A Primer on Mathematical Modelling

Preface......Page 7
Octave Functions and Scripts......Page 8
Contents......Page 10
Main Symbols......Page 13
1 A Warm-Up to Models......Page 14
1.2 Numerical Modelling and Scientific Computing......Page 15
1.3.1 Different Physical Problems, the Same Mathematical Model (a Subtle, Yet Crucial, Difference)......Page 17
1.3.2 One Physical Problem, Many Mathematical Models......Page 19
The Numerical Error......Page 21
The Validation of the Process and the Verification of the Numerical Model......Page 23
What Do Errors Depend On?......Page 24
1.5 There Is Error and Error......Page 25
1.6 What We Have Learnt......Page 27
2 Getting Started......Page 28
2.1 When the Solution to the Mathematical Model Is Not Known......Page 29
How Do We Compute the Area of the Trapezoid T?......Page 30
`Divide et impera\': Let Us Subdivide the Interval......Page 31
Up and Down, Step by Step......Page 32
Quadrature Formula at Midpoints, or by Rectangles......Page 33
…and Answers......Page 35
2.1.3 The Numerical Solution of Problem ``Heights\'\'......Page 36
Can We Determine How Many Intervals to Use Without Wasting Time?......Page 37
2.2 Vectors and Matrices to Handle Complexity......Page 38
Vectors: Orderly Queues......Page 40
Matrices, or Close-Order Setting?......Page 41
Equality of Vectors......Page 43
The Determinant of a 22 Matrix......Page 44
2.2.4 The Gauss Elimination Method (GEM) for a 22System......Page 45
2.2.5 From 2 to 3......Page 46
The Product of an n  n Matrix with a Column Vector of Dimension n......Page 47
2.2.6 The 3  3 Linear System in Matrix Form......Page 48
2.2.7 The Cramer Method for a 33 System......Page 49
2.2.8 for Loops in Algorithms......Page 52
2.2.9 Variable Assignment in an Algorithm......Page 53
2.2.10 The Gauss Elimination Method (GEM) for a 33System......Page 54
2.2.11 We Reached 100!......Page 58
2.2.12 How Many Operations Must We Do?......Page 59
2.2.13 The Computer Comes into Play......Page 60
2.2.14 GEM, Too, Adapts to the Computer......Page 61
Mathematical Tools......Page 63
Numerical Methods......Page 64
3 Reckoning with the Computer......Page 65
3.1 Programming with Octave......Page 66
3.1.1 Arithmetic Operations and Variables......Page 67
3.1.2 Mathematical Functions and Their Graphical Representation......Page 69
3.1.3 Vectors and Matrices......Page 71
Operations with Matrices......Page 74
Row Vectors via the Instruction ``:\'\'......Page 76
Simple Vector Operations......Page 77
Augmenting Arrays (Adding Components to Vectors)......Page 78
``Element-by-Element\'\' Operations......Page 79
Printing Instructions......Page 81
The Script: A Shopping List of Sorts......Page 83
User-Defined Functions or, for Short, Functions......Page 84
The for Loop......Page 87
Selection Blocks......Page 89
Relational Operators......Page 91
While Loops......Page 93
3.1.6 Tidbits of Graphics......Page 95
Graphs in Logarithmic Scale......Page 97
3.2 How Computers Represent Numbers......Page 98
An Elementary, But Inevitable, Mistake......Page 99
3.2.1 Floating-Point Numbers......Page 100
3.2.2 Roundoff Errors......Page 102
Exact Arithmetic and Floating-Point Arithmetic......Page 104
A Recursive Formula for Computing π......Page 105
3.4 Script of This Chapter......Page 107
4.1 Surfing with Google......Page 108
4.2 Understanding the Problem......Page 109
Ordering Pages......Page 110
A Computer for Searching, Many Computers for Finding......Page 111
4.3.1 The Directed Graph as a Model of the Web......Page 112
