Through a thick$-$walled cylindrical metal pipe, a hot fluid flows with a constant temperature $\backslash ($ T$\_$i $=450\backslash ,[\backslash$deg C$]\backslash ).$ The cylinder wall has an inner radius of $\backslash (1\mathrm{.}0\backslash ,[$cm$]\backslash )$ and an outer radius of $\backslash (2\mathrm{.}0\backslash ,[$cm$]\backslash ).$ The temperature distribution $\backslash ($ u$($r$)\backslash )$ in the metal is determined by the differential equation

$\backslash [$

r $\backslash$frac$\{$d$^2$ u$\}\{$dr$^2\}+\backslash$frac$\{$du$\}\{$dr$\}=0$

$\backslash ]$

with $\backslash ($ u $=$ T$\_$i $\backslash )$ at $\backslash ($ r $=1\backslash )($length unit cm$).$ The surrounding temperature $($outer temperature$)$ is $\backslash ($ T$\_$e $=20\backslash ,[\backslash$deg C$]\backslash ).$ At $\backslash ($ r $=2\backslash ),$ the temperature gradient $\backslash (\backslash$frac$\{$du$\}\{$dr$\}\backslash )$ is proportional to the temperature difference, i$.$e$.,$ it holds that

$\backslash [$

$\backslash$frac$\{$du$\}\{$dr$\}=-$K$($u $-$ T$\_$e$).$

$\backslash ]$

Here, $\backslash ($ K $\backslash )$ is a material constant, which depends on the heat transfer coefficient $\backslash (\backslash$alpha $\backslash )($in the unit $\backslash ([$W$/($m$^2\backslash$cdot K$)]\backslash ))$ between the metal and air and the metal's thermal conductivity $\backslash ($ k $\backslash )($in the unit $\backslash ([$W$/($m $\backslash$cdot K$)]\backslash ))$ according to $\backslash ($ K $=\backslash$frac$\{\backslash$alpha$\}\{$k$\}\backslash ).$ Let in the test case $\backslash ($ K $=1\backslash ).$

a$)$ According to the finite difference method, discretize the interval $\backslash (1\backslash$leq r $\backslash$leq $2\backslash )$ divided into $\backslash ($ N $\backslash )$ subintervals. Discretize the boundary condition $(5)$ with a second$-$order difference approximation $($e$.$g$.,$ skewed stencil or central difference and ghost point$).$ Show how the boundary value problem can be approximated by a matrix problem. Solve this first for $\backslash ($ N $=25\backslash ),$ continuing with successive doublings of $\backslash ($ N $\backslash )$ until the desired precision is achieved$$e$.$g$.,$ four correct digits in the temperature value at the cylinder's outer radius. Plot the temperature distribution in the metall.