臺灣大學數學系

 八十六學年度第一學期碩博士班資格考試試題

微分方程式

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1. Solve the following PDEs.
   1. $yu_x + zu_y + u_z = zu^3, u(x,y,0) = e^{x-y}.$
   2. $u_y + u_x^2 + u = 0, u(x,0) = x.$
2. Let $\phi(x)=[\sin(2\pi x)]^4$. Assume $u$ satisfies
   
   $$\begin{array}{l} u_{tt} - u_{xx} = 0 \hbox{ for }-1<x<1,0<t,... ...x,0)=0 \hbox{ for }x\in [-1,0]\cup[{1 \over 2},1]. \end{array}$$
   
   Find the functions $u(x,2)$ and $u(x,4)$.
3. (Liouville's theorem) Prove that a harmonic function defined and bounded in all of $\mathbb{R}^n$ is a constant.
4. Let $\Omega=\{(x,t):-1<x<1, t>0\}$ and let $u\in C^2(\Omega)\cap C^0(\bar\Omega)$ on Ω satisfy
   
   $$\begin{array}{l} u_t = u_xx \hbox{ on } \Omega\\ u(-1,t)=u... ...or }t>0,\\ u(x,0)=f(x) \hbox{ for }-1\le x\le 1. \end{array}$$
   
   Assume $f(-1)=0$, $f(-x)=f(x)$ and $0\le f(x)\le 1$.
   1. Show that $0\le u(x,t)\le 1$ on Ω.
   2. Show that $u(x,t)=u(-x,t)$ on Ω.
   3. Prove that the energy $E(t)=\int_{-1}^{1}u^2(x,t)\,dx$ is decreasing in $t$ for $t>0$.
5. Let Ω denote an open bounded set in $n$-dimensional $x$-space described by an inequality $\phi(x)>0$, so that $\phi(x)=0$ on $\partial\Omega$. Let $S_{\lambda}$ for $\lambda>0$ denote the hyper-surface in $xt$-space given by $t=\lambda \phi(x)$ for $x \in \Omega$. On $S_\lambda$ define
   
   $$E(\lambda)=\int_{S_{\lambda}} Q_{\lambda}\,dx,$$
   
   where
   
   $$Q_{\lambda}={1 \over 2}\big( u_t^2+c^2\sum_i u_{x_i}^2 \big) + \lambda c^2u_t\sum_i u_{x_i}\phi_{x_i}.$$
   
   
   1. Prove $E(\lambda)=$ constant when $u_{tt}-c^2u_{xx}=0$.
   2. Show that $Q_{\lambda}$ as a quadratic form in $u_t,u_{x_1},...,u_{x_n}$ is positive definite, when $S_{\lambda}$ is spacelike.
   3. Show that the initial data on $S_0$ of a solution of $u_{tt}-c^2u_{xx}=0$ uniguely determine $u$ on all $S_{\lambda}$ with sufficiently small λ.
6. Let $f(\theta)$ be a $C^4$ function of period $2\pi$ with Fourier series
   
   $$f(\theta)=\sum_{n=0}^{\infty}[a_n\cos(n\theta)+b_n\sin(n\theta)].$$
   
   
   1. Prove that
      
      $$u=\sum_{n=0}^{\infty}[a_n\cos(n\theta)+b_n\sin(n\theta)]r^n$$
      
      represents in polar coordinates $r,\theta$ the solution of the Laplace equation $\triangle u=0$ in the disk $x^2+y^2<1$ with boundary values $f$.
   2. Derive Poisson's integral formula
      
      $$u(r,\theta)= {{1-r^2}\over{2\pi}}\int_0^{2\pi}{{f(\alpha)} \over{1+r^2-2r\cos(\theta-\alpha)}}\,d\alpha$$
      
      from (a) by substituting for the $a_n,b_n$ their Fourier expressions in terms of $f$ and interchanging summation and integration.

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