Scilab/C4/ODE-Euler-methods/English-timed

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 practice 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 parenthesis 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 parenthesis minus two asterisk t close parenthesis minus y close single quote close parenthesis
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 parenthesis f comma t init comma y init comma h comma capital N close parenthesis
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 parenthesis 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 parenthesis
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 parenthesis f comma t init comma y init comma h comma capital N close parenthesis
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 summarizes 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.

Contributors and Content Editors

Gaurav, PoojaMoolya, Sandhya.np14