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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2011 • Question 9

Question

Math 6410 • ODE Fall 2011 • Question 9

Compute the unique entropy solution of \[ \begin {aligned} u_t+\left (\frac {u^2}{2}\right )_x &= 0 && \text {in }\mathbb {R}\times (0,\infty ),\\ u &= g && \text {on }\mathbb {R}\times \{0\}, \end {aligned} \] where \[ g(x)= \begin {cases} 0, & \text {if }x<0,\\ x, & \text {if }0\le x\le 1,\\ 0, & \text {if }x>1. \end {cases} \] Draw a sketch documenting your answer, illustrating what happens at all times \(t>0\).

Raw LaTeX

Compute the unique entropy solution of
\[
\begin{aligned}
  u_t+\left(\frac{u^2}{2}\right)_x &= 0 && \text{in }\R\times(0,\infty),\\
  u &= g && \text{on }\R\times\{0\},
\end{aligned}
\]
where
\[
  g(x)=
  \begin{cases}
    0, & \text{if }x<0,\\
    x, & \text{if }0\le x\le1,\\
    0, & \text{if }x>1.
  \end{cases}
\]
Draw a sketch documenting your answer, illustrating what happens at all times \(t>0\).

Topic Tags

Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Fundamental Solutions Poincare Maps and Periodic Solutions