differential equations - Implement finite Fourier transforms

Recently I came across finite Fourier transforms, which can be used for solving certain type of boundary value problem (BVP) of linear partial differential equation (PDE) with constant coefficient. The definition of the transforms and their properties are as follows.

Finite Fourier sine transform and its inversion:

$$\mathcal{F}_s \{f (x)\}=\tilde{f}_s(n)=\int_0^a f (x) \sin (\frac{n\pi x}{a}) \, dx$$ $$\mathcal{F}^{-1}_s \{\tilde{f}_s(n)\}=f (x)=\frac{2}{a} \sum _{n=1}^{\infty } \tilde{f}_s(n) \sin (\frac{n \pi x}{a}) $$

Finite Fourier cosine transform and its inversion:

$$\mathcal{F}_c \{f (x)\}=\tilde{f}_c(n)=\int_0^a f (x) \cos (\frac{n\pi x}{a}) \, dx$$ $$\mathcal{F}^{-1}_c \{\tilde{f}_c(n)\}=f (x)=\frac{1}{a}\tilde{f}_c(0)+\frac{2}{a} \sum _{n=1}^{\infty } \tilde{f}_c(n) \cos (\frac{n \pi x}{a}) $$

Their main properties playing role in PDE solving are:

$$\mathcal{F}_s \{f' (x)\}=-(\frac{n \pi}{a})\tilde{f}_c(n)$$ $$\mathcal{F}_c \{f' (x)\}=(\frac{n \pi}{a})\tilde{f}_s(n)+(-1)^{n}f'(a)-f'(0)$$

For more information you can refer to this material or Chapter 10 of this book.

Can we implement these transforms in Mathematica?