Consider the PDE for u $=$ u$($x$,$ t$)$

utt $=$ c

$2$uxx, c in R$,(1)$

u$(0,$ t$)=$ u$($L$,$ t$)=0,$ L in N

u$($x$,0)=0,$ ut$($x$,0)=2\backslash$pi sin $(\backslash$pi x$)$

a$.$ Give a physical interpretation of the PDE $(1)$ and its boundary and initial conditions.

$(2$ marks$)$

b$.$ Write down an explicit central difference scheme to solve the PDE $(1)$ on the interval

$[0,1],$ making sure all terms and variables are defined consistently. You will need to

include the discrete initial conditions for your algorithm.

$(5$ marks$)$

c$.$ Write a Python script that numerically solves the PDE $(1)$ with the stated initial and

boundary conditions. You may use the codes from tutorials as guidance and a starting

point, if you wish. Take c

$2$

to be the last non$-$zero digit of your university ID number,

and take L to be the penultimate non$-$zero digit of your ID number. You will need to

choose suitable values for other variables. Run your script with a suitable timestep and

grid spacing. Make a two dimensional plot that shows the solution at $5$ equally spaced

moments in time. Your results should look smooth $($not jagged or piecewise$).$ The title

of the plot must include your ID number $-$ it will not be marked without it$.$ Your code

should include comments to aid the reader. Include a copy $($a screenshot or a PDF$)$ of

your code and the plot $($as well as uploading your code$).$

$($