Scilab/C4/ODEApplications/English
Title of script: Solving ODEs using Scilab ode Function
Author: Shamika
Keywords: ODEs



Slide 1  Dear Friends,
Welcome to the Spoken Tutorial on “Solving ODEs using Scilab ode Function” 
Slide 2 Learning Objective Slide  At the end of this tutorial, you will learn how to:

Slide 3 Learning Objective Slide  The typical examples we will be solving are:

Slide 4System Requirement slide  To record this tutorial, I am using

Slide 5 Prerequisites slide  To practise this tutorial, a learner
To learn Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website. 
Slide 6 ode Function  The ode function is an ordinary differential equation solver.

Slide 7 Motion of Simple pendulum  Consider the motion of simple pendulum.

Slide 8 Motion of Simple pendulum  Then the position of the pendulum is described by
theta double dash t minus g by l into sin of theta t equal to zero.

Slide 9 Motion of Simple pendulum  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.

Open Pendulum.sci on Scilab Editor  Let us look at the code for solving this problem.

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y0=[%pi/4 0]'; t0=0; t=0:1:5

The first line of the code defines the initial conditions of the ODE.

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function dy=Pendulum(t, y) dy(1) = y(2); dy(2) = (9.8/0.5)*sin(y(1)); endfunction

Next, we convert the given equation to a system of first order ODEs.

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y=ode(y0,t0,t,Pendulum)

Then we call the ode function with arguments y zero, t zero, t and the function Pendulum. 
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plot(t,y(1,:),'*',t,y(2,:),'')

The solution to the equation will be a matrix with two rows.

Click on Execute and select Save and Execute  Save and execute the file Pendulum dot sci 
Show plot  The plot shows how the values of y and y dash vary with time. 
Switch to Scilab console  Switch to Scilab console 
Type y
Press Enter 
If you want to see the values of y, type y on the console and press Enter.

Slide 10 Van der Pol Equation  Let us solve Van der Pol equation using the ode function.
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.

Open Vanderpol.sci on scilab editor  Let us look at the code for Van der Pol equation.

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y0=[1 0]'; t0=0; t=2:1:10;

We define the initial conditions of the ODEs and time and then define the time range.

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function dy=Vanderpol(t, y) dy(1) = y(2); dy(2) = 0.897*(y(1).^2  1)*y(2)  y(1); endfunction

Then we define the function van der pol and construct a system of first order ODEs.

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y=ode(y0,t0,t,Vanderpol)

Then we call ode function and solve the system of equations. 
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plot(t,y(1,:),'',t,y(2,:),'')

Finally we plot y and y dash versus t. 
Click on Execute and select Save and Execute  Save and execute the file van der pol dot sci. 
Show plot  The plot showing voltage versus time is shown.

Slide 11, 12 Lorenz system

The Lorenz system of equations is given by
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.

Open Lorenz.sci on Scilab editor

Switch to Scilab editor and open Lorenz dot sci 
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x0=[10 10 25]'; t0=0; t=0:1:25;

We start by defining the initial conditions of the ODEs.

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function dx=Lorenz(t, x) sigma = 10; r = 28; b = 8/3; dx(1) = sigma*(x(2)  x(1) ); dx(2) = ((1 + r)  x(3))*x(1)  x(2); dx(3) = x(1)*x(2)  b*x(3); endfunction

We define the function Lorenz and then define the given constants sigma, r and b. Then we define the first order ODEs. 
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x=ode(x0,t0,t,Lorenz)

Then we call the ode function to solve the Lorenz system of equations.

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plot(t,x(1,:),'**',t,x(2,:),'', t,x(3,:),'..')

Then we plot x one, x two and x three versus time. 
Click on Execute and select Save and Execute  Save and execute the file Lorenz dot sci. 
Show plot  The plot of x one, x two and x three versus time is shown. 
Slide 13 Summary  Let us summarize this tutorial.

Show Slide 14
Title: About the Spoken Tutorial Project

* About the Spoken Tutorial Project

Show Slide 15
Title: Spoken Tutorial Workshops The Spoken Tutorial Project Team

The Spoken Tutorial Project Team

Show Slide 16
Title: Acknowledgement

* Spoken Tutorial Project is a part of the Talk to a Teacher project

This is Ashwini Patil from IIT Bombay signing off. Thanks for joining. 