Laplace transform method for parabolic problems with time-dependent coefficients

Laplace transform method has proven to be very efficient to deal with parabolic problems whose coefficients are time-independent, and is easily parallelizable. However, the method has been doubtful to be applicable to any nonlinear or linear problems 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. In this paper, we propose a method of Laplace transform to linear parabolic problems with time-dependent coefficients, which is as efficient as the method for parabolic problems with time-independent coefficients. Several numerical results are provided, which support the efficiency of the proposed scheme.