Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24495320240701Variational problem, Lagrangian and $\mu$-conservation law of the generalized Rosenau-type equation257265519710.22060/ajmc.2023.22352.1154ENKhodayarGoodarziDepartment of Mathematics, Broujerd Branch, Islamic Azad University, Broujerd, IranJournal Article20230421The goal of this article is to compute conservation law, Lagrangian and $\mu$-conservation law of the generalized Rosenau-type equation using the homotopy operator, the $\mu$-symmetry method and the variational problem method. The generalized Rosenau-type equation includes the generalized Rosenau equation, the generalized Rosenau-RLW equation and the generalized Rosenau-KdV equation, which admits the third-order Lagrangian. The article also compares the conservation law and the $\mu$-conservation law of these three equation. https://ajmc.aut.ac.ir/article_5197_8e0d7e98e9d0ca66404437763f4d7253.pdf