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Chapters
Focus Analysis / Calculus
Foundations
Functions of Several Variables
Multivariate Functions
Common Types of Multivariate Functions
Curves
Surfaces
Scalar Fields
Vector Fields
Topology in Higher Dimensions
Complement of a Set
Euclidean Norm
The ε-Ball
Inner Points
Interior
Open Sets
Boundary Points
Boundary
Closure
Closed Sets
Bounded Sets
Compact Sets
Convex Sets
Continuity
Convergence of Vector Sequences
Continuity of Multivariate Functions
Partial Differentiation
Directional Derivative
Partial Derivatives
The Gradient
Steepest Ascent and Descent
Isolines
Second-Order Partial Derivatives
Smoothness
Hessian Matrix
Stationary Points and Extrema
Multivariate Functions
Jacobian Matrix
Calculation Rules
Differential Operators
Nabla Operator
Laplace Operator
Divergence
Rotation
Taylor Expansion
Multivariate Taylor Polynomials
Zeroth Order: A Flat Plateau
First Order: The Tangent Hyperplane
Second Order: A Quadratic Bowl
Matching f at the Base Point
Coordinate Transformations
Cartesian Coordinates
Definition
Coordinate Transformation Matrix and the Jacobian
Common Coordinate Systems
Polar Coordinates
Cylindrical Coordinates
Spherical Coordinates
Roots and Optima
Roots
The 1D Reminder
The Geometric Idea
Why a Neighborhood, and Why Quadratic
Lifting the Iteration to d Dimensions
Solve, Don’t Invert
Stopping Conditions
Local Convergence Theorem
Optima
Global Extrema
Local Extrema
Strict Extrema
Stationary Points
Saddle Points
Definiteness of a Symmetric Matrix
Eigenvalue Criterion
Classifying Stationary Points
Curves and Surfaces
Curves
Image (Trace)
Start, End, and Closed Curves
Smoothness
Surfaces
Regularity
Bézier Surfaces in Computer Graphics
Quadrature
Double and triple integrals
Fubini’s theorem — the order doesn’t matter
Standard domains
Integration over standard domains
Area and volume of a standard domain
Integration under coordinate transformations
Surface integrals
Scalar surface integrals
Vector surface integrals
Surface area
Gauss’s divergence theorem
Applications
Statistics — computing moments
Machine learning — kernel density estimation
Partial Differential Equations
First-order PDEs
Linear with constant coefficients
Linear with variable coefficients
Quasilinear
Second-order PDEs
Stationary vs instationary
Canonical linear examples
Canonical nonlinear examples
Introduction to Mathematical Modeling
Terminology
Model
Mathematical Modeling
Simulation
Acceptance Across Fields
Aims of Simulation
Simulation Pipeline
Modeling
Numerics
Implementation
Visualization
Validation
Embedding
Disciplines along the pipeline
Examples of Applications
A non-exhaustive tour
Hard models and soft models
The questions this raises
Derivation of Models
What exactly should be modeled?
Which quantities matter?
Relationships among the relevant quantities
Descriptive tools
Algebraic equations and inequalities
Systems of ordinary differential equations
Systems of partial differential equations
Automata and state transition diagrams
Graphs
Probability distributions
Fuzzy logic
Neural networks
Algebraic structures
Simulation tasks
Analysis of Models
Existence of solutions
Uniqueness of solutions
Continuous dependency on input data
Well-posed and ill-posed problems
Applicability of the model
Solution approaches
Analytic
Heuristic
Direct-numerical
Approximate-numerical
Model assessment
Validation
Accuracy
Classification of Models
Discrete vs. continuous
Phenomenon character vs. model character
Deterministic vs. stochastic
Phenomenon character vs. model character
Scales, Hierarchy
Where scale shows up in practice
Multiscale problems
Hierarchy centered around humans
Discrete Modeling and Simulation
Decision Models: Games, Strategies, Elections
Generic Decision Model
Three classes of decision situations
The model
Strategy under certainty
Strategies under risk
Caution (max-min payoff)
Full risk (max-max payoff)
Alternative-caution (min-max regret)
Pessimism–optimism (Hurwicz)
Principle of insufficient reason (Laplace)
Quiz
Two-player Zero-sum Game
The model
Sequential play
Best response functions
Simultaneous play
The maximin–minimax inequality
Saddle points
Mixed strategies
Existence of equilibria
Quiz
Group Decision Making
Rank mappings
Preference relations
Many voters
Relations and their properties
The structure of preference relations
Collective choice function
Majority decision
Rank addition
Dictatorships
The democratic basic rules
Rule 1: Universal domain
Rule 2: Collective rationality
Rule 3: Pareto condition
Rule 4: Independence of irrelevant alternatives
Rule 5: No-dictator condition
Arrow’s impossibility theorem
A weakness of the model, or of reality?
Quiz
Scheduling
Problem setup
Model parameters
Goal
Machine characterization
Process scheduling
The precedence graph
