5.2.2 The General Euler's Method

The method used for obtaining approximate solutions to the Velocity-Squared Model on the preceding page is a special case of Euler's Method, a general procedure for obtaining numerical solutions of initial value problems.

Note 2 – Euler

Suppose we want to approximate the solution of the initial value problem

on the interval `[a,b].` Here `ftext[(]t,y text[)]` represents some expression in `y` or `t` or both. [In the Velocity-Squared Model, `f` is a function of the dependent variable `v` alone: `ftext[(]vtext[)]=g-cv^2`, where `g` and `c` are constants.]

In this general setting, Euler's Method consists of the following steps:

1. Select the number `n` of equal subdivisions of the interval `[a,b].` Then

   `Delta t = (b-a)/n.`

2. Set `t_k=a+kDelta t` for `k=0, 1, 2, ..., n`.

3. Let

   `y_1=y_0+ftext[(]t_0,y_0text[)]Delta t`

   `y_2=y_1+ftext[(]t_1,y_1text[)]Delta t`

   ...

   `y_k=y_(k-1)+ftext[(]t_(k-1),y_(k-1)text[)]Delta t`

   ...

   `y_n=y_(n-1)+ftext[(]t_(n-1),y_(n-1)text[)]Delta t

Then `y_k` approximates the solution values `ytext[(]t_ktext[)]` for `k=1, 2, ..., n.` In general, the accuracy of the approximation improves as the number of subdivisions increases, i.e., as the step size `Delta t` decreases.

Checkpoint 1

Activity 2

1. Use a computer algebra tool to determine the Euler's Method approximation to the solution of the drizzle drop initial value problem

   `(dv)/(dt)=g-Kv` with `vtext[(]0text[)]=0`

   on the interval `[0,0.2]` with `n=10.` Use units of feet and seconds, and let `K=52.6 text[sec]^(-1)`.

2. Repeat with `n=20.`

3. Repeat with `n=40.`

4. Repeat with `n=80.`

5. Compare your approximate solutions in (a)-(d) with the symbolic solution

   ${\displaystyle v\left(t\right)=\frac{g}{K}(1-e^{-Kt})}$

   obtained in Activity 2 in Section 5.1.

Comment on Activity 2