\\title{Sample Questions from Past Qualifying Exams}<\/p>\n
This list may give the impression that the exams consist of a series of questions fired at the student one after another. In fact most exams have more the character of a conversation with considerable give and take. Hence this list cannot be expected to indicate accurately the difficulties involved.<\/p>\n
The list indicates the professor associated to each question where available. Some have been in the MGSA files for a while, and this information has been lost (if it was ever there).<\/p>\n
The listing by section is approximate, since some questions may fit under more than one heading.
\n
\n\\section{Partial Differential Equations}
\n\\begin{enumerate}
\n\\item Let $\\Omega\\subset \\R^n$ be an open, bounded, smooth domain. Let $f\\colon \\partial\\Omega\\to \\R$ be continuous. Is there a solution to $\\Delta u=0$ in $\\Omega$, $u=f$ on $\\partial\\Omega$? If there is a solution, explain how it is found.
\n\\item Let $\\{v_n\\}$ be a sequence of harmonic functions in $\\Omega$. Assume $v_n\\to v$ uniformly in norm and that $v_n(x)\\nearrow v(x)\\forall x\\in \\Omega$. Show that $v$ is harmonic in $\\Omega$.
\n\\item Consider $u_{tt}=u_{xx}+u_{yy}$ with $u(x,y,0)=0$ in $x^2+y^2< 1$ and $u_y(x,y,0)=0$ everywhere. What is $u(0,0,{1\\over 2})$?\n\\item Consider $(\\ast )\\ u_t=u_{xx}$, $u(x,0)=f(x)$. Do you know a solution? Suppose you don't know the fundamental solution of the heat equation, how would you derive a solution of $(\\ast)$? What conditions would you impose on $f(x)$ for uniqueness?\n\\item For one dimensional Laplace's, heat and wave equations give initial and\/or boundary conditions that allow you to find solutions.\n\\item State the mean value property for harmonic functions and explain how you prove it.\n\\item Consider the inviscid Burgers' equation. Assume there is a curve across which the solution is discontinuous. State the Rankine-Hugoniot condition. State the entropy condition analytically.\n\\item Follow the steps below to give a heuristic derivation of the entropy condiiont (that a shoc will occur if $u_r< u_l$):\n \\begin{enumerate}\n \\item Consider $(\\ast)\\ v_t+vv_x-\\epsilon v_{xx}=0$. Let $w=v_x$ and differentiate $(\\ast)$ with respect to $x$. write the result in terms of $w$.\n \\item Use $w^2\\ge 0$ and the maximum principle for subsolutions of the heat equation to conclude that $w$ is bounded for all $x$ and $t > 0$.
\n \\item Thus $v_x\\le M$. Explain how this implies that if $u_r > u_l$ you cannot get a shock.
\n \\end{enumerate}
\n\\end{enumerate} <\/p>\n","protected":false},"excerpt":{"rendered":"
\\title{Sample Questions from Past Qualif …<\/p>\n