Solving Differential Equations in R

• Soetaert, Karline, Cash, Jeff, Mazzia, Francesca
• Paperback: 264 pages
• Publisher: Springer; 2012 edition (June 7, 2012)
• Language: English
• ISBN-10: 3642280692
  

1 Differential Equations...................................................... 1
1.1 Basic Theory of Ordinary Differential Equations ................. 1
1.1.1 First Order Differential Equations ...................... 1
1.1.2 Analytic and Numerical Solutions ...................... 2
1.1.3 Higher Order Ordinary Differential Equations ......... 3
1.1.4 Initial and Boundary Values ............................. 4
1.1.5 Existence and Uniqueness of Analytic Solutions ...... 5
1.2 Numerical Methods ................................................. 6
1.2.1 The Euler Method........................................ 6
1.2.2 Implicit Methods......................................... 7
1.2.3 Accuracy and Convergence of Numerical Methods .... 8
1.2.4 Stability and Conditioning .............................. 9
1.3 Other Types of Differential Equations ............................. 11
1.3.1 Partial Differential Equations ........................... 11
1.3.2 Differential Algebraic Equations........................ 12
1.3.3 Delay Differential Equations ............................ 13
References.................................................................... 13
2 Initial Value Problems ..................................................... 15
2.1 Runge-Kutta Methods .............................................. 15
2.1.1 Explicit Runge-Kutta Formulae ......................... 15
2.1.2 Deriving a Runge-Kutta Formula ....................... 17
2.1.3 Implicit Runge-Kutta Formulae ......................... 22
2.2 Linear Multistep methods .......................................... 22
2.2.1 Convergence, Stability and Consistency ................ 23
2.2.2 Adams Methods.......................................... 25
2.2.3 Backward Differentiation Formulae .................... 27
2.2.4 Variable Order – Variable Coefficient
Formulae for Linear Multistep Methods ................ 29
2.3 Boundary Value Methods........................................... 30
2.4 Modified Extended Backward Differentiation Formulae ......... 31
2.5 Stiff Problems ....................................................... 32
2.5.1 Stiffness Detection ....................................... 33
2.5.2 Non-stiffness Test ........................................ 34
2.6 Implementing Implicit Methods.................................... 34
2.6.1 Fixed-Point Iteration to Convergence ................... 34
2.6.2 Chord Iteration ........................................... 35
2.6.3 Predictor-Corrector Methods ............................ 36
2.6.4 Newton Iteration for Implicit Runge-Kutta
Methods .................................................. 36
2.7 Codes to Solve Initial Value Problems ............................. 37
2.7.1 Codes to Solve Non-stiff Problems ..................... 38
2.7.2 Codes to Solve Stiff Problems .......................... 38
2.7.3 Codes that Switch Between Stiff and
Non-stiff Solvers ......................................... 38
References.................................................................... 39
3 Solving Ordinary Differential Equations in R........................... 41
3.1 Implementing Initial Value Problems in R......................... 41
3.1.1 A Differential Equation Comprising One Variable..... 42
3.1.2 Multiple Variables: The Lorenz Model ................. 44
3.2 Runge-Kutta Methods .............................................. 45
3.2.1 Rigid Body Equations ................................... 47
3.2.2 Arenstorf Orbits.......................................... 49
3.3 Linear Multistep Methods .......................................... 51
3.3.1 Seven Moving Stars...................................... 52
3.3.2 A Stiff Chemical Example .............................. 56
3.4 Discontinuous Equations, Events .................................. 59
3.4.1 Pharmacokinetic Models ................................ 60
3.4.2 A Bouncing Ball ......................................... 64
3.4.3 Temperature in a Climate-Controlled Room ........... 66
3.5 Method Selection ................................................... 68
3.5.1 The van der Pol Equation................................ 70
3.6 Exercises ............................................................ 75
3.6.1 Getting Started with IVP ................................ 75
3.6.2 The Robertson Problem ................................. 76
3.6.3 Displaying Results in a Phase-Plane Graph ............ 76
3.6.4 Events and Roots......................................... 78
3.6.5 Stiff Problems ............................................ 79
