Help with every problem would be greatly apperciated. $(20$ pts$)$ Problem $2($Alternative derivations of finite$-$difference schemes$)$

Suppose that in the neighborhood of the points $x_i=0,x_{i+1}=x,$ and $x_{i+2}=2x,$ the scalar $u(x)$

is represented by a second degree polynomial of the form

$u(x)=a+bx+c{x}^2$

Using the formula above, derive the following upwind $($one$-$sided$)$ schemes

$($i$)u^{\text{'}}(x_i)=u_i^{\text{'}}=\frac{-3u_i+4u_{i+1}-u_{i+2}}{2x},$

$($ii$)u^{\text{'}\text{'}}(x_i)=u_i^{\text{'}\text{'}}=\frac{u_{i+2}-2u_{i+1}+u_i}{x^2}.$

Grading rubric

$20$ pts for the correct derivation

$(20$ pts$)$ Problem $3$

Consider the finite differencing scheme to calculate the first derivative of a quantity $u$ at a certain

point

$u_i^{\text{'}}=\frac{u_{i+1}-u_{i-1}}{2x}$

Suppose that the measurements tilde$(u{)}_{i+1},$tilde$(u{)}_{i-1},$ and widetilde$(x)$ and their corresponding uncertainties $tilde(u{)}_{i+1},$

$tilde(u{)}_{i-1},$ and $widetilde(x)$ are available.

$($i$)$ Using the first$-$order analysis based on Taylor series expansion, find the fractional uncertainty

$\frac{tilde(u{)}_i^{\text{'}}}{|tilde(u{)}_i^{\text{'}}|}=f(tilde(u{)}_{i+1},$tilde$(u{)}_{i-1},(widetilde(x)),tilde(u{)}_{i+1},tilde(u{)}_{i-1},(widetilde(x)))$

$($ii$)$ Find the numerical value of the fractional uncertainty given the following data: tilde$(u{)}_{i+1}=2\mathrm{.}1,$

tilde$(u{)}_{i-1}=1\mathrm{.}1,$widetilde$(x)=0\mathrm{.}005,tilde(u{)}_{i+1}=0\mathrm{.}01,tilde(u{)}_{i-1}=0\mathrm{.}01,widetilde(x)=0\mathrm{.}001$