A new analytic method with a convergence-control parameter for solving nonlinear problems

In this paper, a new analytic method with a convergence-control parameter $c$ is first proposed. The parameter $c$ is used to adjust and control the convergence region and rate of the resulting series solution. It turns out that the convergence region and rate can be greatly enlarged by choosing a proper value of $c$. Furthermore, a numerical approach for finding the optimal value of the convergence-control parameter is given. At the same time, it is found that the traditional Adomian decomposition method is only a special case of the new method. The effectiveness and applicability of the new technique are demonstrated by several physical models including nonlinear heat transfer problems, nano-electromechanical systems, diffusion and dissipation phenomena, and dispersive waves. Moreover, the ideas proposed in this paper may offer us possibilities to greatly improve current analytic and numerical techniques.