Difference between revisions of "Scilab/C4/ODE-Applications/Gujarati"

|Dear Friends,  

| Welcome to the Spoken Tutorial on '''Solving ODEs using Scilab ode function'''  

| At the end of this tutorial, you will learn how to:

|* Use Scilab ode function

|* Solve typical examples of '''ODEs''' and

|* Plot the solution.  

|The typical examples will be:

|* Motion of '''simple pendulum'''

|* '''Van der Pol equation'''

|* and ''' Lorenz system'''.

|To record this tutorial, I am using

| '''Ubuntu 12.04''' as the operating system

|and '''Scilab 5.3.3''' version.

|To practice this tutorial, a learner should have basic knowledge of '''Scilab'''

|and should know how to solve '''ODEs.'''

|To learn '''Scilab,''' please refer to the relevant tutorials available on the '''Spoken Tutorial''' website.  

|The '''ode''' function is an '''ordinary differential equation solver'''.

||The syntax is '''y equal to ode''' within parenthesis '''y zero, t zero, t''' and '''f'''.

|| Here '''y zero''' is the initial condition of the '''ODEs''',

| '''t zero''' is the '''initial time''',  

|'''t''' is the '''time range''',   

|and '''f''' is the '''function'''.  

||Consider the motion of '''simple pendulum.'''  

|Let '''theta t''' be the angle made by the '''pendulum''' with the '''vertical''' at time '''t.'''

|We are given the initial conditions-  

|'''theta of zero''' is equal to '''pi by four''' and

| Then the position of the '''pendulum''' is given by:

| Here '''g equal to 9.8 m per second square''' is the '''acceleration due to gravity''' and

|''' l equal to zero point five meter''' is the length of the '''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.'''  

|We also have to '''plot''' the solution.  

|Let us look at the code for solving this problem.  

|Open '''Pendulum dot sci''' on '''Scilab editor.'''  

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

|Then we define the intial time value. And we provide the '''time range.'''  

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

|We substitute the values of '''g''' and '''l''' .

||Here we take '''y''' to be the given '''variable theta''' and '''y dash''' to be '''theta dash.'''  

|Then we call the '''ode''' function with arguments '''y zero, t zero, t''' and the function '''Pendulum.'''  

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

| The first '''row''' will contain the values of '''y''' in the given '''time range.'''

| The second '''row''' will contain the values of '''y dash ''' within the '''time range. '''

|Hence we plot both the '''rows''' with respect to '''time. '''

|Save and execute the file '''Pendulum dot sci'''.

|The plot shows how the values of '''y''' and '''y dash''' vary with '''time. '''

|Switch to '''Scilab console'''.

| If you want to see the values of '''y,''' type '''y''' on the '''console''' and press '''Enter.'''

|The values of '''y''' and '''y dash''' are displayed.  

|Let us solve '''Van der Pol equation''' using the '''ode''' function.  

|We are given the '''equation '''-

|The initial conditions are '''v of two equal to one''' and '''v dash of two equal to zero. '''

|Assume '''epsilon is equal to zero point eight nine seven. '''

|We have to find the solution within the '''time range two less than t less than ten''' and then '''plot'''  the solution.  

|Let us look at the code for '''Van der Pol equation. '''

| Switch to '''Scilab editor''' and open '''Vander pol dot sci.'''

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

|Since the '''inital time value''' is given as '''two''', we start the time range at two.  

| Then we define the '''function Vander pol''' and construct a system of '''first order ODEs.'''  

|We substitute the value of '''epsilon''' with '''zero point eight nine seven. '''

|Here '''y''' refers to the '''voltage v.'''  

| Then we call '''ode''' function and solve the system of '''equations.'''

| Finally we plot '''y''' and '''y dash versus t.'''

| Save and execute the file '''Vander pol dot sci.'''  

|The '''plot''' showing '''voltage versus time''' is shown.

|Let's move onto '''Lorenz system of equations.'''

|The '''Lorenz system of equations''' is given by: 

|'''x two dash equal to one plus r minus x three into x one minus x two''' and

|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.'''  

|Let '''sigma''' be equal to '''ten,  r''' be equal to '''twenty eight''' and '''b''' equal to '''eight by three.'''

|Switch to '''Scilab editor''' and open '''Lorenz dot sci'''.

|We start by defining the initial conditions of the '''ODEs.'''  

|Since there are three different '''ODEs,''' there are three initial conditions.  

|Then we define the '''inital time''' condition and next the '''time range.'''

|We define the '''function Lorenz''' and then define the given constants '''sigma, r''' and '''b.'''

|Then we define the '''first order ODEs.'''

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

|We equate the solution to '''x.'''  

|Then we '''plot x one, x two''' and '''x three versus time.'''

|Save and execute the file '''Lorenz dot sci.'''  

|The '''plot''' of '''x one, x two''' and '''x three versus time''' is shown.  

|Let us summarize this tutorial.  

|In this tutorial we have learnt to develop '''Scilab code''' to solve an '''ODE''' using '''Scilab ode function'''.  

|Then we have learnt to '''plot''' the solution.  

| Watch the video available at the  link shown below.

| It summarizes the Spoken Tutorial project.  

||If you do not have good bandwidth, you can download and watch it.

||The spoken tutorial project Team:

||Conducts workshops using spoken tutorials.  

||Gives certificates to those who pass an online test.

||For more details, please write to contact@spoken-tutorial.org.  

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

| It is supported by the National Mission on Eduction through ICT, MHRD, Government of India.  

|More information on this mission is available at the link shown below.

|This is Ashwini Patil, signing off.

| Thank you for joining.