Modelling with Ordinary Differential Equations: A Comprehensive Approach (Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series)

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