Numerical Methods for Chemical Engineering: Applications in MATLAB Book

Preface ix

1 Linear algebra 1

Linear systems of algebraic equations 1

Review of scalar, vector, and matrix operations 3

Elimination methods for solving linear systems 10

Existence and uniqueness of solutions 23

The determinant 32

Matrix inversion 36

Matrix factorization 38

Matrix norm and rank 44

Submatrices and matrix partitions 44

Example. Modeling a separation system 45

Sparse and banded matrices 46

MATLAB summary 56

Problems 57

2 Nonlinear algebraic systems 61

Existence and uniqueness of solutions to a nonlinear algebraic equation 61

Iterative methods and the use of Taylor series 62

Newton's method for a single equation 63

The secant method 69

Bracketing and bisection methods 70

Finding complex solutions 70

Systems of multiple nonlinear algebraic equations 71

Newton's method for multiple nonlinear equations 72

Estimating the Jacobian and quasi-Newton methods 77

Robust reduced-step Newton method 79

The trust-region Newton method 81

Solving nonlinear algebraic systems in MATLAB 83

Example. 1-D laminar flow of a shear-thinning polymer melt 85

Homotopy 88

Example. Steady-state modeling of a condensation polymerization reactor 89

Bifurcation analysis 94

MATLAB summary 98

Problems 99

3 Matrix eigenvalue analysis 104

Orthogonal matrices 104

A specific example of an orthogonal matrix 105

Eigenvalues and eigenvectors defined 106

Eigenvalues/eigenvectors of a 2 x 2 real matrix 107

Multiplicity and formulas for the trace and determinant 109

Eigenvalues and the existence/uniqueness properties of linear systems 110

Estimating eigenvalues; Gershgorin's theorem 111

Applying Gershgorin's theorem to study theconvergence of iterative linear solvers 114

Eigenvector matrix decomposition and basis sets 117

Numerical calculation of eigenvalues and eigenvectors in MATLAB 123

Computing extremal eigenvalues 126

The QR method for computing all eigenvalues 129

Normal mode analysis 134

Relaxing the assumption of equal masses 136

Eigenvalue problems in quantum mechanics 137

Single value decomposition SVD 141

Computing the roots of a polynomial 148

MATLAB summary 149

Problems 149

4 Initial value problems 154

Initial value problems of ordinary differential equations (ODE-IVPs) 155

Polynomial interpolation 156

Newton-Cotes integration 162

Gaussian quadrature 163

Multidimensional integrals 167

Linear ODE systems and dynamic stability 169

Overview of ODE-IVP solvers in MATLAB 176

Accuracy and stability of single-step methods 185

Stiff stability of BDF methods 192

Symplectic methods for classical mechanics 194

Differential-algebraic equation (DAE) systems 195

Parametric continuation 203

MATLAB summary 207

Problems 208

5 Numerical optimization 212

Local methods for unconstrained optimization problems 212

The simplex method 213

Gradient methods 213

Newton line search methods 223

Trust-region Newton method 225

Newton methods for large problems 227

Unconstrained minimizer fminunc in MATLAB 228

Example. Fitting a kinetic rate law to time-dependent data 230

Lagrangian methods for constrained optimization 231

Constrained minimizer fmincon in MATLAB 242

Optimal control 246

MATLAB summary 252

Problems 252

6 Boundary value problems 258

BVPs from conservation principles 258

Real-space vs. function-space BVP methods 260

The finite difference method applied to a 2-D BVP 260

Extending the finite difference method 264

Chemical reaction and diffusion in a spherical catalyst pellet 265

Finite differences for a convection/diffusion equation 270

Modeling a tubular chemical reactor with dispersion; treating multiple fields 279

Numerical issues for discretized PDEs with more than two spatial dimensions 282

The MATLAB 1-D parabolic and elliptic solver pdepe 294

Finite differences in complex geometries 294

The finite volume method 297

The finite element method (FEM) 299

FEM in MATLAB 309

Further study in the numerical solution of BVPs 311

MATLAB summary 311

Problems 312

7 Probability theory and stochastic simulation 317

The theory of probability 317

Important probability distributions 325

Random vectors and multivariate distributions 336

Brownian dynamics and stochastic differential equations (SDEs) 338

Markov chains and processes; Monte Carlo methods 353

Genetic programming 362

MATLAB summary 364

Problems 365

8 Bayesian statistics and parameter estimation 372

General problem formulation 372

Example. Fitting kinetic parameters of a chemical reaction 373

Single-response linear regression 377

Linear least-squares regression 378

The Bayesian view of statistical inference 381

The least-squares method reconsidered 388

Selecting a prior for single-response data 389

Confidence intervals from the approximate posterior density 395

MCMC techniques in Bayesian analysis 403

MCMC computation of posterior predictions 404

Applying eigenvalue analysis to experimental design 412

|Bayesian multi response regression 414

Analysis of composite data sets 421

Bayesian testing and model criticism 426

Further reading 431

MATLAB summary 431

Problems 432

9 Fourier analysis 436

Fourier series and transforms in one dimension 436

1-D Fourier transforms in MATLAB 445

Convolution and correlation 447

Fourier transforms in multiple dimensions 450

Scattering theory 452

MATLAB summary 459

Problems 459

References 461

Index 464<