Abstract
Exact solutions were derived for a time-fractional Levi equation with Riemann–Liouville fractional derivative. The methods involve, first, the reduction of the time-fractional Levi equation to fractional ordinary differential equations with Erdélyi-Kober fractional differential operator with respect to point symmetry groups, and second, use of the invariant subspace to reduce the time-fractional Levi equation into a system of fractional ordinary differential equations, which were solved by the symmetry group method. The obtained explicit solutions have interesting analytic behaviors connected with blow-up and dispersion. The conservation laws generated by the point symmetries of the time-fractional Levi equation are shown via nonlinear self-adjointness method.
Highlights
Exact solutions were derived for a time-fractional Levi equation with Riemann–Liouville fractional derivative
It has been shown that the fractional order models are much more adequate than the integer order models
In [14], the authors presented the algorithm for the systematic calculation of Lie point symmetries for Fractional differential equations (FDEs), which was implemented in the MAPLE package FracSym
Summary
Fractional differential equations (FDEs) [1,2,3,4] are of considerable importance due to their connection with real-world problems that depend on the instant time and on the previous time, in particular, modeling the phenomena by means of fractals, random walk processes, control theory, signal processing, acoustics, etc. Inspired by the preliminary results, several works [23,24] were devoted to investigating the conserved vectors to FDEs. The method of invariant subspace initially presented by Galaktionov and his collaborators [25] is an effective way to perform reductions of nonlinear PDEs to finite-dimensional dynamical systems. There it is concluded that the dimension of invariant subspaces admitted by such kind of operator is bounded by 2mk + 1 Such an estimate was further extended to a two-component nonlinear vector differential operators with two different orders [28].
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