On exact solution of Poisson equation with boundary conditions by the homotopy analysis method

In this paper, we use the homotopy analysis method (shortly HAM) to obtain the numerical solutions of Poisson equation with 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 exact solutions is made, the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of solution series.