Modelling With Ordinary Differential Equations: A Comprehensive Approach (Chapman & Hall/Crc Numerical Analysis and Scientific Computing)

✍ Scribed by Alfio Borzì

• Publisher: CRC Press Inc
• Year: 2020
• Tongue: English
• Leaves: 405
• Edition: Comprehensive
• Category: Library

✦ Synopsis

Modelling with Ordinary Differential Equations: A Comprehensive Approach aims to provide a broad and self-contained introduction to the mathematical tools necessary to investigate and apply ODE models. The book starts by establishing the existence of solutions in various settings and analysing their stability properties. The next step is to illustrate modelling issues arising in the calculus of variation and optimal control theory that are of interest in many applications. This discussion is continued with an introduction to inverse problems governed by ODE models and to differential games.

The book is completed with an illustration of stochastic differential equations and the development of neural networks to solve ODE systems. Many numerical methods are presented to solve the classes of problems discussed in this book.

Features:

Provides insight into rigorous mathematical issues concerning various topics, while discussing many different models of interest in different disciplines (biology, chemistry, economics, medicine, physics, social sciences, etc.)

Suitable for undergraduate and graduate students and as an introduction for researchers in engineering and the sciences

Accompanied by codes which allow the reader to apply the numerical methods discussed in this book in those cases where analytical solutions are not available

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Author
1. Introduction
1.1 Ordinary differential equations
1.2 The modelling process
2. Elementary solution methods for simple ODEs
2.1 Simple ODEs
2.1.1 Simple ODE of 1. type, y' = f(x)
2.1.2 Simple ODE of 2. type, y' = f(y)
2.1.3 Simple ODE of 3. type, y' = f(x) g(y)
2.1.4 Simple ODE of 4. type, y' = f(ax + by + d)
2.1.5 Simple ODE of 5. type, y' = f(y/x)
2.2 Linear ODEs
2.3 Method of variation of the constants
2.4 Bernoulli’s differential equation
2.5 Riccati’s differential equation
2.6 Exact differential equations
3. Theory of ordinary differential equations
3.1 The Cauchy problem and existence of solutions
3.2 Euler’s method
3.3 Uniqueness of solutions
3.4 The Carathéodory theorem
4. Systems of ordinary differential equations
4.1 Systems of first-order ODEs
4.2 Dependence of solutions on the initial conditions
4.3 Systems of linear ODEs
4.4 Systems of linear homogeneous ODEs
4.5 The d’Alembert reduction method
4.6 Nonhomogeneous linear systems
4.7 Linear systems with constant coefficients
4.8 The exponential matrix
4.9 Linear systems with periodic coefficients
5. Ordinary differential equations of order n
5.1 Ordinary differential equations of order n in normal form
5.2 Linear differential equations of order n
5.3 The reduction method of d’Alembert
5.4 Linear ODEs of order n with constant coefficients
5.5 Nonhomogeneous ODEs of order n
5.6 Oscillatory solutions
6. Stability of ODE systems
6.1 Local stability of ODE systems
6.2 Stability of linear ODE systems
6.3 Stability of nonlinear ODE systems
6.4 Remarks on the stability of periodic ODE problems
6.5 Autonomous systems in the plane
6.6 The Lyapunov method
6.7 Limit points and limit cycles
6.8 Population dynamics
6.9 The Lorenz model
6.10 Synchronisation
7. Boundary and eigenvalue problems
7.1 Linear boundary-value problems
7.2 Sturm-Liouville eigenvalue problems
8. Numerical solution of ODE problems
8.1 One-step methods
8.2 Motion in special relativity
8.3 The Kepler problem
8.4 Approximation of Sturm-Liouville problems
8.5 The shape of a drop on a flat surface
9. ODEs and the calculus of variations
9.1 Existence of a minimum
9.2 Optimality conditions
9.2.1 First-order optimality conditions
9.2.2 Second-order optimality conditions
9.3 The Euler-Lagrange equations
9.3.1 Direct and indirect numerical methods
9.3.2 Unilateral constraints
9.3.3 Free boundaries
9.3.4 Equality constraints
9.4 The Legendre condition
9.5 The Weierstrass-Erdmann conditions
9.6 Optimality conditions in Hamiltonian form
10. Optimal control of ODE models
10.1 Formulation of ODE optimal control problems
10.2 Existence of optimal controls
10.3 Optimality conditions
10.4 Optimality conditions in Hamiltonian form
10.5 The Pontryagin’s maximum principle
10.6 Numerical solution of ODE optimal control problems
10.7 A class of bilinear optimal control problems
10.8 Linear-quadratic feedback-control problems
11. Inverse problems with ODE models
11.1 Inverse problems with linear models
11.2 Tikhonov regularisation
11.3 Inverse problems with nonlinear models
11.4 Parameter identification with a tumor growth model
12. Differential games
12.1 Finite-dimensional game problems
12.2 Infinite-dimensional differential games
12.3 Linear-quadratic differential Nash games
12.4 Pursuit-evasion games
13. Stochastic differential equations
13.1 Random variables and stochastic processes
13.2 Stochastic differential equations
13.3 The Euler-Maruyama method
13.4 Stability
13.5 Piecewise deterministic processes
14. Neural networks and ODE problems
14.1 The perceptron and a learning scheme
14.2 Approximation properties of neural networks
14.3 The neural network solution of ODE problems
14.4 Parameter identification with neural networks
14.5 Deep neural networks
Appendix: Results of analysis
A.1 Some function spaces
A.1.1 Spaces of continuous functions
A.1.2 Spaces of integrable functions
A.1.3 Sobolev spaces
A.2 The Arzelà-Ascoli theorem
A.3 The Gronwall inequality
A.4 The implicit function theorem
A.5 The Lebesgue dominated convergence theorem
Bibliography
Index