Approximate conservation laws for fractional differential equations

An approach to construction of approximate conservation laws for a certain class of fractional differential equations (FDEs) is proposed. It is assumed that the FDEs can contain the left-sided and the right-sided Riemann-Liouville as well as the Caputo fractional derivatives, and the orders of all these fractional derivatives have the same small deviation from the nearest integers. Then a corresponding small parameter can be introduced into the orders of fractional differentiation. First order expansions of the Riemann-Liouville and Caputo fractional derivatives with respect to this small parameters are presented. Using these expansions, a FDE belonging to the considered class can be approximated by a perturbed integer-order differential equation with the small parameter. It is shown that the concept of nonlinear self-adjointness is applicable for such approximate equations without approximate Lagrangians. This gives the opportunity to construct approximate conservation laws for such perturbed equations using their approximate symmetries. The proposed approach is illustrated by several examples of finding approximate conservation laws for nonlinear ordinary and partial FDEs without Lagrangians.