Introduction
Many engineering applications describe the evolution of some process with time. In order to define such an application we require two distinct pieces of information: we need to know what the process is and also when or where the application started.
In this Section we begin with a discussion of some of these so-called initial value problems . Then we look at two numerical methods that can be used to approximate solutions of certain initial value problems. These two methods will serve as useful instances of a fairly general class of methods which we will describe in Section 32.2.
Prerequisites
- revise the trapezium method for approximating integrals in HELM booklet 31.2
- review the material concerning approximations to derivatives in HELM booklet 31.3
Learning Outcomes
- recognise an initial value problem
- implement the Euler and trapezium $$ method to approximate the solutions of certain initial value problems
Contents
1 Initial value problems2 Numerical solutions
3 An explicit method
3.1 Accuracy of Euler’s method
4 An implicit method
4.1 Accuracy of the trapezium method