Nancyvarkey at 18:29, 1 February 2014

2014-02-01T18:29:28Z

Lavitha Pereira: Created page with ''''Title of script''': '''Solving ODEs using Euler Methods''' '''Author: Shamika''' '''Keywords: ODEs, Euler method, modified Euler method, video tutorial''' {| style="border…'

2013-12-24T11:56:06Z

Created page with ''''Title of script''': '''Solving ODEs using Euler Methods''' '''Author: Shamika''' '''Keywords: ODEs, Euler method, modified Euler method, video tutorial''' {| style="border…'

New page

'''Title of script''': '''Solving ODEs using Euler Methods'''

'''Author: Shamika'''

'''Keywords: ODEs, Euler method, modified Euler method, video tutorial'''

{| style="border-spacing:0;"
! Visual Cue
! <center>Narration</center>

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 1
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Dear Friends,

Welcome to the Spoken Tutorial on “'''Solving ODEs using Euler Methods'''”

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 2 -Learning Objective Slide
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| At the end of this tutorial, you will learn how to:

* Solve '''ODEs '''using''' Euler '''and''' Modified Euler methods '''in '''Scilab'''
* Develop '''Scilab''' code to solve '''ODEs'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 3-System Requirement slide
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| To record this tutorial, I am using

'''Ubuntu 12.04''' as the operating system

and '''Scilab 5.3.3''' version

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 4- Prerequisites slide
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| To practise this tutorial, a learner

* should have basic knowledge of '''Scilab '''
* and should know how to solve''' ODEs.'''

To learn '''Scilab''', please refer to the relevant tutorials available on the '''Spoken Tutorial '''website.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 5- Euler Method
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| In '''Euler''' '''method''', we get an accurately approximate solution of the '''ODE'''.

It is used to solve initial value problems where initial values of the '''differential equation''' are given.

It can be used to solve '''continuous functions'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 6- Example
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us solve an example using '''Euler method.'''

We are given an initial value problem

'''y dash is equal to minus two t minus y'''.

The initial value of '''y '''is given as '''minus one '''and the '''step length '''is given as '''zero point one'''.

We have to find the value of '''y '''at time '''t''' '''equal to zero point five'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Open Euler_ode.sci on Scilab Editor
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us look at the code for '''Euler method'''.

Open '''Euler underscore o d e dot sci''' on '''Scilab editor'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

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

| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We define the function '''Euler underscore o d e '''with '''arguments f, t init, y init, h '''and''' n'''

where

* '''F '''denotes the function to be solved,
* '''t init''' is the initial value of time '''t''',
* '''y init '''is the initial value of '''y''',
* '''h''' is the '''step length''', and '''n''' is the number of '''iterations'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

t <nowiki>=</nowiki> zeros(N+1, 1)

y <nowiki>=</nowiki> zeros(N+1,1)
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we initialize the values of '''t '''and '''y to vectors of zeros. '''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

t(1) <nowiki>=</nowiki> tinit

y(1) <nowiki>=</nowiki> yinit

| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We place the initial values of '''t '''and''' y''' in''' t of one''' and '''y of one''' respectively.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

for j <nowiki>=</nowiki> 1:N

t(j+1) <nowiki>=</nowiki> t(j) + h

y(j + 1) <nowiki>=</nowiki> y(j) + h<nowiki>*</nowiki>f(t(j), y(j))

end
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we '''iterate''' from '''one to N''' to find the value of''' y'''.

Here we apply '''Euler method''' to find the value of''' y'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

endfunction

| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Finally we end the '''function'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Click on Execute and select Save and Execute
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Save and execute the file''' Euler underscore o d e dot sci'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Switch to Scilab console
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Switch to '''Scilab console''' the solve the example problem.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

<nowiki>deff('[ydot]=f(t,y)','ydot=(-2*t)-y')</nowiki>
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| 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'''

Press '''Enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

tinit=0
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then type '''t init is equal to zero'''.

Press '''Enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

yinit=-1
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Type '''y init is equal to minus one'''.

Press '''Enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

h=0.1
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Type '''step length h is equal to zero point one'''.

Press '''Enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

N=5
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The '''step length '''is '''zero point one''', and we have to find the value of '''y '''at '''zero point five.'''

So, the number of '''iterations''' should be '''five'''.

At each '''iteration, '''the value of '''t '''will be increased by '''zero point one'''.

So type '''capital n is equal to five'''.

And press '''Enter.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

<nowiki>[t, y] = Euler_ode(f, tinit, yinit, h, N)</nowiki>
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| To '''call''' the '''function,''' type

'''open square bracket t comma y close square bracket equal to Euler underscore o d e open paranthesis f comma t init comma y init comma h comma capital n close paranthesis '''

Press '''Enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Show console
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The value of''' y''' '''at t equal to zero point five''' is shown.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 7- Modified Euler Method
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Now let us look at '''modified''' '''Euler method'''.

It is a '''second order method '''and is a '''stable two step method'''.

We find the '''average '''of the '''function '''at the beginning and end of '''time step'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 8- Example
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us solve this example using '''modified Euler method'''.

We are given a '''function y dash is equal to t plus y plus t y'''.

The initial value of '''y '''is '''one '''and the '''step length '''is '''zero point zero one'''.

