
# 4.E: Hyperbolic Equations (Exercises)

### ﻿Q4.1

Show that $$u(x,t)\in C^2(\mathbb{R}^2)$$ is a solution of the one-dimensional wave equation

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

if and only if

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

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(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

(i) Show that

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

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

Find a solution of Schrödinger's equation

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

which satisfies the side condition

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

provided $$E\in\mathbb{R}^1$$ is an (eigenvalue) of the elliptic equation

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

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

Find solutions of the equation

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

which can be written as

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

with $$f(0)=K$$, $$K$$ a constant.

Hint: Transform equation for $$f(s)$$ by using the substitution $$s:=z^2/A$$ with an appropriate constant $$A$$ into Bessel's differential equation

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

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

• Integrated by Justin Marshall.