.. _homework2:

Homework 2 (Due 6 pm, Wed, 2/15/2017)
##########################################################################

Please submit your homework to your git repo by
**6 pm, Wednesday, 2/15/2017**.

* :download:`Homework 2 <./hw2.pdf>`


Hints for Problem 4 in Homework 2
===========================================

Using the definition of weak solutions, we want to show:

.. math::
   \int_0^\infty \int_{-\infty}^{\infty} [\phi_t u + \phi_x f(u)] dxdt
   = -\int_{-\infty}^{\infty} \phi(x,0) u(x,0) dx.



To show this, prove the following claims in steps:

**Claim 1 - Show the first term in the definition is the right hand side:**

 .. math::
    :nowrap:

       \begin{equation}
       \int_0^\infty \int_{-\infty}^{\infty} \phi_t u dx dt =
       \int_{-\infty}^{\infty} \phi(x,0) u(x,0) dx
       \nonumber
       \end{equation}

**Claim 1a - To prove Claim 1, you need to first prove:**

.. math::
   :nowrap:

      \begin{equation}
      \int_0^\infty \int_{-\infty}^{\infty} u_t \phi dx dt = 0
      \nonumber
      \end{equation}

**Claim 1b - To complete Claim 1, observe that:**

.. math::
   :nowrap:

      \begin{equation}
      \int_{-\infty}^{\infty} \phi(x,0) u(x,0) dx = -
      \int_{-\infty}^{t/2} \phi(x,0)dx
      \nonumber
      \end{equation}

**Claim 2 - Show the second term in the definition is zero:**

.. math::
   :nowrap:

      \begin{equation}
      \int_0^\infty \int_{-\infty}^{\infty} \phi_x f(u) dx dt = 0
      \nonumber
      \end{equation}

**Claim 2a - To prove Claim 2, you need to prove also:**

.. math::
   :nowrap:

      \begin{equation}
      0 = \int_0^\infty \int_{-\infty}^{\infty} \phi (u_t + f_x) dx dt =
      -\int_0^\infty \int_{-\infty}^{\infty} \phi_x f dx dt.
      \nonumber
      \end{equation}



In the above, split the spatial integral into two parts and use the
definition of :math:`u(x,t)`:

.. math::
   \int_0^{\infty}\int_{-\infty}^{\infty} u(x,t) dx dt
   = \int_0^{\infty}\int_{-\infty}^{t/2} u(x,t) dx dt +
   \int_0^{\infty}\int_{t/2}^{\infty} u(x,t) dx dt
   = \int_0^{\infty}\int_{-\infty}^{t/2} u(x,t) dx dt


and also use the fact that :math:`\phi (x,t) \in C^1_0(\mathbb{R}\times \mathbb{R}^+)`.