Problem $2$

Consider the following reaction kinetics

$\frac{dc}{(d)t}=-\sqrt[\phantom{2}]{c},$

with $c(0)=1$ and $=0\mathrm{.}25.$

Show $($by hand$)$ the approximation to the solution to $(3)$ for the first two timesteps using $t=1$

using

$($a$)(20$ pts$)$ Forward Euler. Recall that the forward Euler method is given by ${}^{n+1}={}^n+$

$tR(t_n,{}^n).$

$($b$)(20$ pts$)$ Backward Euler. Recall that the Backward$-$Euler method is given by ${}^{n+1}={}^n+$

$tR(t_{n+1},{}^{n+1}).$ You must derive and typeset the residual function, its derivative, and what

Newton's method looks like for this method. Take two Newton iterations per timestep. Set

your initial guess for the Newton iterations as the final value from the previous timestep. You

can simplify the Newton iteration by renaming the variables which you are solving for, i$.$e$.$

$c^{1}x$ etc...

$($c$)$ pts$)$ Crank$-$Nicolson. The Crank$-$Nicolson method is given by the formula ${}^{n+1}={}^n+$

$\frac{1}{2}t[R(t_n,{}^n)+R(t_{n+1},{}^{n+1})].$ You must derive and typeset the residual function, its deriva$-$

tive, and what Newton's method looks like for this method. Take two Newton iterations per

timestep. Set your initial guess for the Newton iterations as the final value from the previous

timestep. You can simplify the Newton iteration by renaming the variables which you are

solving for, i$.$e$.c^{1}x$ etc...