Admissible schedules and DAGs
Constructing schedules
Critical paths
A worked example
Job-shop problems
Admissibility
Conjunctive and disjunctive edges
Optimization
Precedence-list scheduling
The heuristic
A worked example
Why shortening hurt
Randomization
A worked example
Hypergraphs
Continuous Modeling and Simulation
Population Dynamics: Models and their Numerical Aspects
Models for Population Dynamics
The Malthus model
A discrete model?
An aside: interest rates
Is exponential growth realistic?
A first refinement: Verhulstian saturation
The saturation equation
Is saturation realistic?
A second refinement: logistic growth
Reading the logistic curve
Why quadratic, and not cubic?
An aside: two generalizations
Another possibility: oscillations
More than one species
Stability of equilibria
A worked example: the arms race
Equilibria of the arms race
Trajectories of the arms race
Equilibrium
Explosion
Spiral
The two-species Jacobian
Competition
Competition: worked examples
Equilibrium
Extinction of P
What becomes of Q alone
Extinction of Q
Degeneration: parallel nullclines
Degeneration: a line of equilibria
Predator and prey
The Lotka–Volterra model
Stability and the Lotka–Volterra orbit
Numerical Treatment of ODE
Initial value problems and boundary value problems
Reading the prototype
A short excursus: discretization
Representing numbers
Resolution — accuracy is relative
Beyond numbers
Rounding and round-off error
Arithmetic on machine numbers
Further sources of errors
One-step methods
Two symbols that look alike: y(t) versus yk
The Euler method
Discretization error: local, global, and what “convergence” means
Comparing the two errors, step by step
Order, and getting more of it: the Heun method
Runge–Kutta methods
Multistep methods
Starting up: the first few steps
Implicit methods
Predictor–corrector
Higher-order equations and systems
What makes an ODE hard to solve
Ill-conditioning
Stability
Stiffness
The three phenomena at a glance
Boundary value problems
Discretizing the second derivative
The tridiagonal system
First derivatives: central differences and upwinding
Kinds of boundary conditions
The shooting method
Summary
Control Engineering: Deterministic and Fuzzy Logic Approaches
The Basics of Control Theory
Outer and inner control loops
Notation: the feedback loop
Important types of controllers
Example: linear feedback control
Solution of the ODE system
Dimensioning of the controller (matrix K)
Control via Fuzzy Logic
Crisp sets and fuzzy sets
Shapes of the membership function
Operations on fuzzy sets
Linguistic variables
Fuzzy control systems
Step 1: Fuzzification
Step 2: The rule base
Step 3: Inference
Step 4: Defuzzification
Example: the inverted pendulum
A fuzzy controller for the pendulum
Summary
Heat Transfer: Model and Numerical Solution
Introduction
Resolving Space
Spatially Resolving Population Models
System Layer vs. Process Layer
Modeling with PDE
Where Space Is Essential
With or Without Time?
The Object of Desire: Temperature
The Scenario
A Multi-Piece Model
Deriving the Heat Equation
The equation we are after
A balance equation from energy conservation
From the balance to the PDE
The general recipe
Boundary and Initial Conditions
Types of boundary conditions
Which combinations are admissible
Back to the scenario
How Many Spatial Dimensions?
Solving the Model
Classifying PDEs
The general second-order linear form
Three types of PDE
Numerics of PDE
Three Families of Discretization
The Finite Difference Method
What a stencil is
Setting up the equations
Example: the Poisson equation on the unit square
Discretization of a Dirichlet boundary
Discretization of a Neumann boundary
Accuracy and the curse of dimensionality
The Finite Element Method
Substructuring and grid generation
The weak form
Discrete approximation and the stiffness matrix
Solving the resulting system
Solving the Linear System: Iterative Methods
Basics of iterative methods
Relaxation methods
The classical relaxation iterations
Convergence analysis: the additive decomposition
When the iteration converges: the spectral radius
Convergence results, and why they are not enough
Minimization Methods: Steepest Descent and Conjugate Gradients
Solving a linear system as minimizing a quadratic
The method of steepest descent
Why steepest descent is slow: the condition number
Conjugate directions
Conjugate gradients
Preconditioning
Strategies for preconditioners
Multigrid Methods
The smoothing property: why relaxation stalls
Frequencies are relative to the grid — the correction scheme
The two-grid algorithm: restriction and prolongation
The V-cycle
Further multigrid schemes
Why it works: cost and convergence
Simulation of Traffic
Discrete Modeling
Two Views of Traffic
The Modeling Problem
A Simple Example: The Post Office
Interfaces to the Environment
