solution is given by:
where we note that
In order to prove this, we need the formula for the sum of a series:
For a readable introduction to difference schemes, we refer the reader to Goldberg (1986).
2.8 SUMMARY AND CONCLUSIONS
In this chapter we gave an introduction to scalar ODEs and systems of ODEs. The main goal was to help the reader become acquainted with their mathematical and numerical foundations as well as become familiar with the associated notation. We recommend learning the main concepts in this chapter because many of them will be used and needed when we model one-factor and two-factor convection-diffusion-reaction PDEs such as the Black–Scholes equation, for example.
Contrary to popular thinking, there is more to ODEs than trying to find analytical solutions for them. Very few ODEs have analytical solutions, and we must resort to ODE solvers.
CHAPTER 3 Ordinary Differential Equations (ODEs), Part 2
The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
Paul Halmos
3.1 INTRODUCTION AND OBJECTIVES
In Chapter 2 we discussed both systems of ODEs and scalar ODEs. The focus was mainly concerned with notation, the structure of ODEs and finite difference schemes to approximate them. We implicitly assumed that the solution of the corresponding initial value problem existed in an otherwise unspecified time interval and that the solution was unique. These assumptions constitute a huge leap of faith. In this chapter we discuss existence and uniqueness results for ODEs and stochastic differential equation (SDEs). We also introduce several important numerical schemes and code in C++ and Python.
3.2 EXISTENCE AND UNIQUENESS RESULTS
We turn our attention to a more general initial value problem for a non-linear system of ODEs:
where:
Some of the important questions to be answered are:
Does System (3.1) have a unique solution?
In which interval does this solution exist?
What is the asymptotic behaviour of the solution as ?
To this end, let B be a region of
and:
Theorem 3.1 Let f and
