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Figures from
Modeling and Analysis Principles for Chemical and Biological Engineers

Michael D. Graham James B. Rawlings
Department of Chemical and Biological Engineering Department of Chemical and Biological Engineering
University of Wisconsin University of Wisconsin
Madison, Wisconsin Madison, Wisconsin

Copyright (C) 2018 Nob Hill Publishing, LLC

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Chapter 1: Linear Algebra

Figure 1.3 (page 31):
An iteration of the Newton-Raphson method for solving f(x)=0 in the scalar case.
Figure 1.6 (page 60):
Contours of constant f(x)=x^TAx.
Figure 1.8 (page 74):
Experimental measurements of variable y versus x.
Figure 1.9 (page 76):
Measured rate constant at several temperatures.
Figure 1.10 (page 84):
Plot of Ax as x moves around a unit circle.

Chapter 2: Ordinary Differential Equations

Figure 2.1 (page 103):
Dynamical regimes for the planar system dx/dt = Ax, A \in \mathbb {R}^{2 x2}.
Figure 2.2 (page 104):
Dynamical behavior on the region boundaries for the planar system dx/dt = Ax, A \in \mathbb {R}^{2 x2}.
Figure 2.4 (page 123):
Function f(x)=o{exp}{\right .}\box }-8{\right .}\box }\frac {x}{\pi }{\right .}\box }^{2}{\right .}\box } and truncated trigonometric Fourier series approximations with K=2,5,10. The approximations with K=5 and K=10 are visually indistinguishable from the exact function.
Figure 2.5 (page 125):
Truncated trigonometric Fourier series approximation to f(x)=x, using K=5,10, 50. The wiggles get finer as K increases.
Figure 2.6 (page 127):
Function f(x)=o{exp}{\right .}\box }-8x^{2}{\right .}\box } and truncated Legendre-Fourier series approximations with n=2,5,10.
Figure 2.7 (page 128):
Function f(x)=H(x) and truncated Legendre-Fourier series approximations with n=10,50,100.
Figure 2.12 (page 173):
Leading-order inner U_0, outer u_0, and composite solutions u_{0c}, for Example 2.30 with \epsilon =0.2, K=1, and k_{2}=1.
Figure 2.15 (page 184):
Contours of an energy function V(x_{1},x_{2}) or H(x_{1},x_{2}).
Figure 2.16 (page 188):
Energy landscape for a pendulum; H = \frac {1}{2} p^2 -\kappa o{cos}q; \kappa =2.
Figure 2.17 (page 189):
Landscape for H=\frac {1}{2}p^{2}+\frac {1}{4}q^{4}-\frac {1}{2}q^{2}.
Figure 2.18 (page 191):
A limit cycle (thick dashed curve) and a trajectory (thin solid curve) approaching it.
Figure 2.19 (page 192):
Periodic (left) and quasiperiodic (right) orbits on the surface of a torus. The orbit on the right eventually passes through every point in the domain.
Figure 2.20 (page 194):
A limit cycle for the R\"ossler system, a=b=0.2, c=1.
Figure 2.21 (page 194):
A strange attractor for the R\"ossler system, a=b=0.2, c=5.7.
Figure 2.22 (page 196):
Bifurcation diagram for the saddle-node bifurcation.
Figure 2.23 (page 197):
Bifurcation diagram for the transcritical bifurcation.
Figure 2.24 (page 198):
Bifurcation diagrams for the pitchfork bifurcation.
Figure 2.25 (page 203):
Approximate solutions to \dot {x}=-x using explicit and implicit Euler methods with \Delta t=2.1, along with the exact solution x(t)=e^{-t}.
Figure 2.26 (page 206):
Stability regions for Adams-Bashforth methods; \dot {x}=\lambda x.
Figure 2.27 (page 207):
Stability regions for Adams predictor-corrector methods; \dot {x}=\lambda x.
Figure 2.28 (page 208):
Stability regions for Runge-Kutta methods; \dot {x}=\lambda x.
Figure 2.29 (page 211):
Hat functions for N=2.
Figure 2.30 (page 213):
Approximate solutions to (2.89) using the finite element method with hat functions for N=6 and N=12. The exact solution also is shown.
Figure 2.31 (page 217):
Dependence of \left | c(j) \right | on j for the Legendre-Galerkin approximation of (2.89) with n=10.
Figure 2.34 (page 239):
Stability regions for Adams predictor-corrector methods; \dot {x}= \lambda x; APCn' uses nth-order predictor and nth-order corrector.

Chapter 3: Vector Calculus and Partial Differential Equations

Figure 3.12 (page 315):
Concentration versus membrane penetration distance for different reaction rate constants.
Figure 3.13 (page 336):
Transient heating of slab, cylinder, and sphere.

