Inverse Laplace Transform using Hetnarski's Algorithm

Question

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me and I've used it to solve for temperature. However, I've been unable to lay my hands on the original article and as such have no way extend the idea to the velocity component. The velocity equation is $$\frac {\partial u}{\partial t} = \frac {\partial^2u}{\partial y^2} - Mu + Gr\theta + Gm\phi$$ The initial and boundary conditions are: $$ t \le 0: u = 0 \quad \forall \quad y$$ $$t\gt 0: u = cos{(\omega t) \quad at \quad y = 0}$$ $$t \gt 0: u \to 0 \quad as \quad y \to \infty $$ Any hints on how to go about it (alternate algorithms for inverse laplace transforms, a summary of the (various?) exponential forms in Hetnarski's algorithm etc etc) would be greatly appreciated. I have the solution already but I want to understand the steps that lead to it. My attempts at a solution are shown below. This is where I'm stuck.