Solving Differential Equations in R

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bfm_978-3-642-28070-2_1 Solving Differential Equations in R Preface R as a Problem Solving Environment About the Three Authors Acknowledgment References Contents 1 Chapter 1 Differential Equations 1.1 Basic Theory of Ordinary Differential Equations 1.1.1 First Order Differential Equations 1.1.2 Analytic and Numerical Solutions 1.1.3 Higher Order Ordinary Differential Equations 1.1.4 Initial and Boundary Values 1.1.5 Existence and Uniqueness of Analytic Solutions 1.2 Numerical Methods 1.2.1 The Euler Method 1.2.2 Implicit Methods 1.2.3 Accuracy and Convergence of Numerical Methods 1.2.4 Stability and Conditioning 1.2.4.1 Absolute Stability 1.3 Other Types of Differential Equations 1.3.1 Partial Differential Equations 1.3.2 Differential Algebraic Equations 1.3.3 Delay Differential Equations References 2 Chapter 2 Initial Value Problems 2.1 Runge-Kutta Methods 2.1.1 Explicit Runge-Kutta Formulae 2.1.2 Deriving a Runge-Kutta Formula 2.1.2.1 The Order Conditions 2.1.2.2 Controlling the Error 2.1.2.3 Changing the Step Size 2.1.2.4 Embedded Runge-Kutta Methods 2.1.2.5 Continuous Solutions 2.1.2.6 Stability 2.1.3 Implicit Runge-Kutta Formulae 2.2 Linear Multistep methods 2.2.1 Convergence, Stability and Consistency 2.2.2 Adams Methods 2.2.2.1 Controlling the Error of Adams Methods 2.2.2.2 Stability of Adams Methods 2.2.3 Backward Differentiation Formulae 2.2.3.1 Variable Step Size in BDF Formulae 2.2.3.2 Stability of BDF 2.2.4 Variable Order – Variable Coefficient Formulae for Linear Multistep Methods 2.2.4.1 Variable Order 2.2.4.2 Variable Coefficient Methods 2.3 Boundary Value Methods 2.4 Modified Extended Backward Differentiation Formulae 2.5 Stiff Problems 2.5.1 Stiffness Detection 2.5.2 Non-stiffness Test 2.6 Implementing Implicit Methods 2.6.1 Fixed-Point Iteration to Convergence 2.6.2 Chord Iteration 2.6.3 Predictor-Corrector Methods 2.6.4 Newton Iteration for Implicit Runge-Kutta Methods 2.7 Codes to Solve Initial Value Problems 2.7.1 Codes to Solve Non-stiff Problems 2.7.2 Codes to Solve Stiff Problems 2.7.3 Codes that Switch Between Stiff and Non-stiff Solvers References 3 Chapter 3 Solving Ordinary Differential Equations in R 3.1 Implementing Initial Value Problems in R 3.1.1 A Differential Equation Comprising One Variable 3.1.2 Multiple Variables: The Lorenz Model 3.2 Runge-Kutta Methods 3.2.1 Rigid Body Equations 3.2.2 Arenstorf Orbits 3.3 Linear Multistep Methods 3.3.1 Seven Moving Stars 3.3.2 A Stiff Chemical Example 3.3.2.1 External Variables 3.4 Discontinuous Equations, Events 3.4.1 Pharmacokinetic Models 3.4.1.1 A Two-Compartment Model Describing Oral Drug Intake 3.4.1.2 A One-Compartment Model Describing Drug Injection 3.4.2 A Bouncing Ball 3.4.3 Temperature in a Climate-Controlled Room 3.5 Method Selection 3.5.1 The van der Pol Equation 3.5.1.1 Printing the Diagnostics of the Solutions 3.5.1.2 Timings 3.5.1.3 Work Precision Diagrams and mescd 3.6 Exercises 3.6.1 Getting Started with IVP 