Using a similarity variable, the laminar boundary layer equation for an incompressible steady flow over a flat plate can be written as:

$f^{\text{'}\text{'}}+\frac{1}{2}f{f}^{\text{'}\text{'}}=0$

Boundary conditions for the problem are:

$)=(0)=(0$

where $f$ is the non$-$dimensional stream function of the similarity variable $n.$

$f^{\text{'}}()=g()$

$g()=1-e^{-}$

$r()={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}g()d$

a$-)$ Select an appropriate finite difference method and find the numerical solution by applying Tri$-$diagonal matrix algorithm $($TDMA$).$ Hint: You can use trapezoidal rule for numerical integration.

b$-)$ Solve the same problem by using Shooting Method utilizing Regula$-$Falsi and $4^{th}$ order Runge Kutta $($RK$4).$ Lower and upper bounds for root finding with Regula$-$Falsi are $0\mathrm{.}2$ and $0\mathrm{.}4,$ respectively.

For both solution numerical approaches of a$)$ and b$)$ plot followings:

Comparative plots of $f()$ and $g()$

Comparative plots of velocity profile $u(y)$ at $x=1m.$

Compare wall shear stress and skin friction coefficient distributions $($obtained using $f^{\text{'}}(0))$ up to $x=1m$ over the plate.

Note: Consider air flow with density of $1\mathrm{.}225k\frac{g}{m^3}$ and dynamic viscosity of $1\mathrm{.}78{10}^{-5}k\frac{g}{m}s.$ Free stream velocity of the flow is $U=5\frac{m}{s}.$

Prepare a report on your calculations indicating the code, outputs such as graphics and comment on them.