We present a Fourier-Bessel-based spectral solver for the Cauchy problem featuring the polar Laplacian under uniform Dirichlet boundary conditions. Separate the angular modes using an azimuthal FFT and perform the discrete Hankel transform (DHT) on each mode along the radial direction to obtain the spectral coefficients. The two transformations are connected by numerical interpolation and cardinal interpolation. Analyze the boundary-dependent error bars of the DHT. The worst case is $\sim N^{-3/2}$ dominating methods and the best $\sim e^{-N}$ dominating numerical interpolation. Complexity is $O[N^3]$. Always solves linear equations using Bessel functions, which are eigenfunctions of the Laplacian operator. For nonlinear equations, integrate the solution using the time-sharing method. Examples are given to validate the method for two-dimensional wave equations, which are linear, and two nonlinear problems: the time-dependent Poiseuille flow and the flow of a Bose-Einstein condensate on a disk.
