For the forthcoming exercises consider the $1$D transport equation $\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}+(f(u,x,t){)}_x=0$ where $t$ is time,

and $x$ represents location. We will assume that $f(u,x,t)=au$ where $a=a(x,u,t)$ in general.

$(20$ points$)$ Consider an infinite model domain $($so that we can avoid boundary conditions$).$

Assume that $a$ is a constant, and that the initial condition is given by $u(x,0)=u_0(x).$

a$.$ Develop the general solution exactly using the method of characteristics.

b$.$ On the same chart but using different line colors, plot your solution for $t=0,t=1,$ and

$t=2$ : for the case of $a=1\mathrm{.}5$ for some smooth initial condition of your choice $(u_0=$

const. is not allowed$).$ Clearly label all axes and provide titles and legends on your plots.

c$.$ Repeat $(1$b$)$ for the same initial condition but for the case of $a=-0\mathrm{.}5.$

d$.$ Provide a brief explanation illustrating what these two cases might represent in the real

world, and how one might explain it to an undergraduate fluid mechanics class $($e$.$g$.,$

ENGR $304).$

$(10$ points$)$ For the previous case $(1)$ with $a=$ positive const.,

a$.$ If $u_0(x)=1$ for $0x1$ and $u_0(x)=0$ otherwise, develop the exact solution to the

model using the method of characteristics.

b$.$ On the same chart but using different line colors, plot your exact solution for $t=0,t=1,$

and $t=2$ with $a=2.$

$(10$ points$)$ For the previous case $(1)$ with $a=cos(t),$

a$.$ If $u_0(x)=1$ for $0x1$ and $u_0(x)=0$ otherwise, develop the exact solution to the

model using the method of characteristics.

b$.$ On the same chart but using different line colors, plot your exact solution for $t=0,t=1,$

and $t=2.$

c$.$ Graduate students only: develop a nice $3D$ surface plot of the exact solution with $t=$

$0\mathrm{.}05$ for tin$[0,2].$ Label all axes.

$(15$ points$)$ For the previous case $(1)$ with $a=(1-x)e^{-t},$

a$.$ If $u_0(x)=1$ for $0x0\mathrm{.}1$ and $u_0(x)=0$ otherwise, develop the exact solution to the

model using the method of characteristics.

b$.$ Graduate students only: develop a nice $3D$ surface plot of the exact solution with $t=$

$0\mathrm{.}05$ for tin$[0,2].$ Label all axes.

$(15$ points$)$ Assume that $u(x,t)$ models the density of cars on a highway around location $x$ and at

time $t.$

a$.$ What is the physical interpretation of $a$ in this application of the model?

b$.$ Under what practical conditions could we have $a=$ const., $a=a(t),$ or $a=a(x)?$

c$.$ Is it always true that $f$ should somehow depend on $u-$i$.$e$.,$ is it physically reasonable for

$f=f(x,t)$ for example? Explain.

$(10$ points$)$ Consider the case where $a(u)=\frac{u}{2}$ so that $f(u)=\frac{u^2}{2}.$ The initial condition is

$u(x,0)=u_0(x).$ This is Burgers' equation.

a$.$ What is $\frac{dx}{dt}?$

b$.$ If the initial condition is defined as $u_0(x)=1$ for $0x1$ and $u_0(x)=0$ otherwise

above, roughly sketch the characteristic lines. Label$/$callout the shock and rarefaction

parts of the sketch.