Invariance properties, exact and explicit solutions of time-fractional Gear–Grimshaw model

Abstract

In this study, we consider a time-fractional shallow water wave model which was introduced by Gear and Grimshaw. This model represents the dynamics of two-layered shallow water waves in oceans and rivers. Also it covers many shallow water wave models with special values taken to parameters. By performing the group invariant analysis approach for the governing system with Riemann–Liouville derivative, symmetry generators were found. Similarity reduction was constructed by Erdelyi–Kober sense. In addition, by adapting the Sine-Gordon expansion method, we find some exact traveling wave solutions. Furthermore, explicit solutions in form of power series are examined via the residual power series method. Exact and approximate solutions are compared for some [Formula: see text] values and these are given as tables.