A convergent two-step method to solve a fractional extension of the Rosenau-Kawahara system

In this work, we extend the Rosenau-Kawahara equation (RKE) to the fractional scenario by using space-fractional operators of the Riesz kind. We prove that this system has functional quantities similar to the energy and the mass of the integer-order model, and we show that they are conserved. A discretized form of the model is proposed along with discretized functionals for the energy and the mass. We prove that these quantities are conserved through time. The solvability of the model is proved via Browder's theorem. Moreover, we establish the properties of second-order convergence, stability and consistency. The numerical model is implemented using a fixed-point approach. Our computations demonstrate that the model conserves the energy and the mass, in agreement with our analysis. This is the first article in the literature in which a conservative scheme for a conservative fractional RKE is propose and rigorously analyzed for the conservation of mass and energy, positivity of energy, existence of solutions, consistency, stability and second-order convergence.