4.E: Hyperbolic Equations (Exercises)



$$u_{tt}=c^2u_{xx}$$

$$u(A)+u(C)=u(B)+u(D)$$

holds for all parallelograms $ABCD$ in the $(x,t)$-plane, which are bounded by characteristic lines, see Figure 4.E.1.

Figure 4.1: Figure to the exercise

Q4.2: Method of separation of variables

Let $v_k(x)$ be an eigenfunction to the eigenvalue of the eigenvalue problem

$-v''(x)=\lambda v(x)$ in $(0,l)$, $v(0)=v(l)=0$

and let $w_k(t)$ be a solution of differential equation

$$-w''(t)=\lambda_kw(t)$$

Prove that $v_k(x)w_k(t)$ is a solution of the partial differential equation (wave equation) $u_{tt}=u_{xx}$.

Q4.3

Solve for given $f(x)$ and $\mu\in\mathbb{R}^1$ the initial value problem

$$\begin{eqnarray*} u_t+u_x+\mu u_{xxx} &=& 0\quad \mbox{in}\quad \mathbb{R}^1\times\mathbb{R}^1_+\\ u(x,0) &=& f(x) \ . \end{eqnarray*}$$

Q4.4

Let $S:= \{(x,t);\ t=\gamma x\}$ be space-like, i.e., $|\gamma|<1/c^2$) in $(x,t)$-space, $x=(x_1,x_2,x_3)$. Show that the Cauchy initial value problem $\Box u=0$ with data for $u$ on $S$ can be transformed using the Lorentz-transform

$$x_1={x_1-\gamma c^2t\over\sqrt{1-\gamma^2c^2}}$$

$$\ x_2'=x_2,\ x_3'=x_3$$

$$t'={t-\gamma x_1\over\sqrt{1-\gamma^2c^2}}$$

into the initial value problem, in new coordinates,

$$\begin{eqnarray*} \Box u &=& 0\\ u(x',0) &=& f(x')\\ u_{t'}(x',0) &=& g(x')\ . \end{eqnarray*}$$

Here we denote the transformed function by $u$ again.

Q4.5

$$u(x,t):=\sum_{n=1}^\infty\alpha_n\cos\left(\dfrac{\pi n}{l} t\right)\sin\left(\dfrac{\pi n}{l} x\right)$$

is a $C^2$-solution of the wave equation $u_{tt}=u_{xx}$ if $|\alpha_n|\le c/n^ 4$, where the constant $c$ is independent of $n$.

(ii) Set

$$\alpha_n:=\int_0^ l f(x)\sin\left(\dfrac{\pi n}{l} x\right)\ dx.$$

Prove $|\alpha_n|\le c/n^4$, provided $f\in C^4_0(0,l)$.

Q4.6

Let $\Omega$ be the rectangle $(0,a)\times (0,b)$. Find all eigenvalues and associated eigenfunctions of $-\triangle u=\lambda u$ in $\Omega$, $u=0$ on $\partial\Omega$. Hint: Separation of variables.

Q4.7

$$i\hbar\psi_t=-\dfrac{\hbar^2}{2m}\triangle_x\psi + V(x)\psi\quad \mbox{in}\ \mathbb{R}^n\times\mathbb{R}^1,$$

$$\int_\mathbb{R}^n|\psi(x,t)|^2dx=1\ ,$$

$$\triangle u+\dfrac{2m}{\hbar^2}(E-V(x))u=0\quad\mbox{in}\ \ \mathbb{R}^n$$

under the side condition $\int_\mathbb{R}^n |u|^2 dx =1$, $u:\ \mathbb{R}^n\mapsto {\mathbb C}$.

Here is

$$\psi:\ \mathbb{R}^n\times\mathbb{R}^1\mapsto{\mathbb C}$$

Planck's constant ($\hbar$) is a small positive constant) and $V(x)$ a given potential.

Remark. In the case of a hydrogen atom the potential is $V(x)=-e/|x|$, $e$ is here a positive constant. Then eigenvalues are given by $E_n=-me^4/(2\hbar^2n^2)$, $n\in{\mathbb N}$, see [22], pp. 202.

Q4.8

Find nonzero solutions by using separation of variables of $u_{tt}=\triangle_xu$ in $\Omega\times (0,\infty)$, $u(x,t)=0$ on $\partial\Omega$, where $\Omega$ is the circular cylinder $\Omega=\{(x_1,x_2,x_3)\in\mathbb{R}^n:\ x_1^2+x_2^2<R^2,\ 0<x_3<h\}$.

Q4.9

Solve the initial value problem

$$\begin{eqnarray*} 3u_{tt}-4u_{xx} &=& 0\\ u(x,0) &=& \sin x \\ u_t(x,0) &=& 1\ . \end{eqnarray*}$$

Q4.10

Solve the initial value problem

$$\begin{eqnarray*} u_{tt}-c^2u_{xx} &=& x^2,\ t>0,\ x\in \mathbb{R}^1\\ u(x,0) &=& x \\ u_t(x,0) &=& 0\ . \end{eqnarray*}$$

Hint: Find a solution of the differential equation independent on $t$, and transform the above problem into an initial value problem with homogeneous differential equation by using this solution.

Q4.11

Find with the method of separation of variables nonzero solutions $u(x,t)$, $0\le x\le1,\ 0\le t<\infty,$ of

$$u_{tt}-u_{xx}+u=0\ ,$$

such that $u(0,t)=0$, and $u(1,t)=0$ for all $t\in[0,\infty)$.

Q4.12

$$u_{tt}-c^2u_{xx}=\lambda^2u,\ \lambda=const.$$

$$u(x,t)=f(x^2-c^2t^2)=f(s),\ s:=x^2-c^2t^2$$

$$z^2f''(z)+zf'(z)+(z^2-n^2)f=0,\ z>0$$

with $n=0$.

Remark. The above differential equation for $u$ is the transformed telegraph equation (see Section 4.4).

Q4.13

Find the formula for the solution of the following Cauchy initial value problem $u_{xy}=f(x,y)$, where $S$: $y=ax+b$, $a>0$, and the initial conditions on $S$ are given by

$$\begin{eqnarray*} u&=&\alpha x +\beta y+\gamma,\\ u_x&=&\alpha,\\ u_y&=&\beta, \end{eqnarray*}$$

$a,\ b,\ \alpha,\ \beta,\ \gamma$ constants.

Q4.14

Find all eigenvalues $\mu$ of

$$\begin{eqnarray*} -q''(\theta)&=&\mu q(\theta)\\ q(\theta)&=&q(\theta+2\pi)\ . \end{eqnarray*}$$

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.