Scilab/C4/ODE-Applications/English-timed

Time	Narration
00.01	Dear Friends,
00.02	Welcome to the Spoken Tutorial on “Solving ODEs using Scilab ode Function”
00.09	At the end of this tutorial, you will learn how to:
00.12	Use Scilab ode function
00.15	Solve typical examples of ODEs and
00.18	Plot the solution
00.21	The typical examples we will be
00.24	Motion of simple pendulum
00.26	Van der Pol equation
00.28	and Lorenz system
00.30	To record this tutorial, I am using
00.33	Ubuntu 12.04 as the operating system
00.36	and Scilab 5.3.3 version
00.40	To practise this tutorial, a learner should have basic knowledge of Scilab
00.45	and should know how to solve ODEs.
00.48	To learn Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website.
00.56	The ode function is an ordinary differential equation solver.
01.01	The syntax is y equal to ode within paranthesis y zero, t zero, t and f
01.10	Here y zero is the initial conditon of the ODEs
01.15	t zero is the initial time
01.17	t is the time range
01.19	and f is the function
01.22	Consider the motion of simple pendulum.
01.25	Let theta t be the angle made by the pendulum with the vertical at time t.
01.33	We are given the initial conditions
01.36	theta zero is equal to pi by four and
01.39	theta dash of zero is equal to zero.
01.44	Then the position of the pendulum is given by
01.47	theta double dash t minus g by l into sin of theta t equal to zero.
01.56	Here g equal to 9.8 m per second square is the acceleration due to gravity and
02.03	l equal to zero point five meter is the length of the pendulum.
02.11	For the given initial conditions, we have to solve the ODE within the time range zero less than equal to t less than equal to five.
02.22	We also have to plot the solution.
02.25	Let us look at the code for solving this problem.
02.29	Open pendulum dot sci on Scilab editor.
02.34	The first line of the code defines the initial conditions of the ODE.
02.40	Then we define the intial time value. And we provide the time range.
02.46	Next, we convert the given equation to a system of first order ODEs.
02.52	We substitute the values of g and l .
02.56	Here we take y to be the given variable theta and y dash to be theta dash. '
03.03	Then we call the ode function with arguments y zero, t zero, t and the function Pendulum.
03.12	The solution to the equation will be a matrix with two rows.
03.17	The first row will contain the values of y in the given time range.
03.21	The second row will contain the values of y dash within the time range.
03.27	Hence we plot both the rows with respect to time.
03.31	Save and execute the file Pendulum dot sci
03.37	The plot shows how the values of y and y dash vary with time.
03.44	Switch to Scilab console
03.47	If you want to see the values of y, type y on the console and press Enter.
03.54	The values of y and y dash are displayed.
03.58	Let us solve Van der Pol equation using the ode function.
04.03	We are given the equation
04.06	v double dash of t plus epsilon into v of t square minus one into v dash of t plus v of t equal to zero.
04.20	The initial conditions are v of two equal to one and v dash of two equal to zero.
04.28	Assume epsilon is equal to zero point eight nine seven.
04.32	We have to find the solution within the time range two less than t less than ten and then plot the solution.
04.42	Let us look at the code for Van der Pol equation.
04.47	Switch to Scilab editor and open vander pol dot sci.
04.53	We define the initial conditions of the ODEs and time and then define the time range.
05.01	Since the inital time value is given as two, we start the time range at two.
05.07	Then we define the function vander pol and construct a system of first order ODEs.
05.15	We substitute the value of epsilon with zero point eight nine seven.
05.21	Here y refers to the voltage v.
05.25	Then we call ode function and solve the system of equations.
05.30	Finally we plot y and y dash versus t.
05.35	Save and execute the file vander pol dot sci.
05.41	The plot showing voltage versus time is shown.
05.45	Let's move onto Lorenz system of equations.
05.50	The Lorenz system of equations is given by
05.53	x one dash equal to sigma into x two minus x one,
06.00	x two dash equal to one plus r minus x three into x one minus x two and
06.08	x three dash equal to x one into x two minus b into x three.
06.16	The initial conditions are x one zero equal to minus ten, x two zero equal to ten and x three zero equal to twenty five.
06.29	Let sigma be equal to ten, r be equal to twenty eight and b equal to eight by three.
06.37	Switch to Scilab editor and open Lorenz dot sci
06.44	We start by defining the initial conditions of the ODEs.
06.48	Since there are three different ODEs, there are three initial conditions.
06.54	Then we define the inital time condition and next the time range.
07.00	We define the function Lorenz and then define the given constants sigma, r and b.
07.08	Then we define the first order ODEs.
07.12	Then we call the ode function to solve the Lorenz system of equations.
07.18	We equate the solution to x.
07.21	Then we plot x one, x two and x three versus time.
07.28	Save and execute the file Lorenz dot sci.
07.33	The plot of x one, x two and x three versus time is shown.
07.39	Let us summarize this tutorial.
07.41	In this tutorial we have learnt to develop Scilab code to solve an ODE using Scilab ode function.
07.50	Then we have learnt to plot the solution.
07.53	Watch the video available at the link shown below
07.56	It summarises the Spoken Tutorial project
07.59	If you do not have good bandwidth, you can download and watch it
08.04	The spoken tutorial project Team
08.06	Conducts workshops using spoken tutorials
08.09	Gives certificates to those who pass an online test
08.13	For more details, please write to contact@spoken-tutorial.org
08.20	Spoken Tutorial Project is a part of the Talk to a Teacher project
08.23	It is supported by the National Mission on Eduction through ICT, MHRD, Government of India.
08.31	More information on this mission is available at the link shown below
08.36	This is Ashwini Patil signing off.
08.38	Thank you for joining.

Contributors and Content Editors

Gaurav, PoojaMoolya, Sandhya.np14