We have to find the value of '''y '''at '''time t equal to zero point one '''using '''Modified Euler's method. '''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Open ModiEuler_ode.sci on Scilab Editor

| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us look at the code for '''Modified Euler method '''on '''Scilab Editor'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

ModiEuler_ode(f, tinit, yinit, h, N)

| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We define the '''function''' with '''arguments f, t init, y init, h '''and''' n '''

where

* '''F''' is the '''function''' to be solved,
* '''t init''' is the intial '''time''' value,
* '''y init''' is the inital value of '''y''',
* '''h''' is the '''step length''' and
* '''n''' is the number of '''iterations'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

t <nowiki>=</nowiki> zeros(N+1,1)

y <nowiki>=</nowiki> zeros(N+1,1)
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we initialize the '''arrays''' for '''y''' and '''t.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

t(1) <nowiki>=</nowiki> tinit

y(1) <nowiki>=</nowiki> yinit
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We place the initial values of''' t''' and''' y '''in''' t of one''' and''' y of one''' respectively.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

for j <nowiki>=</nowiki> 1:N

t(j+1) <nowiki>=</nowiki> t(j) + h

y(j+1) <nowiki>=</nowiki> y(j) + h<nowiki>*</nowiki>f(t(j), y(j))

y(j+1) <nowiki>=</nowiki> y(j) + h<nowiki>*</nowiki>(f(t(j),y(j)) + f(t(j + 1),y(j)+h<nowiki>*</nowiki>f(t(j),y(j))))/2<nowiki>;</nowiki>

end
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We implement '''Modified Euler Method''' here.

Here we find the average value of '''y''' at the beginning and end of '''time step'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Click on Execute and select Save and Execute
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Save and execute the file '''Modi Euler underscore o d e dot sci'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Switch to Scilab console
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Switch to '''Scilab console.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type clc

Press enter
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Clear the screen by typing '''c l c'''.

Press '''Enter.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

<nowiki>deff('[ydot]=f(t,y)','ydot=t+y+t*y')</nowiki>
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| 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 '''

Press '''Enter.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

tinit=0
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then type''' t init equal to zero'''

and press '''Enter'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

yinit=1
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Type '''y init equal to one'''

and press '''Enter.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

h=0.01
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then type '''h equal to zero point zero one'''

and press '''Enter.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

N=10
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Type '''capital N equal to ten'''.

Since the number of '''iterations '''should be '''ten to time t equal to zero point one '''with '''step length '''of '''zero point zero one. '''

Press '''Enter.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

<nowiki>[t, y] = ModiEuler_ode(f, tinit, yinit, h, N)</nowiki>
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| 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.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Show console
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The value of '''y '''at''' t equal to zero point one''' is shown.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 9- Example
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us summarize this tutorial.

In this tutorial we have learnt to develop Scilab code for '''Euler '''and '''modified Euler methods'''.

We have also learnt to solve '''ODEs '''using these methods in '''Scilab.. '''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Show Slide 10'''

'''Title: About the Spoken Tutorial Project'''

* Watch the video available at [http://spoken-tutorial.org/What_is_a_Spoken_Tutorial http://spoken-tutorial.org/What_is_a_Spoken_Tutorial]

* It summarises the Spoken Tutorial project

* If you do not have good bandwidth, you can download and watch it

| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * About the Spoken Tutorial Project<br/>

* Watch the video available at [http://spoken-tutorial.org/ http://spoken-tutorial.org]/What_is_a_Spoken_Tutorial <br/>

* It summarises the Spoken Tutorial project<br/>

* If you do not have good bandwidth, you can download and watch it

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Show Slide 11'''

'''Title: Spoken Tutorial Workshops'''

The Spoken Tutorial Project Team

* Conducts workshops using spoken tutorials

* Gives certificates for those who pass an online test

* For more details, please write to contact@spoken-tutorial.org

| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The Spoken Tutorial Project Team

* Conducts workshops using spoken tutorials

* Gives certificates for those who pass an online test

* For more details, please write to contact at spoken hyphen tutorial dot org

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Show Slide 12'''

'''Title: Acknowledgement'''

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

* It is supported by the National Mission on Education through ICT, MHRD, Government of India

* More information on this Mission is available at

* [http://spoken-tutorial.org/NMEICT-Intro http://spoken-][http://spoken-tutorial.org/NMEICT-Intro tutorial.org/][http://spoken-tutorial.org/NMEICT-Intro NMEICT-][http://spoken-tutorial.org/NMEICT-Intro Intro]

| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Spoken Tutorial Project is a part of the Talk to a Teacher project
* It is supported by the National Mission on Education through ICT, MHRD, Government of India
* More information on this Mission is available at
* spoken hyphen tutorial dot org slash NMEICT hyphen Intro <br/>

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"|
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| This is Ashwini Patil signing off. Thanks for joining.

|}

Scilab/C4/ODE-Euler-methods/English - Revision history

Nancyvarkey at 18:29, 1 February 2014

Nancyvarkey at 07:24, 26 December 2013

Lavitha Pereira: Created page with ''''Title of script''': '''Solving ODEs using Euler Methods''' '''Author: Shamika''' '''Keywords: ODEs, Euler method, modified Euler method, video tutorial''' {| style="border…'

@@ Line 140: / Line 140: @@
 |-
 | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Switch to Scilab console
-| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Switch to '''Scilab console''' the solve the example problem.
+| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Switch to '''Scilab console''' to solve the example problem.
 |-