Chapter 4: Probability, Random Variables, and Estimation

Figure 4.1 (page 357):
Normal distribution, with probability density p_\xi (x) = (1/\sqrt {2\pi \sigma ^2}) o{exp}(-(1/2) (x-m)^2/\sigma ^2).
Figure 4.2 (page 362):
Multivariate normal for n=2. The contour lines show ellipses containing 95, 75, and 50 percent probability.
Figure 4.5 (page 375):
A joint density function for the two uncorrelated random variables in Example4.8.
Figure 4.6 (page 379):
A nearly singular normal density in two dimensions.
Figure 4.8 (page 386):
Histogram of 10,000 samples of uniformly distributed x.
Figure 4.9 (page 386):
Histogram of 10,000 samples of \displaystyle y=\DOTSB \sum@ \slimits@ _{i=1}^{10} x_i .
Figure 4.10 (page 401):
The multivariate normal, marginals, marginal box, and bounding box.
Figure 4.11 (page 425):
The sum of squares fitting error (top) and validation error (bottom) for PCR versus the number of principal components \ell ; cross validation indicates that four principal components are best.
Figure 4.12 (page 426):
The sum of squares validation error for PCR and PLSR versus the number of principal components/latent variables \ell ; note that only two latent variables are required versus four principal components.
Figure 4.13 (page 427):
Predicted versus measured outputs for the validation dataset. Top: PCR using \textit {four} principal components. Bottom: PLSR using \textit {two} latent variables. Left: first output. Right: second output.
Figure 4.14 (page 428):
Effect of undermodeling. Top: PCR using \textit {three} principal components. Bottom: PLSR using \textit {one} latent variable.
Figure 4.15 (page 431):
The indicator (step) function f_1(w;x) and its smooth approximation, f(w;x).
Figure 4.16 (page 448):
Typical strain versus time data from a molecular dynamics simulation from data file \texttt {rohit.dat} on the website \jbrawweb .
Figure 4.18 (page 454):
Smooth approximation to a unit step function, H(z-1).

Chapter 5: Stochastic Models and Processes

Figure 5.1 (page 465):
A simulation of the Wiener process with fixed sample time \Delta t= 10^{-6} and D=5x10^5.
Figure 5.2 (page 466):
Sampling faster on the last plot in Figure 5.1; the sample time is decreased to \Delta t = 10^{-9} and the roughness is restored on this time scale.
Figure 5.3 (page 473):
A representative trajectory of the discretely sampled Brownian motion; D=2, V=0, n=500.
Figure 5.4 (page 473):
The mean square displacement versus time; D=2, V=0, n=500.
Figure 5.5 (page 481):
Two first-order reactions in series in a batch reactor, c_{A0}=1, c_{B0}=c_{C0}=0, k_1=2, k_2=1.
Figure 5.6 (page 483):
A sample path of the unit Poisson process.
Figure 5.7 (page 483):
A unit Poisson process with more events; sample path (top) and frequency distribution of event times \tau .
Figure 5.9 (page 490):
Stochastic simulation of first-order series reaction A-> B-> C starting with 100 A molecules.
Figure 5.11 (page 494):
Solution to master equation for A + B <-> C starting with 20 A molecules, 100 B molecules and 0 C molecules, k_1=1/20, k_{-1}=3.
Figure 5.12 (page 495):
Solution to master equation for A + B <-> C starting with 200 A molecules, 1000 B molecules and 0 C molecules, k_1=1/200, k_{-1}=3.
Figure 5.13 (page 497):
The equilibrium reaction extent's probability density for Reactions 5.52 at system volume \Omega =20 (top) and \Omega = 200 (bottom). Notice the decrease in variance in the reaction extent as system volume increases.
Figure 5.14 (page 501):
Simulation of 2 A \mathrel {\mkern 4mu}\mathrel {\smash {\mathchar "392D}}\mathrel {\mkern -2.5mu}-> B for n_0=500, \Omega =500. Top: discrete simulation; bottom: SDE simulation.
Figure 5.15 (page 502):
Cumulative distribution for 2 A \mathrel {\mkern 4mu}\mathrel {\smash {\mathchar "392D}}\mathrel {\mkern -2.5mu}-> B at t=1 with n_0=500, \Omega =500. Discrete master equation (steps) versus omega expansion (smooth).
Figure 5.16 (page 516):
The change in 95% confidence intervals for \hat {x}(k\delimiter "026A30C k) versus time for a stable, optimal estimator. We start at k=0 with a large initial variance P(0), which gives a large confidence interval.
Figure 5.17 (page 526):
Deterministic simulation of reaction A + B <-> C compared to stochastic simulation.