臺灣大學數學系

 八十七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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20 points each.

1. Given two continous functions $f_1$, $f_2$, solve the following hyperbolic linear system:
   
   $$u(x, t)={{u_1(x, t)} \choose {u_2(x, t)}}$$
   
   $\hbox{(i)} \cases{u_t+{\pmatrix{1&0\cr 0&-1\cr}}u_x={1\choose 2} \qquad t>0,\,x \in \mathbb{R}\cr \noalign{\vskip 3pt} u(x,0)={{f_1(x)}\choose{f_2(x)}}; \cr}$ $\hbox{(ii)} \cases{u_t+{\pmatrix{1&0\cr 0&-1\cr}}u_x={1\choose 2} \qquad t>0,\... ...} u_1(0, t)=u_2(0, t)+1 \quad \qquad t>0.\hbox{(This is a mixed problem.)}\cr}$
2. (i) Show that the function $G(x, y)$ defined by
   
   $$G(x, y)=\cases{1 &for $x >a,\,y>b$ \cr \noalign{\vskip 3pt} 0 &for all other $x,y$ \cr}$$
   
   is a fundamental solution with the pole $(a, b)$ of the operator $L=\partial^2\,/\,\partial x \partial y$ in the $x-y$ plane. (i.e. $LG=\delta_a \delta_b$, δ is the Dirac function.)
   (ii) Solve $\cases{\partial^2\,/\,\partial x \partial y = F(x, y) \cr \noalign{\vskip 3pt}... ..., x)=0 (\hbox{Here initial datas are given on the line} \{ (x, y) :x=y \}.\cr}$
   (iii) Using (ii), solve the following non-homogeneous equations:
   
   $$\cases{u_{tt}-u_{xx}=f(x, t) \qquad u=u(x, t) \cr \noalign{\... ...uad(\hbox{Here}\, f,g \,\,\hbox{are continous functions.})\cr}$$
   
   
3. Let $\Omega=\{ (x, y): x>0, y\in \mathbb{R}\}$. Solve the following Dirlichlet problem by constructing a Green's function:
   $\cases{u_{xx}+u_{yy}=0 \,\,\hbox{on} \,\,\Omega \cr \noalign{\vskip 3pt} u(0, y)=f(y), f\,\, \hbox{is bounded and continous}. \cr}$
4. Let Ω be an open set in $C$. Assume $f$ is a conformal map from Ω to $\Omega^{\prime}$. (i.e. $f$ is analytic in Ω. $f^{\prime}(z) \ne 0$ for all comples number $z$ in Ω.) Let $f(x+iy)=u(x,y)+iv(x,y)$ and $U$ is harmonic in $\Omega^{\prime}$. Let $V(x, y)=U(u(x,y), v(x,y))$. Prove that $V$ is harmonic in Ω. i.e. $(\partial\,/\,\partial x)^2 V+(\partial \,/\, \partial y)^2V=0$ on Ω.
   
   
5. (i) Find solutions of $u(x,t)$ of the one-dimentional heat equation $u_t=u_{xx}$ of the form $u=(1\,/\,{\sqrt t})f(x\,/\, 2 {\sqrt t})$.($f$ has to satisfy a linear second-order O.D.E..)
   (ii) Find solutions of $u(x,t)$ of the one-dimensional Burgers' equation $u_t+uu_x=u_{xx}$ of the form $u(x,t)=(1\,/\,\sqrt{t}) f(x\,/\,\sqrt{t})$.

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