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 \(f(t,y)\) represents some expression in \(y\) or \(t\) or both. [In the Velocity-Squared Model, \(f\) is a function of the dependent variable \(v\) alone: \(f(v)=g-cv^2\), 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

   \(\Delta t = \frac{b-a}{n}\).

2. Set \(t_k=a+k\Delta t\) for \(k=0, 1, 2, ..., n\).

3. Let

   \(y_1=y_0+f(t_0,y_0)\Delta t\)

   \(y_2=y_1+f(t_1,y_1)\Delta t\)

   ...

   \(y_k=y_{k-1}+f(t_{k-1},y_{k-1})\Delta t\)

   ...

   \(y_n=y_{n-1}+f(t_{n-1},y_{n-1})\Delta t\)

Then \(y_k\) approximates the solution values \(y(t_k)\) 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 the Euler's Method tool to find an approximation to the solution of the drizzle drop initial value problem

   \(\frac{dv}{dt}=g-Kv\) with \(v(0)=0\)

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

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