Find general solution
\begin{equation}
u_x-3u_y=0;\label{eq-1}
\end{equation}
Solve IVP problem $u|_{x=0}=e^{-y^2}$ 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.
Problem 2
Corrected
Find the general solution of
\begin{equation}
xu_x+4yu_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-4yu_y=0\label{eq-3}
\end{equation}
in $ \{(x,y)\ne (0,0)\}$; when this solution is continuous at $(0,0)$?
Explain the difference.
Problem 3
Find the solution of
\begin{equation}
\left\{\begin{aligned}
&u_x+3u_y=xy,\\
&u|_{x=0}=0.
\end{aligned}
\right.\label{eq-4}
\end{equation}
Problem 4
Corrected
Find the general solution of
\begin{equation}
yu_x-xu_y=xy;\label{eq-5}
\end{equation}
Find the general solution of
\begin{equation}
yu_x-xu_y=x^2+y^2;\label{eq-6}
\end{equation}
In one instanse solution does not exist. Explain why.
Problem 5
Find the general solution of
\begin{equation}
u_{tt}-9u_{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 $\{x>3t, x>-3t\}$ and find a solution to it, satisfying Goursat problemCorrected September 23
\begin{equation}
u|_{x=3t}=t,\quad u|_{x=-3t}=2t.
\end{equation}
Problem 6
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 formual 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 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?