I know how to solve heat equation where it's like $u\_t=k\cdot u\_{xx}$ (using Fourier Transform or using Separation of Variables) but this exercise is really difficult for me.

I have this:

$$u\_t(x,t)=k \cdot u\_{xx}(x,t)-a\cdot k \cdot u(x,t)$$
$$u\_x(0,t)=0$$
$$u(x,0) = f(x)$$

with $x>0, t>0$ and $a, k$ are positive constants.

I have to find $u(x,t)$ and propose a possible $f(x)$

Any help? Thanks

I was told I cannot use Fourier Transform, I have to use Fourier Cosine Transform, and I don't know why