Solution of the model of beam-type micro- and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems

Abstract

In this paper we solve the common nonlinear boundary value problems (BVPs) of cantilever-type micro-electromechanical system (MEMS) and nano-electromechanical system (NEMS) using the distributed parameter model by the Duan–Rach modified Adomian decomposition method (ADM). The nonlinear BVPs that are investigated include the cases of the single and double cantilever-type geometries under the influence of the intermolecular van der Waals force and the quantum Casimir force for appropriate distances of separation. The new Duan–Rach modified ADM transforms the nonlinear BVP consisting of a nonlinear differential equation subject to appropriate boundary conditions into an equivalent nonlinear Fredholm–Volterra integral equation before designing an efficient recursion scheme to compute approximate analytic solutions without resort to any undetermined coefficients. The new approach facilitates parametric analyses for such designs and the pull-in parameters can be estimated by combining with the Padé approximant. We also consider the accuracy and the rate of convergence for the solution approximants of the resulting Adomian decomposition series, which demonstrates an approximate exponential rate of convergence. Furthermore we show how to easily achieve an accelerated rate of convergence in the sequence of the Adomian approximate solutions by applying Duan's parametrized recursion scheme in computing the solution components. Finally we compare the Duan–Rach modified recursion scheme in the ADM with the method of undetermined coefficients in the ADM for solution of nonlinear BVPs to illustrate the advantages of our new approach over prior art.