2 Numerical solutions

We suppose that the initial value problem

$\frac{dy}{dt}=f\left(t,y\right)y\left(0\right)=y_0$

is such that we are unable (or unwilling) to seek a solution analytically (that is, by hand) and that we prefer to use a computer to approximate $y$ instead. We begin by asking what we expect a numerical solution to look like.

Numerical solutions to initial value problems discussed in this Workbook will be in the form of a sequence of numbers approximating $y\left(t\right)$ at a sequence of values of $t$ . The simplest methods choose the $t$ -values to be equally spaced, and we will stick to these methods. We denote the common distance between consecutive $t$ -values as $h$ .

In Figure 1 the exact solution $y\left(t\right)$ is shown as a thick curve and approximations to $y\left(nh\right)$ are shown as crosses.

Figure 1

The general idea is to take the given initial condition $y_0$ and then use it together with what we know about the physical process (from the differential equation) to obtain an approximation $y_1$ to $y\left(h\right)$ . We will have then carried out the first time step .

Then we use the differential equation to obtain $y_2$ , an approximation to $y\left(2h\right)$ . Thus the second time step is completed.

And so on, at the $n^{\text{th}}$ time step we find $y_n$ , an approximation to $y\left(nh\right)$ .

The way in which we choose to “use the differential equation" will define a particular numerical method, and some ways are better than others. We begin by looking at the simplest method.