* Solving ODEs

*** Outline

***** Introduction

******* What are we going to do?

******* How are we going to do?

******* Arsenal Required

********* working knowledge of arrays

********* working knowledge of functions

*** Script

    Welcome. 

    In this tutorial we shall look at solving Ordinary Differential Equations

    using odeints in Python.

    Let's consider a classical problem of the spread of epidemic in a

    population.

    This is given by dy/dt = ky(L-y) where L is the total population.

    For our problem Let us use L=25000, k=0.00003.

    Let the boundary condition be y(0)=250.

    First of all run the magic command to import odeint to our program.

    In []: from scipy.integrate import odeint

    For now just remember this as a command that does some magic to obtain

    the function odeint in to our program.

    We will come back to the details of this command in subsequent sessions.

    We can represent the given ODE as a Python function.

    This function takes the dependent variable y and the independent variable t

    as arguments and returns the ODE.

    Our function looks like this:

    (Showing the slide should be sufficient)

    In []: def epid(y, t):

      ....     k = 0.00003

      ....     L = 25000

      ....     return k*y*(L-y)

    Independent variable t can have be assigned the values in the interval of

    0 and 12 with 61 points using linspace:

    In []: t = linspace(0, 12, 61)

    Now obtaining the odeint of the ode we have already defined is as simple as

    calling the Python's odeint function which we imported:

    In []: y = odeint(epid, 250, t)

    We can plot the the values of y against t to get a graphical picture our ODE.

    Let us move on to solving a system of two ordinary differential equations.

    Here we shall take the example ODEs of a simple pendulum.

    The equations can be written as a system of two first order ODEs

    d(theta)/dt = omega

    and

    d(omega)/dt = - g/L sin(theta)

    Let us define the boundary conditions as: at t = 0, 

    theta = theta 0 (10 degrees) and omega = 0

    Let us first define our system of equations as a Python function, pend_int.

    As in the earlier case of single ODE we shall use odeint function of Python

    to solve this system of equations by passing pend_int to odeint.

    pend_int is defined as shown:

    In []: def pend_int(initial, t):

      ....     theta = initial[0]

      ....     omega = initial[1]

      ....     g = 9.81

      ....     L = 0.2

      ....     f=[omega, -(g/L)*sin(theta)]

      ....     return f

      ....

    It takes two arguments. The first argument is a 2-tuple containing the two

    dependent variables in the system, theta and omega.

    The second argument is the independent variable t.

    In the function we assign theta and omega to first and second values of the

    initial argument respectively.

    Acceleration due to gravity, as we know is 9.8 meter per second sqaure.

    Let the length of the the pendulum be 0.2 meter.

    We create a list, f, of two equations which corresponds to our two ODEs,

    that is d(theta)/dt = omega and d(omega)/dt = - g/L sin(theta).

    We return this list of equations f.

    Now we can create a set of values for our time variable t over which we need

    to integrate our system of ODEs. Let us say,

    In []: t = linspace(0, 20, 101)

    We shall assign the boundary conditions to the variable initial.

    In []: initial = [10*2*pi/360, 0]

    Now solving this system is just a matter of calling the odeint function with

    the correct arguments.

    So first let us import odeint function into our program using the magic

    import command

    In []: from scipy.integrate import odeint

    We can call ode_int as:

    In []: pend_sol = odeint(pend_int, initial,t)

    Plotting theta against t and omega against t we obtain the plots as shown

    in the slide.

    Thus we come to the end of this session on solving ordinary differential

    equations in Python. Thanks for listening to this tutorial.

*** Notes