Problem $1($c$)$: Results

a$.$ Grid of $11$ points

b$.$ Grid of $21$ points

c$.$ Grid of $41$ points

d$.$ Grid of $81$ points

Fig $1$: Results for all grids

Following table is generated:

h error ratio observed order

$0\mathrm{.}300004\mathrm{.}77973$e$-02$ NaN NaN

$0\mathrm{.}150001\mathrm{.}19896$e$-023\mathrm{.}986541\mathrm{.}99514$

$0\mathrm{.}075003\mathrm{.}00274$e$-033\mathrm{.}992911\mathrm{.}99744$

$0\mathrm{.}037507\mathrm{.}51011$e$-043\mathrm{.}998261\mathrm{.}99937$

Write a python code to the following matlab code.

clc;clear;close all;

$\%$ Problem$1($c$)$

a $=0$;

b $=3$;

sigma $=1$;

beta $=3$;

mplus$1=[10204080]$;

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

$\%$ True solution

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

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

$\%$ 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$\_1$c$($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$\_$loglog$($hvals$,$error$)$; $\%$ produce log$-$log plot of errors and least squares

fit