Modeling the environment
Modeling the system behavior
Analyzing the results
The Instruments: Random Variables
Single-number summaries
The dynamical view
A Simple Example: The Uniform Distribution
A More Exciting Example: Constant Hazard Rate
Stochastic Processes
The Hitchhiker’s Paradox
The Outside Observer
Model Parameters
Little’s Law
An Intuitive Proof
Elementary Queuing Systems
Arrival and Service Processes
The Poisson Counting Process
Classifying Elementary Queuing Systems
Sequence Planning: Service Strategy
First Results
Queuing Networks
Load and Task Streams
Parameters in a Queuing Network
Locating the Traffic Bottleneck
Asymptotic Analysis of Closed Networks
The Saturation Bend
Service Time and Maximum Throughput
What the Asymptotics Tell Us
Stochastic Processes Revisited
Processes, Time, and State
From Independence to the Markov Property
Homogeneity and Birth–Death
The Family Tree
Markov Chains in Discrete Time
Classifying States by Reachability
Long-Run Behavior
Markov Chains in Continuous Time
The M | M | 1 Queue
Performance of the Queue
Random Walk
Random and Pseudo-Random Numbers
Pseudo-Random Numbers
The Linear Congruential Generator
Choosing the Parameters
Functions of Random Variables
The Logarithmic Transform
The Sine–Cosine Transform
Inverse Transform Sampling
Generating Negative-Exponential Numbers
Tabulated Inversion
Tailored Methods for Special Distributions
The Gaussian via the Central Limit Theorem
Poisson Numbers by Counting
Testing Random Numbers
Frequency Tests: Uniformity
Runs Tests: Independence
Other Tests of Independence
Traffic Flow Models
Why Model Traffic Flow
Possible Characterizations
Analogies Beyond the Road
The Role of the Individual
Steady Traffic Flow
Two Ways to Watch a Stream
The Scenario and the Three Quantities
A First Relationship: Flow, Density and Velocity
A Second Relationship: Velocity and Density
A Third Relationship: The Fundamental Diagram
Equilibrium States
The Cubic Model
Fitting the Cubic to Real Data
Separating the Two Phases
Unsteady Situations
What We Assume to Get There
A Conservation Law
The Traffic Equation
Reading the Traffic Equation
Signal Velocity
Tangent and Secant on the Fundamental Diagram
The End of a Tailback
Driving Off at a Traffic Light
Formation and Dissolution of Congestions
A Density Drop Spreads Out
Appendix
Notation
Greek Letters
Letters used in this course
Visual variants
Epsilon: ε vs ϵ
Theta: θ vs ϑ
Phi: ϕ vs φ
Rho: ρ vs ϱ
Pronunciation notes
Algebra
Quadratic Formula
The Formula
The Discriminant
Derivation by Completing the Square
Vieta’s Formulas
Functions
Bijection
Injective (one-to-one)
Surjective (onto)
Bijective (both)
Calculus
Differentiation
Core Rules
Common Derivatives
Trigonometric Derivatives
Chain Rule in Detail
Translation Invariance
Partial Derivatives via These Rules
Integration
The basic question
Why the +C?
Building the rules
The power rule, by reversal
Linearity
Putting it together
Common Integrals
Trigonometric Integrals
Substitution: reversing the chain rule
The diagnostic: what to look for
Fixing an off-by-constant factor
Examples
Integration by parts: reversing the product rule
The Definite Integral
Riemann sums: where the area definition comes from
Arc length: integrating speed
Linear Algebra
Matrices
Transpose
Symmetric Matrix
Skew-Symmetric Matrix
Diagonal Matrix
Trace
Standard Basis
Definition
Key Properties
Inner Product
Inner Product vs Dot Product
Algebraic Definition
Geometric Interpretation
Projection
Key Properties
Cross Product
Definition
Cross-Product Matrix
Geometric Interpretation
Key Properties
Determinant
2×2 Determinant
3×3 Determinant
Determinant from Eigenvalues
Key Properties
Eigenvalues and Eigenvectors
Definition
Finding Eigenvalues
Finding Eigenvectors
Worked Example
Sum and Product of Eigenvalues
Diagonal Matrices: A Free Lunch
Differential Equations
Solving ODEs
Separation of Variables
What makes an ODE separable
The method
Worked example: the Malthus equation
Worked example: the saturation equation
When separation isn’t enough
Phase-Plane Analysis
Directional Fields
The directional field
Drawing the field by hand
1. Draw the nullclines
2. Label the steady states
3. Consider the signs in each region
4. Combine the arrows
5. Sketch the trajectories
Classifying the steady state
Alternative: eigenvalues instead of the arrows
Worked example
Linear Stability Analysis
Setup
Classification
Real eigenvalues
Complex eigenvalues
Why the classification works
Step 1 — Linearization
Step 2 — Why eigenvalues matter
Step 3 — Complex eigenvalues
Connection to the broad categories
Summary
Probability
Monty Hall and the Coupon Collector
The Monty Hall Problem
The Coupon Collector’s Problem
Simulation and Real Data
About
The Course
Lecturer
Tutors
Book
Who Made This
How It Was Made
A Note on Content
License

