I am trying to solve the diffusion-reaction equation analytically (PDE of space and time) with a first order reaction term using laplace transformation:
\begin{align} \frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2} - \lambda_{t}\cdot c \end{align}
IC: c(x,0) = $\text{T}_{0}$
BC1: c(0,t) = $\text{C}_{0}$
BC2: \begin{align} \frac{\partial c(\infty,t)}{\partial x} = 0 \end{align}
However, I have reached a dead end when trying to use inverse laplace of the solution in the s domain:
\begin{align} c(x,s) = \frac{ c_{0}}{s}\cdot \exp\left(-x\cdot\sqrt{\frac{s+\lambda_{t}}{D}} \right) + \frac{ T_{0}}{s+\lambda_t}\cdot \exp\left(-x\cdot\sqrt{\frac{s+\lambda_{t}}{D}} \right) - \frac{ T_{0}}{s+\lambda_t} \end{align}
I can't find an inverse laplace transform of the above terms (beside the last) in any lookup tables. I prefer if someone is able to this using LT as I haven't tried or learned other methods yet but other methods are also welcomed.