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 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.

Contributors and Content Editors

Gaurav, PoojaMoolya, Sandhya.np14