The Laplace transformation method has proven to be very efficient for dealing with parabolic problems whose coefficients are time-independent, and is easily parallelizable. However, it seems problematic to apply the method to any nonlinear or linear problem whose coefficients are time-dependent. The reason is that the Laplace transform of two time-dependent functions leads to a convolution of the Laplace transformed functions in the dual variable.
However, under certain conditions, we can propose a method of Laplace transformation to linear parabolic problems with time-dependent coefficients, which is as efficient as in the time-independent case.
Some numerical results will be presented to demonstrate this claim.