Chapters
Continuous Modeling and Simulation
Heat Transfer: Model and Numerical Solution

Heat Transfer: Model and Numerical Solution

Introduction
Modeling with PDE
Numerics of PDE

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Index

A

abstract model
accuracy
Adams–Moulton method
admissible schedule
admissible schedule (job-shop)
alternative-caution strategy (min-max regret)
analytic solution
approximate-numerical method
arbitrary machines
arc length
arms race of two superpowers
arrival process
arrival rate
Arrow's impossibility theorem
asymmetric (relation)
attractive equilibrium
augmented precedence graph

B

backward recurrence time
best response function
bijective function
binary relation
boundary
boundary point
boundary value problem (BVP)
bounded set

C

canal
capacity
carrying capacity
Cartesian coordinates
caution strategy (max-min payoff)
central difference
central limit theorem
change of variables
characteristic polynomial
chi-square test
Cᵏ class
Cᵏ-curve
closed curve
closed queuing network
closed set
closed subset
closure
coefficient of variation
collective choice function
compact set
condition of a problem
conditional probability
congestion phase
conjugate (A-orthogonal) vectors
conjugate gradient method
conjunctive edge
connex
consistency (numerical method)
continuity (multivariate function)
continuous model
continuously partially differentiable
convergence (numerical method)
convergence (vector sequence)
convex set
coordinate transformation
coordinate transformation matrix
correction scheme
crisp set
critical path
Critical Path Method (CPM)
critical vertex
cross product
cross-product matrix
cubic model
curse of dimensionality
curve
cut fuzzy set
cylindrical coordinates

D

DAG (directed acyclic graph)
data error
decision under certainty
decision under risk
decision under uncertainty
definite integral
definiteness of a symmetric matrix
defuzzification
degree of freedom
density jump (shock)
determinant (2×2)
determinant (3×3)
determinant from eigenvalues
deterministic model
diagonal matrix
difference quotient
differentiable (multivariate)
differentiation rules
direct-numerical method
directional derivative
directional field
Dirichlet boundary condition (ODE)
Dirichlet boundary condition (PDE)
discrete model
discretization
discretization error
discriminant
disjunctive edge
disjunctive edge assignment
divergence
double integral (rectangular domain)
double integral (standard domain)
doubling time
dwelling time

E

eigenvalue and eigenvector
eigenvalues of a diagonal matrix
elementary queuing system
elliptic PDE
embed-test definition
embed-test theorem
equilibrium point
Euclidean norm
Euler description
Euler method
event rate
expected payoff (mixed strategies)
expected value
explicit method
external dictator

F

feedback control
filling
finite difference method
finite element method
floating-point number
forward recurrence time
free-flow phase
Fubini's theorem
full-risk strategy (max-max payoff)
function of a random variable
functional unit
fundamental diagram
fundamental relationship (traffic flow)
fundamental theorem of calculus
fuzzification
fuzzy complement
fuzzy control system
fuzzy inference
fuzzy intersection
fuzzy rule base
fuzzy set
fuzzy union

G

Gauss's divergence theorem
generic decision model
geometric distribution
global discretization error
global extremum
gradient
growth rate

H

harmonic number
hazard rate
heat equation
Hessian matrix
Heun method
heuristic
homogeneous birth–death process
homogeneous Markov process
hyperbolic PDE

