$(60$ pts $)$ Consider the $1$D continuous heat transfer model

$\backslash$rho cu$\_($t$)-$Ku$\_(\backslash$times $)=$f$,($x$,$t$)$in$[0,$L$]\backslash$times $[0,$T$],$

subject to the initial condition u$($x$,0)=$g$($x$)$ and boundary conditions u$(0,$t$)=$h$\_(1)($t$)$ and

u$($L$,$t$)=$h$\_(2)($t$).$ Given n and maxk, define $\backslash$Delta x$=($L$)/($n$),\backslash$Delta t$=($T$)/($maxk$)$ and u$\_($i$)^($k$)=$u$($i$\backslash$Delta x$,$k$\backslash$Delta t$).$

$($a$)$ Use the finite difference approximations

u$\_($t$)($i$\backslash$Delta x$,$k$\backslash$Delta t$)$~~$($u$\_($i$)^($k$+1)-$u$\_($i$)^($k$))/(\backslash$Delta t$)$

and

u$\_(\backslash$times $)($i$\backslash$Delta x$,$k$\backslash$Delta t$)$~~$($u$\_($i$+1)^($k$)-2$u$\_($i$)^($k$)+$u$\_($i$-1)^($k$))/((\backslash$Delta x$)^(2))$

to derive the $1$D discrete heat transfer model

The $1$D heat transfer FD model is:

u$\_($i$)^($k$+1)=,+,($u$\_($i$-1)^($k$)+$u$\_($i$+1)^($k$))+$cdots,u$\_($i$)^($k$),$

for i$=1,$dots,n$-1,$k$=0,$dots,maxk$-1.$

$($b$)$ Give the corresponding stability condition for the explicit finite difference scheme just

derived. The stability condition is: