1 Initial value problems

In HELM booklet  19.4 we saw the following initial value problem which arises from Newton’s law of cooling

$\frac{d\theta }{dt}=-k\left(\theta -{\theta }_s\right),\theta \left(0\right)={\theta }_0.$

Here $\theta =\theta \left(t\right)$ is the temperature of some liquid at time $t$ , ${\theta }_0$ is the initial temperature at $t=0$ and ${\theta }_s$ is the surrounding temperature. The constant of proportion $k$ has units ${\text{s}}^{-1}$ and depends on the properties of the liquid.

This initial value problem has two parts: the differential equation $\frac{d\theta }{dt}=-k\left(\theta -{\theta }_s\right)$ , which models the physical process, and the initial condition $\theta \left(0\right)={\theta }_0$ .

It should be noted that there are applications in which initial value problems do not model processes that are time dependent, but we will not dwell on this fact here.

The initial value problem above is such that we can write down an exact or analytic solution (it is $\theta \left(t\right)={\theta }_s+\left({\theta }_0-{\theta }_s\right)e^{-kt}$ ) but there are many applications where it is impossible or undesirable to seek such a solution. The aim of this Section is to begin to describe numerical methods that can be used to find approximate solutions of initial value problems.

Rather than using the application-specific notation given above involving $\theta$ we will consider the following initial value problem in this Section. We seek $y=y\left(t\right)$ (or an approximation to it) that satisfies the differential equation

$\frac{dy}{dt}=f\left(t,y\right),\left(t>0\right)$

and which is subject to the initial condition

$y\left(0\right)=y_0,$

a known quantity.

Some of the examples we will consider will be such that an analytic solution is readily available, and this fact can be used as a check on the accuracy of the numerical methods that follow.