3.6.2 The Robertson Problem 3.6.3 Displaying Results in a Phase-Plane Graph 3.6.3.1 The Rossler Equations 3.6.3.2 Josephson Junctions 3.6.4 Events and Roots 3.6.5 Stiff Problems References 4 Chapter 4 Differential Algebraic Equations 4.1 Introduction 4.1.1 The Index of a DAE 4.1.2 A Simple Example 4.1.3 DAEs in Hessenberg Form 4.1.4 Hidden Constraints and the Initial Conditions 4.1.4.1 Hidden Constraints in Index 2 DAEs in Hessenberg Form 4.1.4.2 Hidden Constraints in Index 3 DAEs 4.1.5 The Pendulum Problem 4.2 Solving DAEs 4.2.1 Semi-implicit DAEs of Index 1 4.2.1.1 The -embedding Method 4.2.1.2 The State Space Form Method 4.2.2 General Implicit DAEs of Index 1 4.2.3 Discretization Algorithms 4.2.4 DAEs of Higher Index 4.2.4.1 Index Reduction 4.2.4.2 Higher Index Solvers 4.2.5 Index of a DAE Variable References 5 Chapter 5 Solving Differential Algebraic Equations in R 5.1 Differential Algebraic Equation Solvers in R 5.2 A Simple DAE of Index 2 5.2.1 Solving the DAEs in General Implicit Form 5.2.2 Solving the DAEs in Linearly Implicit Form 5.3 A Nonlinear Implicit ODE 5.4 A DAE of Index 3: The Pendulum Problem 5.5 Multibody Systems 5.5.1 The Car Axis Problem 5.6 Electrical Circuit Models 5.6.1 The Transistor Amplifier 5.7 Exercises 5.7.1 A Simple DAE 5.7.2 The Robertson Problem 5.7.3 The Pendulum Problem Revisited 5.7.4 The Akzo Nobel Problem 5.7.4.1 Problem Formulation 5.7.4.2 Task References 6 Chapter 6 Delay Differential Equations 6.1 Delay Differential Equations 6.1.1 DDEs with Delays of the Dependent Variables 6.1.2 DDEs with Delays of the Derivatives 6.2 Difficulties when Solving DDEs 6.2.1 Discontinuities in DDEs 6.2.2 Small and Vanishing Delays 6.3 Numerical Methods for Solving DDEs References 7 Chapter 7 Solving Delay Differential Equations in R 7.1 Delay Differential Equation Solvers in R 7.2 Two Simple Examples 7.2.1 DDEs Involving Solution Delay Terms 7.2.2 DDEs Involving Derivative Delay Terms 7.3 Chaotic Production of White Blood Cells 7.4 A DDE Involving a Root Function 7.5 Vanishing Time Delays 7.6 Predator-Prey Dynamics with Harvesting 7.7 Exercises 7.7.1 The Lemming Model 7.7.2 Oberle and Pesch 7.7.3 An Epidemiological Model 7.7.4 A Neutral DDE 7.7.5 Delayed Cellular Neural Networks With Impulses References 8 Chapter 8 Partial Differential Equations 8.1 Partial Differential Equations 8.1.1 Alternative Formulations 8.1.2 Polar, Cylindrical and Spherical Coordinates 8.1.3 Boundary Conditions 8.2 Solving PDEs 8.3 Discretising Derivatives 8.3.1 Basic Diffusion Schemes 8.3.1.1 Non-negativity of Diffusion Schemes 8.3.2 Basic Advection Schemes 8.3.2.1 Stability 8.3.2.2 Non-negativity of Advection Schemes 8.3.3 Flux-Conservative Discretisations 8.3.4 More Complex Advection Schemes 8.3.4.1 Advection of a Square Pulse 8.3.4.2 Time Step Conditions 8.3.4.3 Flux Limiters 8.4 The Method Of Lines 8.5 The Finite Difference Method References 9 Chapter 9 Solving Partial Differential Equations in R 9.1 Methods for Solving PDEs in R 9.1.1 Numerical Approximations 9.1.1.1 Setting up a Grid 