4.3.2 The Adjacency Matrix of a Graph......Page 113
The Probability of Clicking on a Link......Page 114
``If You Feel You Are in a Black Hole, Don\'t Give Up. There\'s a Way Out\'\' (S. Hawking)......Page 115
The Possibility of Changing Path......Page 116
It\'s a Matter of Probability......Page 117
Matrices and Vectors to Help Us Model......Page 118
4.6 The Numerical Model: How Do We Solve the System p = Gp?......Page 119
Is the Milkshake Ready? The Stopping Test......Page 121
4.7.1 While Loops in Algorithms......Page 122
4.9 A Bit of History and a Glimpse Beyond......Page 125
4.10 List of Functions and Scripts of the Chapter......Page 127
5 A Network of Capillaries......Page 129
5.1 Heart, Arteries, Veins and Capillaries......Page 130
It\'s Easy to Say `Blood\'......Page 131
5.2.1 The Beat That Keeps Us Alive......Page 133
5.2.2 Fluid Flow and Viscosity......Page 134
Non-Newtonian Fluids......Page 136
5.4 The Model for One Capillary......Page 137
5.5 Blood Flow in a Small Capillary Bed......Page 140
The Adjacency Matrix of a Graph......Page 141
5.5.2 To Compute the Speeds We Need the Pressures......Page 142
5.5.3 To Compute the Pressures We Need the BalanceEquations......Page 143
5.5.4 A System of Balance Equations......Page 144
5.5.5 From the Matrix G of the Graph to the Matrix A of the System......Page 146
5.5.7 From the Mathematical Model to the Solution: GEM......Page 148
5.5.8 Volumetric Flow Rates and Velocities in EachCapillary......Page 149
Computing Pressures......Page 150
After the Volumetric Flow Rates Compute the Average Speeds......Page 151
The Graph......Page 152
The Adjacency Matrix of the Graph......Page 154
Graph → Pressures → Volumetric Flow Rates → Speeds......Page 155
5.8 What Lies Beyond......Page 157
5.9 List of Functions and Scripts of the Chapter......Page 158
6 Predators and Preys in the Maths Ocean......Page 162
6.1 Fishing in the Adriatic Sea......Page 163
6.2 Simplify in Order to Model......Page 164
6.3 Understanding the Problem......Page 166
6.4.1 The Natural Dynamics......Page 167
6.4.2 The Interaction with the Environment: Constant Harvest in Time......Page 171
6.4.3 The Interaction with the Environment: Time-Dependent Harvest (the Model)......Page 173
6.4.4 A Numerical Method to Approximate First Derivatives......Page 174
The Linear Scale and the Logarithmic Scale......Page 178
Reading the Order of Convergence Off the Plot......Page 179
6.4.6 The Euler Method for Approximating a Differential Equation......Page 180
6.4.7 Error of the Euler Method......Page 184
6.4.8 The Interaction With the Environment: Time-Dependent Harvest (the Numerical Solution)......Page 185
6.4.9 An Even More Realistic Model: The Competition Within the Population......Page 188
6.5 The Dynamics of Two Populations......Page 191
6.5.1 The Prey-Predator System (the Model)......Page 192
6.5.2 Numerical Approximation of a System of Differential Equations......Page 193
6.5.3 The Prey-Predator System (the Numerical Solution)......Page 194
6.5.4 An Even More Realistic Model: The Prey-Predator System with Harvest......Page 197
6.5.5 The Prey-Predator Model with Harvest: The Validation......Page 199
The Way to Proceed......Page 200
6.7 Looking Beyond......Page 201
6.8 List of Functions and Scripts of the Chapter......Page 203
7 Take-Home Message......Page 209
8.1 Exercises of Chapter 2......Page 210
8.2 Exercises of Chapter 3......Page 218
8.3 Exercises of Chapter 4......Page 223
8.4 Exercises of Chapter 5......Page 227
8.5 Exercises of Chapter 6......Page 233
Index......Page 241