optimal third-kind chebyshev collocation algorithm for solving beam-type micro- and nanoscale bvps

In this paper, we develop a numerical scheme for solving nonlinear boundary value problems (bvps) arising in cantilever-type micro-electromechanical (mems) and nano-electromechanical (nems) systems. the method is based on a novel third-kind chebyshev collocation approach. we derive an operational matrix of derivatives using shifted third-kind chebyshev polynomials, which enables efficient spectral approximations of the governing equations. numerical experiments confirm the accuracy and efficiency of the proposed method in handling the nonlinearities present in mems/nems actuator models.