Modelling with Ordinary Differential Equations

A Comprehensive Approach

Alfio Borzì

• Publisher: Chapman and Hall/CRC
• Year: 2020
• Print ISBN: 9781032336671
• eBook ISBN: 9781351190374
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

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:

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• 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.)

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

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• 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

Chapter 1

Introduction

1.1Ordinary differential equations

1.2The modelling process

In this chapter, ordinary differential equations are defined and illustrated by means of examples, also with the purpose to introduce the notation and the basic terminology used throughout this book.

In the second part of this introduction, the main principles behind the modelling process are discussed focusing on a simple model of population growth and on Newton’s model of gravitation dynamics.

1.1Ordinary differential equations

An ordinary differential equation (ODE) is an equation relating a function of one independent variable to some of its derivatives with respect to this variable.

Ordinary differential equations represent an important field of mathematics and its story begins in the seventeenth century and still constitutes a very active and broad field of research and application in the sciences and technology. The concept of a differential equation was established with the works of Gottfried Wilhelm Leibniz and Isaac Newton, of the brothers Jakob I and Johann Bernoulli, Daniel Bernoulli, and of Leonhard Euler, Giuseppe Luigi Lagrangia (Joseph-Louis Lagrange), and Pierre-Simon Laplace, among others. These mathematicians also started the development of a general theory for ODEs along with numerous applications in geometry, mechanics, and optimisation. This remarkable mathematical development continued in the 19th century with the works of Augustin-Louis Cauchy and Giuseppe Peano, Charles Émile Picard, Henri Poincarè, Vito Volterra, Constantin Carathéodory, etc., who greatly contributed to the foundation of the modern theory and methodology of ordinary differential equations.

Before giving some introductory examples of ODEs, let us assume that the independent variable represents time t, then it is quite common to denote the derivative of function with respect t...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Author
• 1. Introduction
• 2. Elementary solution methods for simple ODEs
• 3. Theory of ordinary differential equations
• 4. Systems of ordinary differential equations
• 5. Ordinary differential equations of order n
• 6. Stability of ODE systems
• 7. Boundary and eigenvalue problems
• 8. Numerical solution of ODE problems
• 9. ODEs and the calculus of variations
• 10. Optimal control of ODE models
• 11. Inverse problems with ODE models
• 12. Differential games
• 13. Stochastic differential equations
• 14. Neural networks and ODE problems
• Appendix: Results of analysis
• Bibliography
• Index