臺灣大學數學系

九十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

(20 points) Solve the following problems using characteristic methods:

(a) $uu_{x_1}+u_{x_2}=1,\ u(x_1,x_1)={x_1\over 2}$

(b) $x_1u_{x_1}+2x_2u_{x_2}=u,\ u(x_1, x_1)=g(x_1)$
where $g\in C^2(R)$.

2

(a) (10 points) Show that the function

$u(x, t)={1\over {\sqrt t}}f{1\over {\sqrt t}}$

is a solution of the heat equation $u_t=u_{xx}$ in $R\times (0, \infty)$ if and only if $f$ satisfies the following ordinary differential equation

$f''(\xi)+\xi f'(\xi)+f(\xi)=0\hskip 1cm\forall \xi \in R \hskip 2cm (*)$

(b) (10 points) Find all solution of the above ordinary differential equation ($*$). Hence or otherwise find a self-simiilar solution of the heat equation in $R^n\times (0, \infty)$.

3

(20 points) Let $u\in C^2$ for $\vert x\vert<a$; $u\in C^0$ for $\vert x\vert\leq a$; $u\geq 0$, $\triangle u=0$ for $\vert x\vert<a$. Show that for $\vert\xi\vert<a$,

${{a^{n-2}(a-\vert\xi\vert)}\over {(a+\vert\xi\vert)^{n-1}}}u(0)\leq u(\xi)\leq {{a^{n-2}(a+\vert\xi\vert)}\over {(a-\vert\xi\vert)^{n-1}}}u(0)$

4

(20 points) Let $\Omega \supset R^n$ be a bounded domain and let $G(x, y)$ be the Green function for the Laplacian in Ω. That is $\triangle_yG(x, y)=-\delta_x$ in Ω and $G(x, y)=0$ for any $x\in \Omega$, $y\in \partial \Omega$ where $\delta_x$ is the delta mass at $x$. Prove that

(a) $G(x, y)\geq 0$ $\forall x,\ y\in \Omega,\ x\ne y$

(b) $G(x, y)=G(y, x)$ $\forall x,\ y\in \Omega,\ x\ne y.$

5

(20 points) Suppose $f\in C(R^n)\cap L^\infty (R^n)$. Show that the function

$u(x, t)={1\over {(4\pi t)^{n/2}}}\int_{R^n}e^{-\vert x-y^2/4t\vert}f(y)dy$
satisfies the heat equation in $R^n\times (0, \infty)$ and

$\lim \limits_{t\searrow 0}u(x, t)=f(x)\hskip 1cm \forall x\in R^n.$

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