9.1.1.2 Numerical Approximation of Advection and Diffusion 9.1.2 Solution Methods 9.2 Solving Parabolic, Elliptic and Hyperbolic PDEs in R 9.2.1 The Heat Equation 9.2.1.1 Problem Definition 9.2.1.2 Solving the Heat Equation in R 9.2.2 The Wave Equation 9.2.2.1 Problem Definition 9.2.2.2 Solving the Wave Equation in R 9.2.3 Poisson and Laplace's Equation 9.2.3.1 Problem Definition 9.2.3.2 Solving the Laplace Equation in R 9.2.4 The Advection Equation 9.2.4.1 Problem Definition 9.2.4.2 Solving the Advection Equation in R 9.3 More Complex Examples 9.3.1 The Brusselator in One Dimension 9.3.2 The Brusselator in Two Dimensions 9.3.3 Laplace Equation in Polar Coordinates 9.3.4 The Time-Dependent 2-D Sine-Gordon Equation 9.3.5 The Nonlinear Schrödinger Equation 9.4 Exercises 9.4.1 The Gray-Scott Equation 9.4.2 A Macroscopic Model of Traffic 9.4.3 A Vibrating String 9.4.4 A Pebble in a Bucket of Water 9.4.5 Combustion in 2-D References 10 Chapter 10 Boundary Value Problems 10.1 Two-Point Boundary Value Problems 10.2 Characteristics of Boundary Value Problems 10.2.1 Uniqueness of Solutions 10.2.2 Isolation of Solutions 10.2.3 Stiffness of Boundary Value Problemsand Dichotomy 10.2.4 Conditioning of Boundary Value Problems 10.2.5 Singular Problems 10.3 Boundary Conditions 10.3.1 Separated Boundary Conditions 10.3.2 Defining Good Boundary Conditions 10.3.3 Problems Defined on an Infinite Interval 10.4 Methods of Solution 10.5 Shooting Methods for Two-Point BVPs 10.5.1 The Linear Case 10.5.2 The Nonlinear Case 10.5.3 Multiple Shooting 10.6 Finite Difference Methods 10.6.1 A Low Order Method for Second Order Equations 10.6.2 Other Low Order Methods 10.6.3 Higher Order Methods Based on Collocation Runge-Kutta Schemes 10.6.4 Higher Order Methods Based on Mono Implicit Runge-Kutta Formulae 10.6.5 Higher Order Methods Based on Linear Multistep Formulae 10.6.6 Deferred Correction 10.7 Codes for the Numerical Solution of BoundaryValue Problems References 11 Chapter 11 Solving Boundary Value Problems in R 11.1 Boundary Value Problem Solvers in R 11.2 A Simple BVP Example 11.2.1 Implementing the BVP in First Order Form 11.2.2 Implementing the BVP in Second Order Form 11.3 A More Complex BVP Example 11.4 More Complex Initial or End Conditions 11.5 Solving a Boundary Value Problem Using Continuation 11.5.1 Manual Continuation 11.5.2 Automatic Continuation 11.6 BVPs with Unknown Constants 11.6.1 The Elastica Problem 11.6.2 Non-separated Boundary Conditions 11.6.3 An Unknown Integration Interval 11.7 Integral Constraints 11.8 Sturm-Liouville Problems 11.9 A Reaction Transport Problem 11.10 Exercises 11.10.1 A Stiff Boundary Value Problem 11.10.2 The Mathieu Equation 11.10.3 Another Swirling Flow Problem 11.10.4 Another Reaction Transport Problem References bbm_978-3-642-28070-2_1 Appendix A A.1 Butcher Tableaux for Some Runge-Kutta Methods A.2 Coefficients for Some Linear Multistep Formulae A.3 Implemented Integration Methods for Solving Initial Value Problems in R A.4 Other Integration Methods in R References Index