  In this paper we present a pseudospectral method in the disk. Unlike the
methods known until now, the disk is not duplicated. Moreover, we solve the
Laplace equation subjected to nonhomogeneous Dirichlet, Neumann and Robin
boundary conditions and the biharmonic equation subjected to nonhomogeneous
Dirichlet conditions by only using the elements of the corresponding
differentiation matrices. It is worth noting that we don not use any
quadrature, do not need to solve any decoupled system of ordinary differential
equations, do not use any pole condition and do not require any lifting. We
solve several numerical examples showing that the spectral convergence is being
met. The pseudospectral method developed in this paper can be applied to
estimate Sherwood numbers integrating the mass flux to the disk and it can be
easily implemented to solve Lotka-Volterra systems and nonlinear problems
involving chemical reactions.
