Find general solution
\begin{equation}
3 u_x + 4u_y=0;
\label{eq-1}
\end{equation}
Solve IVP problem $u|_{x=0}=1/(y^2+1)$ for equation (\ref{eq-1}) in $\mathbb{R}^2$;
Consider equation (\ref{eq-1}) in $\{x>0, y>0\}$ with the initial condition $u|_{x=0}=y$ ($y>0$); where this solution defined? Is it defined everywhere in $\{x>0, y>0\}$ or do we need to impose condition at $y=0$?
In the latter case impose condition $u|_{y=0}=x$ ($x>0$) and solve this IVBP;
Consider equation (\ref{eq-1}) in $\{x<0, y>0\}$ with the initial condition $u|_{x=0}=y$ ($y>0$); where this solution defined? Is it defined everywhere in $\{x<0, y>0\}$ or do we need to impose condition at $y=0$? In the latter case impose condition $u|_{y=0}=x$ ($x<0$) and solve this IVBP.
Find the general solution of
\begin{equation}
xu_x+yu_y=0
\label{eq-2}
\end{equation}
in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$?
Find the general solution of
\begin{equation}
xu_x-yu_y=0
\label{eq-3}
\end{equation}
in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$?
Explain the difference between (\ref{eq-2}) and (\ref{eq-3}).
Find the general solution of
\begin{equation}
u_{tt}-4u_{xx}=0;
\label{eq-7}
\end{equation}
Solve IVP
\begin{equation}
u|_{t=0}=x^2,\quad u_t|_{t=0}=x
\end{equation}
for (\ref{eq-7});
Consider (\ref{eq-7}) in $\{t>0, \, 2t> x > -2t\}$ and find a solution to it, satisfying Goursat problem
\begin{equation}
u|_{x=2t}=t,\quad u|_{x=-2t}=2t.
\end{equation}
Remark.
Goursat problem for wave equation $u_{tt}-c^2u_{xx}=0$ in ${t> 0, -ct<x<ct}$ is $u|_{x=ct, t>0}=\phi(t)$, $u|_{x=-ct, t>0}=\psi(t)$ and one often assumes that compatibility condition $\phi(0)=\psi(0)$ is fulfilled. It is very important that $x=\pm ct$ are characteristics.
Derivation of a PDE describing traffic flow. The purpose of this problem is to derive a model PDE that describes a congested one-dimensional highway. Let
$\rho(x,t)$ denote the traffic density : the number of cars per
kilometer at time $t$ located at position $x$;
$q(x,t)$ denote the traffic flow: the number of cars per hour passing
a fixed place $x$ at time $t$;
$N(t,a,b)$ denote the number of cars between position $x=a $ and $
x=b$ at time $t$.
Derive a formula for $N(t,a,b)$ as an integral of the traffic density.
You can assume there are no exits or entrances between position $a$
and $b$.
Derive a formula for $\frac {\partial N} {\partial t}$ depending on
the traffic flow. Hint: You can express the change in cars between time $t_1=t$ and $t_2= t+h$ in terms of traffic flow;
Differentiate with respect to $t$ the integral form for $N$ from
part (a) and make it equal to the formula you got in part (b). This of
the integral form of conservation of cars;
Express the right hand side of the formula of part (c) in terms of an
integral. Since $a,b$ are arbitrary, obtain a PDE. This PDE is called
the conservation of cars equation;
What equation do you get in part (4) if $ q=c \rho$, for some
constant $c$. What choice of $c$ would be more realistic, i.e. what
should $c$ be function of?