Scilab/C4/ODE-Euler-methods/English-timed
From Script | Spoken-Tutorial
| Time | Narration |
| 00:01 | Dear Friends, |
| 00:02 | Welcome to the Spoken Tutorial on “Solving ODEs using Euler Methods” |
| 00:09 | At the end of this tutorial, you will learn how to: |
| 00:12 | Solve ODEs using Euler and Modified Euler methods in Scilab |
| 00:18 | Develop Scilab code to solve ODEs |
| 00:22 | To record this tutorial, I am using |
| 00:25 | Ubuntu 12.04 as the operating system
|
| 00:28 | and Scilab 5.3.3 version |
| 00:32 | To practise this tutorial, a learner |
| 00:34 | should have basic knowledge of Scilab |
| 00:37 | and should know how to solve ODEs. |
| 00:40 | To learn Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website. |
| 00:48 | In Euler method, we get an accurately approximate solution of the ODE. |
| 00:55 | It is used to solve initial value problems where initial values of the differential equation are given. |
| 01:03 | It can be used to solve continuous functions. |
| 01:08 | Let us solve an example using Euler method. |
| 01:12 | We are given an initial value problem - |
| 01:15 | y dash is equal to minus two t minus y.
|
| 01:20 | The initial value of y is given as minus one |
| 01:25 | and the step length is given as zero point one.
|
| 01:29 | We have to find the value of y at time t equal to zero point five. |
| 01:36 | Let us look at the code for Euler method.
|
| 01:39 | Open Euler underscore o d e dot sci on Scilab editor. |
| 01:47
|
We define the function Euler underscore o d e with arguments f, t init, y init, h and n
|
| 01:58 | 'where f denotes the function to be solved,
|
| 02:01 | t init is the initial value of time t, |
| 02:05 | y init is the initial value of y |
| 02:09 | h is the step length, and n is the number of iterations. |
| 02:14 | Then we initialize the values of t and y to vectors of zeros. |
| 02:21 | We place the initial values of t and y in t of one and y of one respectively. |
| 02:29 | Then we iterate from one to N to find the value of y. |
| 02:33 | Here we apply Euler method to find the value of y. |
| 02:39 | Finally we end the function. |
| 02:42 | Save and execute the file Euler underscore o d e dot sci
|
| 02:49 | Switch to Scilab console to solve the example problem.
|
| 02:54 | We define the function by typing
|
| 02:56 | 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
|
| 03:26 | Press Enter. |
| 03:28 | Then type t init is equal to zero.
|
| 03:31 | Press Enter.
|
| 03:33 | Type y init is equal to minus one.
'
|
| 03:38 | Press Enter
|
| 03:40 | Type step length h is equal to zero point one.
|
| 03:44 | Press Enter
|
| 03:46 | The step length is zero point one, and we have to find the value of y at zero point five.
|
| 03:53 | So, the number of iterations should be five.
|
| 03:59 | At each iteration, the value of t will be increased by zero point one.
|
| 04:05 | So type capital N is equal to five.
|
| 04:09 | And press Enter.
|
| 04:11 | To call the function, type |
| 04:14 | 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 |
| 04:33 | Press Enter.
|
| 04:35 | The value of y at t equal to zero point five is shown. |
| 04:41 | Now let us look at Modified Euler method. |
| 04:45 | It is a second order method and is a stable two step method.
|
| 04:51 | We find the average of the function at the beginning and end of time step.
|
| 04:56 | Let us solve this example using Modified Euler method.
|
| 05:02 | We are given a function y dash is equal to t plus y plus t y.
|
| 05:08 | The initial value of y is one |
| 05:12 | and the step length is zero point zero one. |
| 05:16 | We have to find the value of y at time t equal to zero point one using Modified Euler's method.
|
| 05:25 | Let us look at the code for Modified Euler method on Scilab Editor |
| 05:31 | We define the function with arguments f, t init, y init, h and n |
| 05:39 | where f is the function to be solved
|
| 05:42 | t init is the intial time value, |
| 05:45 | y init is the inital value of y |
| 05:49 | h is the step length and
|
| 05:51 | n is the number of iterations. |
| 05:54 | Then we initialize the arrays for y and t. |
| 05:58 | We place the initial values of t and y in t of one and y of one respectively. |
| 06:07 | We implement Modified Euler Method here.
|
| 06:11 | Here we find the average value of y at the beginning and end of time step. |
| 06:17 | Save and execute the file Modi Euler underscore o d e dot sci. |
| 06:23 | Switch to Scilab console. |
| 06:26 | Clear the screen by typing c l c. |
| 06:30 | Press Enter.
|
| 06:32 | 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 |
| 07:01 | Press Enter.
|
| 07:03 | Then type t init equal to zero, press Enter
|
| 07:08 | Type y init equal to one and press Enter.
|
| 07:12 | Then type h equal to zero point zero one press Enter. |
| 07:19 | Type capital N equal to ten.
|
| 07:22 | Since the number of iterations should be ten to time t equal to zero point one with step length of zero point zero one. |
| 07:34 | Press Enter. |
| 07:36 | Then call the function modi euler underscore o d e by typing. |
| 07:41 | 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
|
| 08:03 | Press Enter.
|
| 08:05 | The value of y at t equal to zero point one is shown. |
| 08:10 | Let us summarize this tutorial. |
| 08:14 | In this tutorial we have learnt to develop Scilab code for Euler and modified Euler methods. |
| 08:21 | We have also learnt to solve ODEs using these methods in Scilab.
|
| 08:28 | Watch the video available at the link shown below |
| 08:32 | It summarises the Spoken Tutorial project
|
| 08:35 | If you do not have good bandwidth, you can download and watch it |
| 08:40 | The spoken tutorial project Team |
| 08:42 | Conducts workshops using spoken tutorials
|
| 08:45 | Gives certificates to those who pass an online test
|
| 08:49 | For more details, please write to contact@spoken-tutorial.org
|
| 08:55 | Spoken Tutorial Project is a part of the Talk to a Teacher project
|
| 09:00 | It is supported by the National Mission on Eduction through ICT, MHRD, Government of India. |
| 09:07 | More information on this mission is available at the link shown below |
| 09:13 | This is Ashwini Patil signing off. |
| 09:15 | Thank you for joining. |
