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In this paper we present a pseudospectral method in the disk. Unlike themethods known until now, the disk is not duplicated. Moreover, we solve theLaplace equation subjected to nonhomogeneous Dirichlet, Neumann and Robinboundary conditions and the biharmonic equation subjected to nonhomogeneousDirichlet conditions by only using the elements of the correspondingdifferentiation matrices. It is worth noting that we don not use anyquadrature, do not need to solve any decoupled system of ordinary differentialequations, do not use any pole condition and do not require any lifting. Wesolve several numerical examples showing that the spectral convergence is beingmet. The pseudospectral method developed in this paper can be applied toestimate Sherwood numbers integrating the mass flux to the disk and it can beeasily implemented to solve Lotka-Volterra systems and nonlinear problemsinvolving chemical reactions.

Polar differentiation matrices for the Laplace equation in the disk  subjected to nonhomogeneous Dirichlet, Neumann and Robin boundary conditions  and the biharmonic equation subjected to nonhomogeneous Dirichlet conditions