$1.$ Consider the problem of finding, on the interval $[$a$,$ b$]$ where a and b are real numbersConsider the problem of finding, on the interval $a,b$ where a and $b$ are real numbers

which are such that $ab,$ the solution $y$ to the initial value problem

$y^{\text{'}}=f(x,y(x)),y(a)=y_0$

where

$f=-s(y-(x+2))+1,$

where $s$ is your student ID number and

$y(0)=1$

For appropriate integers $j,$ let $x_j=a+jh$ where $h$ is the step size and is such that $N=\frac{b-a}{h}$

is an integer.

$[15$ marks$]$ For each of the following mehods $(2),(3),(4),$ write the Python code that

produces the discrete time points in a straight line xin$[0,{10}^{-5}]$ for $h=0\mathrm{.}000002,0\mathrm{.}00000125,$

implements the method and plots numerical solution togehter with the exact one in the same

axes. Save all codes in CW$1$q$1\_$your$\_$name.py

Save all solutions in CW$1\_$your$\_$name.pdf

a Consider a one$-$step finite difference method to find the approximate solution to $(1)$

$y_{n+1}=y_n+hf(x_n+h,y_n+hf(x_n,y_n))$

a$1[5$ marks$]$ Find such values of $$ that the local truncation error of the FDE $(2)$ is

$O(h^2).$

a$2[5$ marks$]$ Find the interval of stability for $(2)$ with the value of $$ obtained in part

$[$a$1]$ and explain why $(2)$ is not absolutely stable.

which are such that a b$,$ the solution y to the initial value problem

y

$=$ f$($x$,$ y$($x$)),$ y$($a$)=$ y$0(1)$

where

f $=$s$($y $($x $+2))+1,$

where s is your student ID number and

y$(0)=1$

For appropriate integers j$,$ let xj $=$ a $+$ jh where h is the step size and is such that N $=$

b$$a

h

is an integer.

$[15$ marks$]$ For each of the following mehods $(2),(3),(4),$ write the Python code that

produces the discrete time points in a straight line x in $[0,105$

$]$ for h $=0\mathrm{.}000002,0\mathrm{.}00000125,$

implements the method and plots numerical solution togehter with the exact one in the same

axes. Save all codes in CW$1$q$1$ your name.py

Save all solutions in CW$1$ your name.pdf

a Consider a one$-$step finite difference method to find the approximate solution to $(1)$

yn$+1=$ yn $+$ hf$($xn $+\backslash$alpha h$,$ yn $+\backslash$alpha hf$($xn$,$ yn$))(2)$

a$1[5$ marks$]$ Find such values of $\backslash$alpha that the local truncation error of the FDE $(2)$ is

O$($h

$2$

$).$

a$2[5$ marks$]$ Find the interval of stability for $(2)$ with the value of $\backslash$alpha obtained in part

$[$a$1]$ and explain why $(2)$ is not absolutely stable.