a$.$ with $11$ grid points

b$.$ with $21$ grid points

c$.$ with $41$ grid points

d$.$ with $81$ grid points

Fig $4.$ Results for all grid points for approach $2$

Write a python code of the following matlab code. clc;clear;close all;

$\%$ Problem $2($c$)-$second approach

a $=0$;

b $=3$;

sigma $=-5$;

beta $=3$;

mplus$1=[10204080]$;

f $=$ @$($x$)$ exp$($x$)$;

$\%$ True solution

u$\_$true $=$ @$($x$)$ exp$($x$)+($sigma$-$exp$($a$))*$x$+$beta$-$exp$($b$)...$

$+$ b$*$exp$($a$)-$b$*$sigma;

$\%$ True solution on fine grid for plotting

xfine $=$ linspace$($a$,$b$,101)$;

ufine $=$ u$\_$true$($xfine$)$;

$\%$ Solve for multiple grids

hvals $=$ zeros$($numel$($mplus$1),1)$;

error $=$ zeros$($numel$($mplus$1),1)$;

for i $=1$:numel$($mplus$1)$

$\%$ run routine for finite diff solve

$[$x$,$hvals$($i$),$u$]=$ finite$\_$difference$\_2$c$2($a$,$b$,$sigma,beta,mplus$1($i$),$f$)$;

$\%$ get error at grid points

u$\_$hat $=$ u$\_$true$($x$)$;

err $=$ u $-$ u$\_$hat;

error$($i$)=$ max$($abs$($err$))$;

disp$(\text{'}\text{'})$

disp$($sprintf$(\text{'}$Error with $\%$i points is $\%9\mathrm{.}5$e$\text{'},$mplus$1($i$)+1,$error$($i$)))$

clf

plot$($x$,$u$,\text{'}$s$\text{'})\%$ plot computed solution

title$($sprintf$(\text{'}$Computed solution with $\%$i grid points',mplus$1($i$)+1))$;

hold on

plot$($xfine$,$ufine$)\%$ plot true solution

xlabel$(\text{'}$x$\text{'})$

ylabel$(\text{'}$U$\text{'})$

hold off

drawnow

input$(\text{'}$Hit to continue $\text{'})$;

clear x u

end

error$\_$table$($hvals$,$error$)$; $\%$ print tables of errors and ratios

Error with $11$ points is $1\mathrm{.}64966$e$-01$