So I've got an equation
\begin{gather}
\frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial y^2}\\
\begin{cases}
u(y,0) = 0\\
u(0,t) = U\\
u(h,t) = 0
\end{cases}
\end{gather}
I need to change variable $u(y,t)$ to $v(y,t)$, where
\[
u(y,t) = U\Bigg(1-\frac{y}{h}\Bigg)+v(y,t)
\]
and use the Fourier basis function $sin(\frac{n\pi y}{h})$ in order to show that the solution is
\[
u = U\Bigg(1-\frac{y}{h}\Bigg)-\frac{2U}{\pi}\sum_{n=1}^{\infty}\frac{1}{n} e^{-n^2\pi^2\frac{\nu t}{h^2}}\sin\frac{n\pi y}{h}
\]
The new PDE we obtained is \begin{gather} {\frac{\partial U}{\partial t}\Bigg(1-\frac{y}{h}\Bigg)}+\frac{\partial v}{\partial t} = \nu\frac{\partial}{\partial y}\Bigg(-\frac{U}{h}+\frac{\partial v}{\partial y}\Bigg)\\ \frac{\partial v}{\partial t} = \nu\frac{\partial^2 v}{\partial y^2} \end{gather} New set of homogeneous boundary conditions \begin{cases} v(0,t) = 0\\ v(h,t) = 0 \end{cases}
But somehow I do not know how to proceed exactly and use Fourier basis functions. I tried a bunch of series representation but do not seem to arrive at an answer. Could anyone outline the next steps for me please? In more detail if possible. I am looking at the theory but cannot put things together. Thanks!
Basically, the Fourier expansion leads to a differential equation instead of PDE, because the second derivative of sin(k.y) is -k²sin(k.y).
Likewise, the time derivative will also simplify into algebraic operation, so that finally you get a linear algebraic equation.
Your first line does not look like an equation but it looks like a possible general solution of the diffusion equation below, according to the method of variable separation, but you need to add unknown coefficients $u_n(t)$ in front of sin(n.k.y).
U(1-y/h) is a particular solution (steady with uniform shear) so you rewrite the equation on the difference.
Your initial and boundary conditions seem inconsistent unless U=0. There must be something wrong.
Actually, you are studying the linear stability of plane 2D Couette (shear) viscous flow. Searching on the internet, I found the very source of your exercise, with the same notations. I let you do the search.
The source is apparently Batchelor An introduction to fluid dynamics but I don't have it at home.
Assuming a parallel flow between two infinite plates with no slip conditions in the Navier-Stokes equations leads to your very equation. The initial condition is indeed singular but the singularity is smoothed out by diffusion.
You may also be interested in Squire's theorem (see Wikipedia) and
Orszag, S., & Patera, A. (1983). Secondary instability of wall-bounded shear flows. Journal of Fluid Mechanics, 128, 347-385. doi:10.1017/S0022112083000518