Pınar, Zehra2024-10-292024-10-2920222147-6268https://doi.org/10.36753/mathenot.799596https://search.trdizin.gov.tr/tr/yayin/detay/1186892https://hdl.handle.net/20.500.11776/12516The size-dependent longitudinal and torsional dynamic problems for small-scaled rods have importance in two-phase media. The special case of the elastic rod equation such as magneto-electro circular equation are seen in the literature commonly, but in this work, the generalized form of the nonlinear elastic circular equation, which was not studied in the literature, is considered. The exact solutions are obtained via Mathieu approximation method with a novel proposed ansatz. Obtained solutions are discussed and illustrated in details. We believe that the proposed results will be key part of further analytical and numerical studies for waves in the dispersive medium with reaction. © MSAEN.en10.36753/mathenot.799596info:eu-repo/semantics/openAccessMathieu approximation methodthe elastic rod equationtravelling wave solutionsArticle10145492-s2.0-852049943791186892N/A