A novel analytic approximation method with a convergence acceleration parameter for solving nonlinear problems

Abstract

In this paper, a new analytic method with a convergencecontrol 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.Mathematics Subject Classification (2010). 34E10; 35B20; 76M45.

Periodic Solution of Nonlinear Conservative Systems

“…Nonperturbation methods also have various limitations. For instance, a study [51] showed that the Adomian decomposition method does not converge to the correct solution in some cases, and the study proposed an optimal convergence acceleration parameter to deal with this issue.…”