Assignment $3$ :

PDE: ID

Write python code only for the following question

Burger's equation $=>$ study the propagation

$**\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}+\frac{\text{u}\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}-v\frac{de{l}^{2}u}{del{x}^2}=0$

$=>\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}+u\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}-\frac{1}{Re}\frac{de{l}^{2}u}{del{x}^2}=0$

$Re=\frac{1}{v},Re=$ Reynolds number of a shock wave generated by the iD "viscous" Burger's equation.

At $t=0$ the shock wave is located at $x=0.$

The initial conditions are

$u_0(x)\frac{?}{b}=ar(u)(x,0)=1\mathrm{.}0,fw,-x_{\text{m}\text{a}\text{x}}x0$

representis initial volue

The following BCS are applied at $x=+-x_{\text{m}\text{a}\text{x}}$

$\frac{?}{\text{b}\text{a}\text{r}}(u)(-x_{\text{m}\text{a}\text{x}},t)=1\mathrm{.}0\frac{,}{b}ar(u)(x_{\text{m}\text{a}\text{x}},t)=0$

$Re=10$

$Re=50$

For the given combination of initial and boundary conditions the exact solution of the $1D$ viscons Durger equation is

where $G($;$x,t)={\displaystyle \underset{0}{\overset{}{}}}{u}_0({}^{\text{'}})d{}^{\text{'}}+\frac{0\mathrm{.}5(x-{)}^2}{t}$

and $Re=\frac{1}{v}$

Solve equation $(1)$ numerically using explicit and impsicit schemes and report the results with proper explanation. # Compare it with the exact solution.