Scilab/C4/ODEEulermethods/English
Title of script: Solving ODEs using Euler Methods
Author: Shamika
Keywords: ODEs, Euler method, modified Euler method, video tutorial
Visual Cue  

Slide 1  Dear Friends,
Welcome to the Spoken Tutorial on “Solving ODEs using Euler Methods” 
Slide 2 Learning Objective Slide  At the end of this tutorial, you will learn how to:

Slide 3System Requirement slide  To record this tutorial, I am using
Ubuntu 12.04 as the operating system and Scilab 5.3.3 version 
Slide 4 Prerequisites slide  To practise this tutorial, a learner
To learn Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website. 
Slide 5 Euler Method  In Euler method, we get an accurately approximate solution of the ODE.

Slide 6 Example  Let us solve an example using Euler method.
y dash is equal to minus two t minus y.

Open Euler_ode.sci on Scilab Editor  Let us look at the code for Euler method.

Highlight
Euler_ode(f, tinit, yinit, h, N)

We define the function Euler underscore o d e with arguments f, t init, y init, h and n
where

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t = zeros(N+1, 1)

Then we initialize the values of t and y to vectors of zeros. 
Highlight
t(1) = tinit

We place the initial values of t and y in t of one and y of one respectively. 
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for j = 1:N t(j+1) = t(j) + h y(j + 1) = y(j) + h*f(t(j), y(j)) end 
Then we iterate from one to N to find the value of y.

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endfunction

Finally we end the function. 
Click on Execute and select Save and Execute  Save and execute the file Euler underscore o d e dot sci 
Switch to Scilab console  Switch to Scilab console the solve the example problem. 
Type on console
deff('[ydot]=f(t,y)','ydot=(2*t)y') 
We define the function by typing
d e f f open paranthesis open single quote open square bracket y dot close square bracket equal to f of t comma y close single quote comma open single quote y dot equal to open paranthesis minus two asterisk t close paranthesis minus y close single quote close paranthesis

Type on console
tinit=0 
Then type t init is equal to zero.
Press Enter 
Type on console
yinit=1 
Type y init is equal to minus one.
Press Enter 
Type on console
h=0.1 
Type step length h is equal to zero point one.
Press Enter 
Type on console
N=5 
The step length is zero point one, and we have to find the value of y at zero point five.
At each iteration, the value of t will be increased by zero point one.

Type on console
[t, y] = Euler_ode(f, tinit, yinit, h, N) 
To call the function, type

Show console  The value of y at t equal to zero point five is shown. 
Slide 7 Modified Euler Method  Now let us look at modified Euler method.

Slide 8 Example  Let us solve this example using modified Euler method.
The initial value of y is one and the step length is zero point zero one.

Open ModiEuler_ode.sci on Scilab Editor

Let us look at the code for Modified Euler method on Scilab Editor. 
Highlight
ModiEuler_ode(f, tinit, yinit, h, N)

We define the function with arguments f, t init, y init, h and n
where

Highlight
t = zeros(N+1,1) y = zeros(N+1,1) 
Then we initialize the arrays for y and t. 
Highlight
t(1) = tinit y(1) = yinit 
We place the initial values of t and y in t of one and y of one respectively. 
Highlight
for j = 1:N t(j+1) = t(j) + h y(j+1) = y(j) + h*f(t(j), y(j)) y(j+1) = y(j) + h*(f(t(j),y(j)) + f(t(j + 1),y(j)+h*f(t(j),y(j))))/2; end 
We implement Modified Euler Method here.

Click on Execute and select Save and Execute  Save and execute the file Modi Euler underscore o d e dot sci. 
Switch to Scilab console  Switch to Scilab console. 
Type clc
Press enter 
Clear the screen by typing c l c.
Press Enter. 
Type on console
deff('[ydot]=f(t,y)','ydot=t+y+t*y') 
Define the function by typing
d e f f open paranthesis open single quote open square bracket y dot close square bracket equal to f of t comma y close single quote comma open single quote y dot equal to t plus y plus t asterisk y close single quote close paranthesis

Type on console
tinit=0 
Then type t init equal to zero
and press Enter 
Type on console
yinit=1 
Type y init equal to one
and press Enter. 
Type on console
h=0.01 
Then type h equal to zero point zero one
and press Enter. 
Type on console
N=10 
Type capital N equal to ten.

Type on console
[t, y] = ModiEuler_ode(f, tinit, yinit, h, N) 
Then call the function modi euler underscore o d e by typing
open square bracket t comma y close square bracket equal to modi euler underscore o d e open paranthesis f comma t init comma y init comma h comma capital n close paranthesis Press Enter. 
Show console  The value of y at t equal to zero point one is shown. 
Slide 9 Example  Let us summarize this tutorial.

Show Slide 10
Title: About the Spoken Tutorial Project

* About the Spoken Tutorial Project

Show Slide 11
Title: Spoken Tutorial Workshops The Spoken Tutorial Project Team

The Spoken Tutorial Project Team

Show Slide 12
Title: Acknowledgement

* Spoken Tutorial Project is a part of the Talk to a Teacher project

This is Ashwini Patil signing off. Thanks for joining. 