$25$ points $1$D Heat Conduction: In the class, we discussed using the Fourier method $($Separation of Vari$-$

ables$)$ to arrive at the solution of PDE's such as the $1$D heat conduction equation. The very same equation

can also be used to describe the behavior of boundary layer during active flow control on aircraft wings,

drones, rail cars, wind turbines, etc. In the following problem, assume $=1.$

Consider the unsteady boundary layer flow over the surface of an aircraft wing with a chord length of $.$

To enhance aerodynamic performance, lets assume boundary layer control is implemented through suction

near the leading edge and blowing near the trailing edge of the wing. Assume that initially, the boundary

layer is in equilibrium with no active control, such that the velocity disturbance along the surface is zero:

$u(x,0)=0,0x$

At the leading edge $(x=0),$ a time$-$dependent suction is then applied to reduce the thickness of the

boundary layer and delay separation. The strength of this suction increases linearly with time and is

modeled by the boundary condition:

$u_x(0,t)=t,t0$

At the trailing edge $(x=),$ a corresponding blowing is applied to enhance flow attachment near the

trailing edge. The strength of the blowing also increases with time, given by:

$u_x(,t)=t,t0$

Your task is to determine the velocity profile $u(x,t)$ of the boundary layer as it evolves over time due to

the applied suction and blowing $($basically the solution of the heat equation with the given BC$\text{'}$s and IC's

using separation of variables$).$ Furthermore, analyze the long$-$term effects of this boundary layer control,

specifically how the velocity distribution stabilizes for large times, and plot the velocity profile as a function

of the chord for large times.