Fourier series expansion

Question

Is it possible to have a Fourier sine series expansion like $$ \sin\left(\left(\frac{\pi}{2} + n\pi\right)x\right) $$ instead of the normal $$\sin(n \pi x)$$

Answer 1

Not only possible, but sometimes necessary. This is the expansion one uses to deal with the mixed boundary value problem on $[0,1]$: $$u(0,t)=0,\quad u_x(1,t)=0 \tag{1}$$ The sine functions you listed satisfy these boundary conditions. Subject to (1) and some initial condition, the wave and diffusion equations will have solutions in the form $$ u(x,t) = \sum_{n=0}^\infty \sin\left(\left(\frac{\pi}{2} + n\pi\right)x\right) T_n(t) $$
The factors $T_n(t)$ are determined by the PDE and the initial condition.