$15$pts p$($x$,$t$)5$pts q$($x$,$t$)($volumetric rate$)$ solutions to predict the volumetric flow rate inside the

system.

$($c$)(30$ pts$)$ You may consider time values at t$=10^(-2),10^(-1),1\mathrm{.}0,10,100,1000,10^(4),10^(5)$ hours. You

must use the Stehfest $($or Gaver$-$Stehfest$)$ algorithm with the parameter N$\_($stef $)=12$ to compute

your p$($x$,$t$)$ and q$($x$,$t$)$ as a function of x and t$()/($bar$)($p$)($x$,$s$)$

and $()/($bar$)($q$)($x$,$s$),$ where s is the Laplace variable with respect to time t $,$ and overbar represent Laplace

space solutionp$($x$,$t$)$ and q$($x$,$t$)$p$($x$,$t$)=$L$^(-1)[()/($bar$)($p$)($x$,$s$)]$ and q$($x$,$t$)=$L$^(-1)[()/($bar$)($q$)($x$,$s$)]$ in the Laplace

space. Here, L$^(-1)$ represents the inverse Laplace operator, and would be Stehfest numerical

inversion algorithm in your case. You can use any method of coding that you are able to

generate your results from the Stehfest Numerical Inversion algorithm. Also, compare your

solutions with the solutions generated by using your real$-$time solutions of p$($x$,$t$)$ and q$($x$,$t$)$ to

see how accurately you can invert the Laplace space solutions with the numerical Laplace inverter.

Present a discussion of your results after you prepare log$-$log plots of $\backslash$Delta p$=$p$^(0)-$p$($x$,$t$)$ and

q$($x$,$t$)$ versus x for a few time values or versus time for a few fixed values of x on the plot. You

can use the values given in Table $4\mathrm{.}1.$

Table $4\mathrm{.}1-$ Input Data for Computing Solutions