On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method

In this Letter, we present the homotopy analysis method (shortly HAM) for obtaining the numerical solutions of Laplace equation with Dirichlet and Neumann boundary conditions. The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. The comparison of the HAM results with the variational iteration method (shortly VIM) results is made. The HAM contains the auxiliary parameter ℏ, therefore we control with a simple way the convergence region of solution series.