactor 282 Lagrangian function
stiffness 191 classical mechanics 136
stochastic PDEs 358–360 optimization 234
state vector 155 Landau free energy model 358
Stochastic Differential Equations (SDEs) (see also Langevin equation 340, 343
Stochastic simulation) 342–353 Lennard-Jones interaction model 368
explicit Euler SDE method 343 Levenberg-Marquardt method 389
Mil’shtein SDE method 346 Line searches 216–217
Integration backtrack (Armijo) line search 216
Initial value problems (IVPs) (see also Initial value strong line search 216
problems) 155 weak line search 216
MATLAB quad 163 Linear algebraic systems 1–57
MATLAB trapz 140 as linear transformation 23
Monte Carlo method (see also Monte Carlo) 168, BVPs of high dimension 282–294
360–361 dimension theorem 31
numerical (see also Quadrature) 162 Example. 1-D laminar flow of Newtonian fluid
orthogonal functions 164 47–54
scalar product 164 Example. modeling a separation system 45–46
square integrable functions 164 existence of solution 30, 110, 143
weighted integrals 164 least-squares approximation solution 145
Interpolation MATLAB mldivide ‘/’ 53, 56
Hermite method 160 null space, kernel 29, 144
Lagrange method 157 range 30, 144
MATLAB interp1 100, 161 solution by Gaussian elimination (see also
Newton method 157 Gaussian elimination) 10–23, 284
polynomial methods 156–161 solution by iterative methods (see also Iterative
support points 156 linear solvers) 285–291
Iterative linear solvers solution by SVD 143
Conjugate Gradient (CG) method (see also uniqueness of solution 30, 110, 143
Conjugate Gradient (CG) method) 286–287 LU factorization 38
Gauss-Seidel method 285–286 incomplete LU factorization 290
Generalized Minimum RESidual (GMRES) method MATLAB lu 57
287–288 MATLAB luinc 291
Jacobi method 114, 285–286 use in calculating matrix inverse 37
Krylov subspace 287
MATLAB bicg 287 Macosko-Miller method 322
MATLAB bicgstab 287 Markov chain 354
MATLAB gmres 287 Markov process 353
MATLAB pcg 285 Mass matrix
preconditioners (see Preconditioner matrix) classical mechanics 136
288–291 of DAE system 195
Successive Over-Relaxation (SOR) method MATLAB commands
285 adaptmesh 310
Symmetric SOR (SSOR) method 286 bicg 287
use for BVPs of high dimension 282–294 bicgstab 287
Itˆo-type SDE 343 binocdf 330
Itˆo’s lemma 345 binofit 330
binoinv 330
Jacobi method 114, 285 binopdf 330
Jacobian matrix 73 binornd 330
approximating by Broyden’s method 77 binostat 330
dynamic stability 172 chol 43, 57
estimating by finite differences 77 cholinc 291
Joint probability 320 cond 113