University of Guilan Journal of Mathematical Modeling 2345-394X 13 4 2025 12 01 Optimal third-kind Chebyshev collocation algorithm for solving beam-type micro- and nanoscale BVPs 883 898 8871 10.22124/jmm.2025.30690.2747 EN Youssri Hassan Youssri Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt Ahmed Gamal Atta Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt Journal Article 2025 05 17 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. https://jmm.guilan.ac.ir/article_8871_5590dcc3bb2a985505795664edfb1fbd.pdf