Exam
UNIVERSITY OF UTAH DEPARTMENT OF MATHEMATICS Ph.D. Preliminary Examination in Differential Equations January 4, 2013.
Spring 2013 · 11 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
-
Question 1
Let \(f(x)\) be continuously differentiable for all \(x\in \mathbb {R}^n\) and let \(x_0\in \mathbb {R}^n\) be a point. Show that there is and \(\epsilon >0\) and a function \(y\in C^1([0,\epsilon ],\mathbb {R}^n)\) that satisfies the initial value problem \[ \begin {cases} \dfrac {dx}{dt}=f(x),\\ x(0)=x_0. \end {cases} \]
-
Question 2
Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda _i\) satisfy \(\Re \lambda _i<0\) for all \(i=1,\ldots ,n\). Answer the questions about the linear system \begin{equation} \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \tag {1} \end{equation}
- Define: \(V(x)\) is a Liapunov Function.
- Find a Liapunov Function.
- Define: \(z(t)=0\) is an Asymptotically Stable solution for (1).
- Using your Liapunov Function, show that \(z(t)=0\) is an asymptotically stable solution.
-
Question 3
Show that there is a solution that goes to \(\infty \) as \(t\to \infty \). \[ y''-\cos ^2(t)y'+\sin ^2(t)y=0. \]
-
Question 4
Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda _i\) satisfy \(\Re \lambda _i<0\) for all \(i=1,\ldots ,n\). Let \(g(x)\in C^1(\mathbb {R}^n,\mathbb {R}^n)\) such that \(g(x)=O(\norm {x}^{\beta })\) for some \(\beta >1\) as \(x\to 0\). Consider the initial value problem for \(x_0\in \mathbb {R}^n\), \begin{equation} \begin {cases} \dfrac {dx}{dt}=Ax+g(x),\\ x(0)=x_0. \end {cases} \tag {2} \end{equation} Show that if \(\norm {x_0}\) is small enough, there is a solution to (2) which exists for all time.
[Hint: Suppose \(g(t)\) and \(u(t)\) are nonnegative functions and \(c_0\ge 0\) is a constant that satisfy \(u(t)\le c_0+\int _0^t g(s)u(s)\dd s\) for all \(t\ge 0\). Then Gronwall’s Inequality implies \[ u(t)\le c_0\exp \left (\int _0^t g(s)\dd s\right ) \] for all \(t\ge 0\).]
-
Question 5
Show that the planar system \[ \begin {aligned} \dot {x} &= x(x^2+y^2-3x-1)-y,\\ \dot {y} &= y(x^2+y^2-3x-1)+x \end {aligned} \] has at least one nonconstant periodic solution.
-
Question 6
- State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
- Using an approximate center manifold, determine the nature of the bifurcation at the origin when \(\epsilon =0\). \[ \begin {aligned} \dot {x} &= \epsilon x-x^3+y^2,\\ \dot {y} &= -y+x^2. \end {aligned} \]
-
Question 7
Suppose that \(\rho (x,t)\) is the number density of cars per unit length along a road, \(x\) being the distance along the road such that \[ \frac {\partial \rho }{\partial t} +\frac {\partial [\rho (1-\rho )]}{\partial x}=0. \]
- Show that \(\rho \) is constant along characteristics \[ \frac {dx}{dt}=1-2\rho , \] and derive the following Rankine-Hugoniot conditions for the speed of a shock \(x=S(t)\): \[ \frac {dS}{dt}=\frac {[\rho (1-\rho )]_-^+}{[\rho ]_-^+}. \]
- A queue is building up at traffic light \(x=1\) so that, when the light turns to green at \(t=0\), \[ \rho (x,0)= \begin {cases} 0, & \text {if }x<0\text { and }x>1;\\ x, & \text {if }0<x<1. \end {cases} \] Solve the characteristic equations, and sketch the resulting characteristic curves. Deduce that a collision first occurs at \(x=1/2\) when \(t=1/2\), and that thereafter there is a shock such that \[ \frac {dS}{dt}=\frac {S+t+1}{2t}. \]
-
Question 8
Let \(D\in \mathbb {R}^n\) be a bounded, connected domain with smooth boundary. Let \(\phi \) be an eigen-funcion of the Laplacian on \(D\) with Dirichlet boundary conditions (\(\phi =0\) on \(\partial D\)) and with eigenvalue \(\lambda \). \begin{equation} \Delta \phi +\lambda \phi =0. \tag {3} \end{equation}
- Show that there is a finite constant \(C\) so that for every nonzero continuously differentiable \(u\in C^1(D)\) such that \(u=0\) on \(\partial D\) satisfies \[ \int _D u^2 \le C\int _D \norm {Du}^2. \]
- By integrating eigenfunctions of (3), conclude that eigenvalues are positive \(\lambda >0\).
