Symmetry analysis, conservation laws and invariant solutions of the time-fractional equal width wave equation

Lie symmetry analysis provides an efficient method to get the analytical and exact solutions of the fractional differential equations. In this paper, we discuss Lie symmetry analysis for the time-fractional equal width wave equation with Riemann–Liouville derivative. This equation is used to describe the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. By employing classical and nonclassical Lie symmetry analysis and some technical calculations, new infinitesimal generators are obtained. Then we reduce the fractional equal width wave equation to the ordinary fractional differential equation by changing the coordinates and find invariant solutions to this equation. By means of Ibragimov’s new conservation theorem and the generalization of the Noether operators, we construct the conservation laws for the equation. Also, we derive the adjoint equation and infinitesimal generator associated with Lie symmetries of the underlying equation and we reduce this equation to the ordinary fractional differential equation. In the reduced equations the derivative is in Erdelyi–Kober sense.