臺灣大學數學系

八十七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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We assume the given functions are all $C^\infty$ if not stated in other way.

1. ($5\%$) Solve $\cases{ u_t + cu_x = 0, \ \ u = u(t,x), \ \hbox{is constant,}\cr \noalign{\vskip 6pt} u(0,x) = f(x).}$
2. ($10\%$) Solve $\cases{ u_t + uu_x = 0, \ \ u = u(t,x),\cr \noalign{\vskip 6pt} u(0,x) = h(x),\cr}$
   show that $u_x$ will become infinite in finite time $t$.
3. ($10\%$) (i) Solve $\cases{ {{\partial^2}\over{\partial\xi\partial\eta}}u(\xi,\eta)= F(\xi,\eta),\... ...eta)= 0,\ u_\eta(\eta,\eta) = 0, \ \hbox{for all}\ \ \eta \in \mathbb{R}. \cr}$
   (ii) Use (i), solve $\cases{ u_{tt} - u_{xx} = G(t,x),\ u = u(t,x)\cr \noalign{\vskip 6pt} u(0,x) = 0 \cr \noalign{\vskip 6pt}u_t(0,x) = 0, \cr}$
   (iii) Solve $\cases{ u_{tt} - u_{xx} = G(t,x),\ u = u(t,x)\cr \noalign{\vskip 6pt} u(0,x) = f(x) \cr \noalign{\vskip 6pt} u_t(0,x) = g(x). \cr}$
   Hint: use linear property.
4. ($10\%$) Find by power series expansion with respect to $y$ the solution of the initial-value problem:
   
   $$\cases{ u_{yy} = u_{xx} + u,\ u = u(x,y)\cr \noalign{\vskip 6pt} u(x,0) = e^x,\ \ u_y(0,x) = 0. \cr}$$
   
   
5. ($10\%$) Let $f$ be a scalar analytic function of $z\in {\mathbb{C}}$. Put $z = x+iy$ with $x,y \in \mathbb{R}, \ f(z) = u(x, y) + iv(x, y)$. [You need to choose a definition of analytic property.]
   1. Show that $u,v$ satisfy the system of Cauchy-Riemann equations: $u_x = v_y, \ u_y = -v_x.$
   2. Show that $u(x,y), v(x,y)$ are real analytic in $(x, y)$ variables.
   (Hint: Use $\vert a\vert+\vert b\vert \le \sqrt 2\vert a+ib\vert$ for $a,b \in \mathbb{R}$)
6. ($15\%$) Let $L$ be a differential operator operator from $C^\infty(\mathbb{R},\mathbb{R})$ to itself, the formal adjoint $\widetilde L$ is defined as $\int\int v(Lu)dxdy = \int\int(\widetilde Lv)u dxdy$, for all $u,v \in C^\infty$. Compute the formal adjoint of the following cases:
   1. $L[u] = ({\partial \over {\partial y}} - a(x,y){\partial \over {\partial x}})[u] = u_y + a(x,y)u_x,$
   2. $L[u] = ({{\partial^2} \over {\partial x^2}} + {{\partial^2} \over {\partial y^2}})[u] = u_{xx}+ u_{yy},$
   3. $L[u] = a(x,y)u_{xx} + b(x,y)u_{yy}.$
7. ($15\%$)
   1. Let $u = u(x, y), \Delta u = u_{xx} + u_{yy},\ v(r,\theta) = u(r\cos\theta, r\sin\theta)$. Find the Laplace operator $L$ in polar coordinates $Lv = \Delta u$.
   2. Let $f(\theta)$ be a $C^\infty$-function of period $2\pi$ with Fourier Series . Prove that represents in polar coordinates $r,\theta$ the solution of the Laplace equation $Lv = 0$ in the disk $r < 1$ with boundary values $f$.
   3. Derive Poisson's integral formular for $v$ by substituting for $A_n$ their Fourier expressions in terms of $f$ (i.e. $A_n = {1 \over {2\pi}}\int^{2\pi}_0f(\theta)e^{-in\theta}d\theta$) and interchanging summation and integration.
8. ($10\%$) Solve $\cases{ u_{tt} - u_{xx} = 0\cr \noalign{\vskip 6pt} u_t(0,x) = g(x), g \in {\... ...}) \,\hbox{[Schwartz function class]}\cr \noalign{\vskip 6pt} u(0,x) = 0 \cr}$
   by performing Fourier transform with respect to the variable $x$
   
   $$\hat u(t,\xi) = {1\over{\sqrt{2\pi}}}\int^\infty_{-\infty}e^{-ix\xi}u(t,x)dx.$$
   
   
9. ($15\%$) (i) Solve $\cases{ u_t = u_{xx}\cr \noalign{\vskip 6pt} u(0,x) = f(x), f\in {\cal S}(\mathbb{R}) \ \ \hbox{[Schwartz function class]}\cr}$
   by performing Fourier transform with respect to the variable $x, \ \hat u(t,\xi) = { 1\over {\sqrt{2\pi}}}\int^\infty_{-\infty}e^{-ix\xi}u(t,x)dx,$ show $u(t,x) = { 1\over {\sqrt{2\pi}}}\int e^{ix\xi-\vert\xi\vert^2t}\hat f(\xi)d\xi$.
   (ii) Show that $u(t, x)$ may be expressed as
   $$\begin{eqnarray*} u(t, x) = \int K(x,y,t)f(y)dy &&\\ K(x,y,t) = { 1\over {\sqrt{4\pi t}}} e^{-(x-y)^2/4t} &&\\ \end{eqnarray*}$$
   
   (iii) Show that $\lim_{t\rightarrow 0}K(x, y, t) = \delta_y \ \ \hbox{[Dirac function]}$ in the sense of distribution.

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