Laplace and Fourier transforms essentially turn differential equations into algebraic equations.

In my experience, the biggest problem with Laplace transforms is the inversion - it's not always that easy to transform back to the (x,t) domain. I assume the PDE is first-order in time and second-order in space. The Laplace transform is tailor made to initial value problems (in fact, I don't see a straight forward way to implement boundary conditions), so just apply the transform to the time component. This will give you a second order ODE in space, which you can hopefully solve analytically. Invert the solution to get the time-dependent result.