- Denote by \[ \lambda _1(D)=\inf _{u\in C_0^1(D),\ u\not \equiv 0} \frac {\int _D \norm {Du}^2}{\int _D u^2} \] the least value of the Rayleigh quotient (which is at least \(1/C\)). Assuming that \(\lambda _1(D)\) is an eigenvalue of \(D\) with Dirichlet boundary conditions, show that if two such domains satisfy \(D_1\subset D_2\), then \(\lambda _1(D_1)\ge \lambda _1(D_2)\).
-
Question 9
Consider the wave equation \[ u_{tt}-c^2u_{xx}=0,\qquad t>0,\ x\in \mathbb {R}. \] along with initial conditions \[ u(x,0)=g(x),\qquad u_t(x,0)=h(x). \]
- Assuming \(c\) is constant, derive d’Alembert’s Formula \[ u(x,t)=\frac 12[g(x+ct)+g(x-ct)] +\frac {1}{2c}\int _{x-ct}^{x+ct}h(\xi )\dd \xi . \]
- Briefly explain the concepts of domain of dependence, and finite propogation speed, and prove that these properties follow d’Alembert’s Formula.
- Suppose that we restrict the spatial domain to \(x>0\) and impose the boundary condition \(u(0,t)=0\) for \(t\ge 0\). If the initial conditions satisfy \(g(0)=h(0)=0\), using d’Alembert’s Formula as a starting point, find a closed form solution for \(u(x,t)\) with \(x,t>0\).
-
Question 10
Let \(D\in \mathbb {R}^d\) be a bounded domain with smooth boundary \(\partial D\). Assume that that a complete orthonormnal set of eigenfunctions \(\{\phi _n\}_{n=1}^\infty \) for the Laplacian on \(D\) with Dirichlet boundary conditions are known, in other words \begin{equation} -\Delta \phi _n=\lambda _n\phi _n,\qquad \text {in }D,\ n=1,2,3\ldots , \tag {4} \end{equation} with \(\phi _n(x)=0\) for \(x\in \partial D\), and \(\int _D\phi _n^2\dd x=1\).
- Given \(f\in L^2(D)\), find a closed form series solution \(u(x,t)\) for the initial, boundary value problem for the heat equation \[ \begin {aligned} -u_t+\Delta u &= 0, && t>0,\ x\in D,\\ u(x,t) &= 0, && x\in \partial D,\\ u(x,0) &= 0, && x\in D. \end {aligned} \]
- Show that there is a constant \(c\) such that \[ \left |\int _D u(x,t)\phi _n(x)\dd x\right |\le c e^{-\lambda _n t} \] for all \(n\). Find \(\lim _{t\to \infty }u(x,t)\).
- Give an example (the case \(d=1\) will suffice) which shows that solutions of the heat equation do not exhibit finite propagation speed.
-
Question 11
Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain \(\Omega \subset \mathbb {R}^2\), so \(\Delta u=0\) in \(\Omega \).
- Prove the mean value property: if \(y=(y_1,y_2)\in \Omega \) and \(r>0\) is chosen such that \(B_r(y)\subset \Omega \) (ball of radius \(r\) centered at \(y\)) then \[ u(y_1,y_2)=\frac {1}{2\pi }\int _0^{2\pi }u(y_1+r\cos \theta ,y_2+r\sin \theta )\dd \theta . \]
- Assuming \(\Omega \) is connected, prove that \(u\) can attain its maximum value at an interior point \(x\in \Omega \), only if \(u\) is constant.