  This study introduces pre-orthogonal adaptive Fourier decomposition (POAFD)
to obtain approximations and numerical solutions to the fractional Laplacian
initial value problem and the extension problem of Caffarelli and Silvestre
(generalized Poisson equation). The method, as the first step, expands the
initial data function into a sparse series of the fundamental solutions with
fast convergence, and, as the second step, makes use the semigroup or the
reproducing kernel property of each of the expanding entries. Experiments show
effectiveness and efficiency of the proposed series solutions.
