When it comes to solving the heat diffusion equation u_t=u_xx the two most important solutions are a) a combination (sum) of sin-terms to resemble the function of the initial condition (that is essentially a fourier series) b) a convolution-integral of the function of the initial condition with the Gauss-curve

In most books you only find a) or b).

My question: How does it come that you get to such different solutions? Why is it that some books end up with a) and others with b)? What is the essential difference of deriving a) or b) in the end?