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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Winter 2007 • Question 9

Question

Math 6410 • ODE Winter 2007 • Question 9

Denote \(U=\{x\in \mathbb {R}^2:x_1>0\}\), and \(\Gamma =\{x\in \mathbb {R}^2:x_1=1\}\). Use the method of characteristics to solve the first-order problem \[ \begin {aligned} x_1u_{x_1}+uu_{x_2}-x_2 &= 0, && \text {in } U,\\ u(1,x_2) &= 2x_2, && \text {on } \Gamma . \end {aligned} \] (Hint: to solve the characteristic equations, note that the linear combinations \(x_2+z\) and \(x_2-z\) decouple.) Determine the region around \(\Gamma \) upon which the solution is well-defined.

Raw LaTeX

Denote \(U=\{x\in\R^2:x_1>0\}\), and \(\Gamma=\{x\in\R^2:x_1=1\}\). Use the
method of characteristics to solve the first-order problem
\[
\begin{aligned}
  x_1u_{x_1}+uu_{x_2}-x_2 &= 0, && \text{in } U,\\
  u(1,x_2) &= 2x_2, && \text{on } \Gamma.
\end{aligned}
\]
\textit{(Hint: to solve the characteristic equations, note that the linear combinations
\(x_2+z\) and \(x_2-z\) decouple.)} Determine the region around \(\Gamma\) upon which the
solution is well-defined.

Topic Tags

Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Method of Characteristics Poincare Maps and Periodic Solutions