I

ideal arithmetic
identical machines
identity matrix
ill-posed problem
image (trace) of a curve
implicit Euler method
implicit method
independence of irrelevant alternatives
independent process
initial condition
initial value problem (IVP)
injective function
inner point
inner product
inner product (geometric form)
instationary (PDE)
integration by parts
integration rules
interior
internal dictator
interpolation polynomial (multistep)
inverse problem
inverse transform sampling
irreducible chain
isoline
iteration matrix

J

Jacobian determinant
Jacobian matrix
job
job-shop model

K

k-times continuously partially differentiable
Kendall notation
Kolmogorov–Smirnov test

L

Lagrange description
Laplace equation
Laplace operator
lead time
linear 1st-order PDE (in 2D)
linear birth and death rates
linear congruential generator
linear feedback control
linear PDE
linear PDE of 2nd order
linguistic variable
Little's law
load
local convergence of Newton's method
local discretization error
local extremum
logistic growth model

M

majority decision
makespan
Malthus model
Markov process
mathematical modeling
maximum throughput
memorylessness
method of steepest descent
midpoint rule
minimax theorem
mixed strategy
model
model hierarchy
multigrid method
multiscale problem
multistep method
multivariate function

N

nabla operator
negative exponential distribution
Neumann boundary condition (ODE)
Neumann boundary condition (PDE)
Newton iteration (multivariate)
no-dictator condition
non-feedback control
normal distribution
null hypothesis
nullcline
number of visits

O

one-step method
open queuing network
open set
order (of a method)
ordinary differential equation (ODE)

P

P controller
parabolic model
parabolic PDE
Pareto condition
partial derivative
partial differential equation (PDE)
partially differentiable
periodic boundary condition
periodic state
pessimism–optimism strategy (Hurwicz)
phase portrait
PID controller
piecewise C¹-curve
plane curve
Poisson distribution
Poisson equation
Poisson process
polar coordinates
Pollaczek–Khinchine formula
positively recurrent state
practical model
precedence condition
precedence graph
preconditioning
predictor–corrector method
preference relation
principle of insufficient reason (Laplace)
priority-list scheduling
process layer
prolongation
pseudo-random numbers

Q

q-th moment
quadratic formula
qualitatively identical machines
quasi-order
quasi-stationary (traffic flow)
quasilinear 1st-order PDE (in 2D)
queuing network
queuing system

R

random variable
random walk
rank addition
rank mapping
recurrence probability
recurrent state
reduction to a first-order system
reflexive (relation)
regret
regular surface
relaxation method
remaining time
repulsive equilibrium
residual
resolution (floating-point)
restriction
root
rotation
round-off error
rounding
Runge–Kutta method (RK4)
runs test

S

saddle point (calculus)
saddle point (game theory)
saddle point (ODE system)
saturation number of jobs
scalar field
scalar projection
scalar surface integral
scale
schedule
scheduling problem
second difference quotient
second-order partial derivative
separable ODE
separation of variables
service process
service rate
service strategy
service time
service unit
set of connex preorders
set of preference relations
shooting method
signal velocity
simulation
simulation pipeline
single machine
skew-symmetric matrix
slack
smooth function
smoothing property
space curve
spectral condition number
spectral radius
spherical coordinates
stability (numerical method)
stable and unstable equilibrium
standard basis
standard domain (R²)
standard domain (R³)
start and completion time
state probability
state space
stationary (PDE)
stationary limit distribution
stationary point
stationary point (ODE system)
stationary process
stencil
stiffness
stiffness matrix
stochastic model
stochastic process
strictly diagonally dominant matrix
support (fuzzy set)
surface
surface area
surface normal
surjective function
symmetric (relation)
symmetric matrix
system
system layer

T

tangent vectors
task stream
Taylor polynomial
theorem of underestimation (Fulkerson 1962)
throughput
topological sort
trace
trace from eigenvalues
traffic bottleneck
traffic density
traffic equation
traffic flow
traffic velocity
trajectory
transition probability
transition rate
transitive (relation)
translation invariance (differentiation)
transpose
trial function
trigonometric derivatives
trigonometric integrals
triple integral (box domain)
triple integral (standard domain)
truncation error
two-player zero-sum game
two-species model

U

uniform distribution
upwind discretization
utilization

V

V-cycle
validation
variance
vector field
vector projection
vector surface integral
Verhulst saturation model
Vieta's formulas
voting paradox

W

well-posed problem

#

1st-order linear PDE with constant coefficients
α-cut
ε-ball

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