Write code to solve the wave equation $u_t+u_x=0$ using

$($a$)$ Lax scheme

$($b$)$ Lax$-$Wendroff scheme

for the initial condition

$u(x,0)=sin[2n(\frac{x}{40})],0x40,$

and periodic boundary conditions. Choose a $41$ grid point mesh with $x=1$ and compute to $t=18.$

For each scheme:

Solve this problem for $n=1,3$ and

$u=1\mathrm{.}0,0\mathrm{.}6,0\mathrm{.}3$ and compare graphically with the exact

solution $($generate graphs for each method, $n,$ and value of

$u).$

Determine $$ for $n=1$ and $n=3,$ and calculate the errors in amplitude and phase for each scheme

with

$u=0\mathrm{.}6.$ Compare these errors with the errors appearing on the graphs.

Show your soultions step by step and be precise. Provide the computer programming codes in Matlab please.