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This book deals with the numerical solution of differential equations, a very important branch of mathematics. The solution of differential equations using R is the main focus of this book. The aim is to give a practical and theoretical account of how to solve a large variety of differential equations, comprising ordinary differential equations, initial value problems and boundary value problems, differential algebraic equations, partial differential equations and delay differential equations. Each chapter that deals with R examples is preceded by a chapter where the theory behind the numerical methods being used is introduced. In the sections that deal with the use of R for solving differential equations, we have taken examples from a variety of disciplines, including biology, chemistry, physics, pharmacokinetics. Many examples are well-known test examples, used frequently in the field of numerical analysis. Differential Equations. Basic Theory of Ordinary Differential Equations. First Order Differential Equations. Analytic and Numerical Solutions. Higher Order Ordinary Differential Equations. Initial and Boundary Values. Existence and Uniqueness of Analytic Solutions. Numerical Methods. The Euler Method. Implicit Methods. Accuracy and Convergence of Numerical Methods. Stability and Conditioning. Other Types of Differential Equations. Partial Differential Equations. Differential Algebraic Equations. Delay Differential Equations. Initial Value Problems. Runge-Kutta Methods. Explicit Runge-Kutta Formulae. Deriving a Runge-Kutta Formula. Implicit Runge-Kutta Formulae. Linear Multistep methods. Convergence, Stability and Consistency. Adams Methods. Backward Differentiation Formulae. Variable Order – Variable Coefficient Formulae for Linear Multistep Methods. Boundary Value Methods. Modified Extended Backward Differentiation Formulae. Stiff Problems. Stiffness Detection. Non-stiffness Test. Implementing Implicit Methods. Fixed-Point Iteration to Convergence. Chord Iteration. Predictor-Corrector Methods. Newton Iteration for Implicit Runge-Kutta Methods. Codes to Solve Initial Value Problems. Codes to Solve Non-stiff Problems. Codes to Solve Stiff Problems. Codes that Switch Between Stiff and Non-stiff Solvers. Solving Ordinary Differential Equations in R. Implementing Initial Value Problems in R. A Differential Equation Comprising One Variable. Multiple Variables: The Lorenz Model. Runge-Kutta Methods. Rigid Body Equations. Arenstorf Orbits. Linear Multistep Methods. Seven Moving Stars. A Stiff Chemical Example. Discontinuous Equations, Events. Pharmacokinetic Models. A Bouncing Ball. Temperature in a Climate-Controlled Room. Method Selection. The van der Pol Equation. Exercises. Getting Started with IVP. The Robertson Problem. Displaying Results in a Phase-Plane Graph. Events and Roots. Stiff Problems. Differential Algebraic Equations. The Index of a DAE. A Simple Example. DAEs in Hessenberg Form. Hidden Constraints and the Initial Conditions. The Pendulum Problem. Solving DAEs. Semi-implicit DAEs of Index 1. General Implicit DAEs of Index 1. Discretization Algorithms. DAEs of Higher Index. Index of a DAE Variable. Solving Differential Algebraic Equations in R. Differential Algebraic Equation Solvers in R. A Simple DAE of Index 2. Solving the DAEs in General Implicit Form. Solving the DAEs in Linearly Implicit Form. A Nonlinear Implicit ODE. A DAE of Index 3: The Pendulum Problem. Multibody Systems. The Car Axis Problem. Electrical Circuit Models. The Transistor Amplifier. Exercises. A Simple DAE. The Robertson Problem. The Pendulum Problem Revisited. The Akzo Nobel Problem. Delay Differential Equations. Delay Differential Equations. DDEs with Delays of the Dependent Variables. DDEs with Delays of the Derivatives. Difficulties when Solving DDEs. Discontinuities in DDEs. Small and Vanishing Delays. Numerical Methods for Solving DDEs. Solving Delay Differential Equations in R. Delay Differential Equation Solvers in R. Two Simple Examples. DDEs Involving Solution Delay Terms. DDEs Involving Derivative Delay Terms. Chaotic Production of White Blood Cells. A DDE Involving a Root Function. Vanishing Time Delays. Predator-Prey Dynamics with Harvesting. Exercises. The Lemming Model. Oberle and Pesch. An Epidemiological Model. A Neutral DDE. Delayed Cellular Neural Networks With Impulses. Partial Differential Equations. Partial Differential Equations. Alternative Formulations. Polar, Cylindrical and Spherical Coordinates. Boundary Conditions. Solving PDEs. Discretising Derivatives. Basic Diffusion Schemes. Basic Advection Schemes. Flux-Conservative Discretisations. More Complex Advection Schemes. The Method Of Lines. The Finite Difference Method. Solving Partial Differential Equations in R. Methods for Solving PDEs in R. Numerical Approximations. Solution Methods. Solving Parabolic, Elliptic and Hyperbolic PDEs in R. The Heat Equation. The Wave Equation. Poisson and Laplace’s Equation. The Advection Equation. More Complex Examples. The Brusselator in One Dimension. The Brusselator in Two Dimensions. Laplace Equation in Polar Coordinates. The Time-Dependent 2-D Sine-Gordon Equation. The Nonlinear Schrodinger Equation. Exercises. The Gray-Scott Equation. A Macroscopic Model of Traffic. A Vibrating String. A Pebble in a Bucket of Water. Combustion in 2-D. Boundary Value Problems. Two-Point Boundary Value Problems. Characteristics of Boundary Value Problems. Uniqueness of Solutions. Isolation of Solutions. Stiffness of Boundary Value Problems and Dichotomy. Conditioning of Boundary Value Problems. Singular Problems. Boundary Conditions. Separated Boundary Conditions. Defining Good Boundary Conditions. Problems Defined on an Infinite Interval. Methods of Solution. Shooting Methods for Two-Point BVPs. The Linear Case. The Nonlinear Case. Multiple Shooting. Finite Difference Methods. A Low Order Method for Second Order Equations. Other Low Order Methods. Higher Order Methods Based on Collocation Runge-Kutta Schemes. Higher Order Methods Based on Mono Implicit Runge-Kutta Formulae. Higher Order Methods Based on Linear Multistep Formulae. Deferred Correction. Codes for the Numerical Solution of Boundary Value Problems. Solving Boundary Value Problems in R. Boundary Value Problem Solvers in R. A Simple BVP Example. Implementing the BVP in First Order Form. Implementing the BVP in Second Order Form. A More Complex BVP Example. More Complex Initial or End Conditions. Solving a Boundary Value Problem Using Continuation. Manual Continuation. Automatic Continuation. BVPs with Unknown Constants. The Elastica Problem. Non-separated Boundary Conditions. An Unknown Integration Interval. Integral Constraints. Sturm-Liouville Problems. A Reaction Transport Problem. Exercises. A Stiff Boundary Value Problem. The Mathieu Equation. Another Swirling Flow Problem. Another Reaction Transport Problem. Appendix. Butcher Tableaux for Some Runge-Kutta Methods. Coefficients for Some Linear Multistep Formulae. Implemented Integration Methods for Solving Initial Value Problems in R. Other Integration Methods in R