.. _homework1:

Homework 1 (Due 6 pm, Friday, Apr 8, 2016)
##########################################################################

Please submit your homework to your git repo by
**6 pm, Fri, Apr 8 2016**.


1. Setup an account from Bitbucket and clone the course repo from the
   remote Bitbucket repository to your local machine (see the sections
   in :ref:`ch01-version-control-systems`).
   
   (a) Generate and edit a file named ``bio.txt`` which includes the
       following information:
     
     * name, your major, email
     * shool year
     * research interests
     * advisor's name (if any)
     * experience in ODEs and PDEs: (1. a lot, 2. somewhat, 3. none)
     * experience in numerical/computational ODEs and PDEs: (1. a lot, 2. somewhat, 3. none)
     * experience with MATLAB (1. a lot, 2. somewhat, 3. none)
     * do you have MATLAB installed on your local machine? (yes/no)
     * if the previous answer is no, do you have an access to a machine that has MATLAB installed? (yes/no)
       
   (b) Check-in (or make a commit) ``bio.txt`` with a comment:

       "my first check in to my own repo on mm/dd/yy as part of homework 1".

   (c) After a successful check-in of ``bio.txt``, modify it by adding
       a new line on your OS and computer:

     * types of OS and machine for the class

   (d) Check-in the updated ``bio.txt`` to the repo again.




       
2. Modify the example MATLAB code in :ref:`EulerMethod` to solve the IVP using the forward Euler's method:

      .. math::
	 :nowrap:
      
	 \begin{equation}
	   \left \{
	   \begin{array}{lll}
	   && \frac{du}{dt} = \sqrt{u(t)} \nonumber \\
	   && u(1) = 0.25.
	   \end{array}
	   \right.
	   \end{equation}

   Note that this has a unique exact solution :math:`u_{exact}(t) = \frac{t^2}{4}` over :math:`[1,\infty]`.
   Please evolve your numerical solution until the maximum time :math:`t_{max} = 5` is reached using
   four different time step sizes, :math:`\Delta t = 1, 0.5, 0.25,
   0.125`. Your grid discretization of :math:`t^n` over
   :math:`[1,t_{max}]` will be such that :math:`t^0 = 1`, and
   :math:`t^n = t^0 + n\Delta t, n=0, \dots, N`.
   
   (a) Plot each of the four cases over :math:`[1,t_{max}] = [1, 5]` along with the exact solution.

   (b) Measure the :math:`L_1` error :math:`||E^N||_1` defined by

       .. math::
	  
	  ||E^N||_1 = \Delta t \sum_{n=1}^{N} | u_{exact}(t^n) - U^n|,

       where :math:`N` is given by :math:`\Delta t = \frac{5 - 1}{N}`.
       Plot the error convergence rates by plotting :math:`||E^n||_1`
       versus :math:`N` in log-log scales (use MATLAB's ``loglog`` command).