臺灣大學數學系

 八十八學年度第二學期碩博士班資格考試試題

微分方程式

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Choose $4$ out of the following $7$ problems

1. Consider the one-dimensional wave equation $u_{tt} = c^{2} u_{xx}$ in the quadrant $x > 0$, $t> 0$, for which
   $$\begin{eqnarray*} u = f(x), \quad u_{t} = g(x) \quad & & \mbox{for} \quad t=0, ... ..._{t} = \alpha u_{x} \quad & & \mbox{for} \quad x=0, \quad t> 0, \end{eqnarray*}$$
   
   where $f$ and $g$ are of class $C^{2}$ for $x > 0$ and vanish near $x=0$.
   1. Let α be a constant $\neq c$. Find the solution $u(x,t)$.
   2. Show that generally no solution exists when $\alpha = -c$.
2. Let $u(x,y)$ be a solution of a quasi-linear equation of the form
   
   $$a(u_{x},u_{y}) u_{xx} + 2 b(u_{x},u_{y}) u_{xy} + c(u_{x},u_{y}) u_{yy} = 0.$$
   
   Introduce new independent variables ξ, η and a new unknown function $\phi$ by
   
   $$\xi = u_{x}(x,y), \qquad \eta = u_{y}(x,y), \qquad \phi = xu_{x}+yu_{y}-u.$$
   
   
   1. Prove that $\phi$ as a function of ξ and η satisfies $x= \phi_{\xi}$, $y= \phi_{\eta}$ and the linear differential equation
      
      $$a(\xi,\eta) \phi_{\eta \eta} - 2 b(\xi,\eta) \phi_{\xi \eta} + c(\xi,\eta) \phi_{\xi \xi} = 0.$$
      
      
   2. [(b)] As an application of the technique discussed in (a), find the general solution of the equation
      
      $$u_{x}^{2} + u_{y} x = 0.$$
      
      
3. Recall that the fundamental solution of Laplace's equation in space is
   
   $$K(X, \xi) = - \frac{1}{4 \pi \vert X-\xi\vert}.$$
   
   The Green's function $G(X,\xi)$ for the Dirichlet problem in a bounded domain $\Omega \subseteq \mathbb{R}^{3}$ has the form
   
   $$G(X, \xi) = K(X, \xi) + h(X,\xi),$$
   
   where
   
   $$\Delta h = 0 \quad \mbox{in} \quad \Omega, \qquad G(X,\xi) = 0 \quad \mbox{for} \quad X \in \partial\Omega.$$
   
   Show that $G(X,\xi) = G(\xi, X)$. That is, $G$ is a symmetric function of the two variables $X$ and ξ.
4. Solve the Poisson equation in two space dimensions
   
   $$u_{xx} + u_{yy} = \phi(x,y)$$
   
   in the quarter plane with the Neumann boundary conditions
   
   $$u_{y}(x,0) = f(x), \quad u_{x}(0,y) = g(y).$$
   
   Give comments to $f$, $g$, and $\phi$ that should be satisfied in order for the solution of the problem to exist.
5. Show that if $u(X,t)$ of the form
   
   $$u(X,t) = \left\{\begin{array}{ll} \frac{1}{2} \; v(X,t) \; ... ...& \quad \mbox{for} \quad \gamma(X) \geq t, \end{array} \right.$$
   solves the wave equation
   
   $$u_{tt} - c^{2} \Delta u = 0$$
   in space, $X \in \mathbb{R}^{3}$, then the surface $S = \{ t = \gamma(X) \}$ must be characteristic. That is, it satisfies the eikonal equation
   
   $$\vert\nabla \gamma\vert = 1/c.$$
   Here $v(X,t)$ is a $C^{2}$ function nonzero on the surface, and satisfies the transport equation
   
   $$v_{t} + c^{2} \nabla\gamma \cdot \nabla v = -\frac{1}{2} c^{2} \left (\Delta \gamma\right ) v.$$
   
   In addition, show that $u(X,t)$ in this case is only a $C^{1}$ function.
6. Let $u(x,t)$ be a positive solution of class $C^{2}$ of
   
   $$u_{t} = \mu u_{xx} \quad \mbox{for} \quad t>0.$$
   
   
   1. Show that $\theta = -2 \mu u_{x}/u$ satisfies Burgers' equation
      
      $$\theta_{t} + \theta \theta_{x} = \mu \theta_{xx} \quad \mbox{for} \quad t>0.$$
      
      
   2. For $f \in C_{0}^{2}\left (\mathbb{R}\right )$, find a solution of the Burgers' equation with initial values $\theta(x,0) = f(x)$.
7. Suppose that $u(X,t)$ is a smooth solution of the heat equation in space $u_{t} = \mu \Delta u$ for $X \in \Omega$, $t \geq 0$, where Ω is a bounded region. Assume that Neumann boundary condition $u_{N} = 0$ on $\partial \Omega$, where $N$ is the unit-outward normal to $\partial \Omega$. Show that
   
   $$\frac{d}{dt} \left [ \int_{\Omega} u(X,t) dX \right ] = 0.$$
   
   That is, $\int_{\Omega} u(X,t) dX$ is constant in time.

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