$1)$ discretize space and time $(10$ points$)$

$-$ space variable x: start value $0,$ end value $10$e$-3,$ interval $1$e$-5$

$-$ time variable t : start value $0,$ end value $1$e$-4,$ number of points $101$

$$

$2)$ generate data $(10$ points$)$

$$compute values for p for each location and each time instance

$$

$$recall:

$-$ all locations are stored in the array x created in $1)$

$-$ all time instances are stored in the array t created in $1)$

$$to generate the data and store it in a $2$d array p follow these steps:

$-$ allocate and initialize a $2$d array called p with as many rows as you

$$have points in space and as many columns as you have time instances

$-$ loop over all time instances and at each time instance loop over all x

$$locations $($hint: nested loops$)$

$-$ use the loop variable it for looping over

$$the time instances and loop variable i for the loop over the x locations

$-$ for each it and i compute p$($i$,$it$)$ as

$$p $=$ exp$(100*$x$)*$cos$(2*$pi$/($xe$-$xs$)*10*$x$-2*$pi$*100$e$3*$t$)$

$+$ exp$(50*$x$)*$cos$(2*$pi$/($xe$-$xs$)*5*$x$-2*$pi$*50$e$3*$t$)$

$+$ cos$(2*$pi$/($xe$-$xs$)*12*$x$-2*$pi$*150$e$3*$t$)$

$+$ exp$(-50*$x$)*$cos$(2*$pi$/($xe$-$xs$)*20*$x$-2*$pi$*200$e$3*$t$)$

$$

$3)$ write data to file $(10$ points$)$

$$write the data in binary format to a file with the name data.bin

$$write the data in the following format

$$column $1$: x

$$columm $2$: t

$$column $3$: p

$$write out the data in double precision format

$$note: p had nt columns, only write nt$-1$ columns to the file

$$

$4)$ visualize data $(15$ points$)$

$$generate a subplot with $2$ rows $(2$ figures in total$)$

$$in the first row plot the values of p for i $=1$ over time

$$in the second row plot the values of p for it $=1,26,51$ and $76$ over x

$$customize the plots as follows

$-$ increase the linewidth to $2$ pts

$-$ set the font size to $20$

$-$ switch on the grid

$-$ for the plot in row $1$:

$*$ set the x$-$axis label to t and the y$-$axis label to p

$*$ set the x$-$axis limits from $0$ to $1$e$-4$

$*$ set the y$-$axis limits from $-2$ to $4$

$-$ for the plot in row $2$:

$*$ set the x$-$axis label to x and the y$-$axis label to p

$*$ set the x$-$axis limits from $0$ to $1$e$-2$

$*$ set the y$-$axis limits from $-5$ to $10$

$*$ add a legend to the figure with the labels $\text{'}1\text{'},\text{'}26\text{'},\text{'}51\text{'}$ and $\text{'}76\text{'}$

$$ for the four curves plotted in this graph

$-$ generate a figure file named project$\_03\_$matlab$\_$figure.png

Matlab Code