This paper systematically investigates the Lie group analysis method of the time-fractional regularized long-wave (RLW) equation with Riemann–Liouville fractional derivative. The vector fields and similarity reductions of the time-fractional (RLW) equation are obtained. It is shown that the governing equation can be transformed into a fractional ordinary differential equation with a new independent variable, where the fractional derivatives are in Erdèlyi–Kober sense. Furthermore, the explicit analytic solutions of the time-fractional (RLW) equation are obtained using the power series expansion method. Finally, some graphical features were presented to give a visual interpretation of the solutions.

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This work is devoted to study the limit behavior of weak solutions of an elliptic problem with variable exponent, in a containing structure, of an oscillating nanolayer of thickness and periodicity parameter depending on ε . The generalized Sobolev space is constructed, and the epiconvergence method is considered to find the limit problem with interface conditions.