Numerical Solution of Ordinary Differential Equations

L.F. Shampine

• Publisher: Routledge
• Year: 2018
• ISBN:
  9781351427548
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

This new work is an introduction to the numerical solution of the initial value problem for a system of ordinary differential equations. The first three chapters are general in nature, and chapters 4 through 8 derive the basic numerical methods, prove their convergence, study their stability and consider how to implement them effectively. The book focuses on the most important methods in practice and develops them fully, uses examples throughout, and emphasizes practical problem-solving methods.

1

The Mathematical Problem

The aim of this chapter is to develop a sense of what kinds of initial value problems can be solved numerically and how to prepare problems for their numerical solution. First we review the “facts of life” about the existence and uniqueness of solutions. It is convenient in the theory of ordinary differential equations (ODEs) to work with problems written in a standard form, and because the codes all expect problems to be presented in this way, we must go into this. Some basic mathematical tools are found in the appendix. The concept of “order” is fundamental to a study of the numerical solution of the initial value problem. Because it may not be familiar to the reader, the elements are developed here. Finally a series of substantial examples are taken up to show how one might be able to deal with problems that do not fit neatly into the standard theory of ODEs and their numerical solution. Some of the examples will be used throughout the book for illustrative purposes.

§1 Existence, Uniqueness, and Standard Form

We begin by considering the initial value problem itself to see what kinds of problems are meaningful and what kinds we might hope to solve numerically. Even very simple problems that can be understood with arguments from calculus show what can happen. If F(x) is continuous on an interval [a, b], the equation

$\frac{dy}{dx}=F\left(x\right)$

(1.1)

has a solution given by the fundamental theorem of calculus,

$y\left(x\right)=B+{\displaystyle {\int }_a^{x}F\left(t\right)}dt.$

This is a solution of the differential equation (1.1) for any value of the constant B, so to select a particular solution some additional information must be supplied. There are a number of ways this might be done. Although several will appear in this chapter, the most common, and the subject of this book, is to specify the initial value

$y\left(a\right)=A.$

(1.2)

The two requirements (1.1) and (1.2) make up an initial value problem for an ordinary differential equation. There is a solution and only one, namely

$y\left(x\right)=A+{\displaystyle {\int }_a^{x}F\left(t\right)}dt.$

In general, an initial value problem for an ordinary differential equation has the form

$\begin{array}{ccc}\frac{dy}{dx}=F\left(x,y\right),& a\le x\le b,& y\left(a\right)\end{array}=A.$

It is assumed that F(x, y) is continuous in both variables. A solution is a function y(x) that is continuous and has a continuous first derivative on [a, b] (in symbols, y ∈ C¹ [a, b]), satisfies y(a) = A, and satisfies

$\frac{dy\left(x\right)}{dx}=F\left(x,y\left(x\right)\right)$

for each x in [a, b]. The contin...