References.................................................................... 79
4 Differential Algebraic Equations ......................................... 81
4.1 Introduction ......................................................... 81
4.1.1 The Index of a DAE ..................................... 82
4.1.2 A Simple Example ....................................... 83
4.1.3 DAEs in Hessenberg Form .............................. 84
4.1.4 Hidden Constraints and the Initial Conditions ......... 85
4.1.5 The Pendulum Problem.................................. 86
4.2 Solving DAEs ....................................................... 87
4.2.1 Semi-implicit DAEs of Index 1 ......................... 87
4.2.2 General Implicit DAEs of Index 1 ...................... 88
4.2.3 Discretization Algorithms ............................... 89
4.2.4 DAEs of Higher Index ................................... 90
4.2.5 Index of a DAE Variable................................. 93
References.................................................................... 94
5 Solving Differential Algebraic Equations in R .......................... 95
5.1 Differential Algebraic Equation Solvers in R ..................... 95
5.2 A Simple DAE of Index 2 .......................................... 96
5.2.1 Solving the DAEs in General Implicit Form ........... 97
5.2.2 Solving the DAEs in Linearly Implicit Form ........... 98
5.3 A Nonlinear Implicit ODE ......................................... 98
5.4 A DAE of Index 3: The Pendulum Problem ....................... 100
5.5 Multibody Systems ................................................. 101
5.5.1 The Car Axis Problem ................................... 102
5.6 Electrical Circuit Models ........................................... 106
5.6.1 The Transistor Amplifier ................................ 107
5.7 Exercises ............................................................ 111
5.7.1 A Simple DAE ........................................... 111
5.7.2 The Robertson Problem ................................. 111
5.7.3 The Pendulum Problem Revisited ...................... 111
5.7.4 The Akzo Nobel Problem ............................... 112
References.................................................................... 115
6 Delay Differential Equations .............................................. 117
6.1 Delay Differential Equations ....................................... 117
6.1.1 DDEs with Delays of the Dependent Variables ........ 118
6.1.2 DDEs with Delays of the Derivatives ................... 118
6.2 Difficulties when Solving DDEs ................................... 119
6.2.1 Discontinuities in DDEs ................................. 119
6.2.2 Small and Vanishing Delays............................. 120
6.3 Numerical Methods for Solving DDEs ............................ 121
References.................................................................... 121
7 Solving Delay Differential Equations in R ............................... 123
7.1 Delay Differential Equation Solvers in R .......................... 123
7.2 Two Simple Examples .............................................. 124
7.2.1 DDEs Involving Solution Delay Terms ................. 124
7.2.2 DDEs Involving Derivative Delay Terms ............... 124
7.3 Chaotic Production of White Blood Cells ......................... 125
7.4 A DDE Involving a Root Function ................................. 127
7.5 Vanishing Time Delays ............................................. 128
7.6 Predator-Prey Dynamics with Harvesting ......................... 130
7.7 Exercises ............................................................ 132
7.7.1 The Lemming Model .................................... 132
7.7.2 Oberle and Pesch......................................... 132
7.7.3 An Epidemiological Model.............................. 133
7.7.4 A Neutral DDE........................................... 134
7.7.5 Delayed Cellular Neural Networks With Impulses..... 134

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References.................................................................... 237
A Appendix .................................................................... 239
A.1 Butcher Tableaux for Some Runge-Kutta Methods ............... 239
A.2 Coefficients for Some Linear Multistep Formulae ................ 239
A.3 Implemented Integration Methods for Solving Initial
Value Problems in R ................................................ 241
A.4 Other Integration Methods in R .................................... 242
References.................................................................... 242
Index ............................................................................... 245