# 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

## TL;DR

This paper introduces a novel pseudospectral method for solving Laplace and biharmonic equations in a disk without duplication, using differentiation matrices exclusively, achieving spectral convergence and applicable to various nonlinear problems.

## Contribution

The method uniquely avoids duplication, quadrature, decoupled systems, pole conditions, and lifting, simplifying computations for boundary value problems in the disk.

## Key findings

- Spectral convergence demonstrated through numerical examples
- Method efficiently solves boundary conditions without additional techniques
- Applicable to nonlinear and reaction-diffusion systems

## Abstract

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.