Scilab/C4/ODE-Applications/English-timed
From Script | Spoken-Tutorial
Time | Narration |
00:01 | Dear Friends, 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 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 practice 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 parenthesis y zero, t zero, t and f. |
01:10 | Here y zero is the initial condition 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 of 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 summarizes 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. |