On the calculation of approximate symmetries of fractional differential equations

A new algorithm for finding approximate symmetries for fractional differential equations with the Riemann–Liouville and Gerasimov–Caputo fractional derivatives, the order of which is close to an integer, is proposed. The algorithm is based on the expansion of the fractional derivative into a series with respect to a small parameter isolated from the order of fractional differentiation. In the first-order, such an expansion contains a nonlocal integro-differential operator with a logarithmic kernel.
As a result, the original fractional differential equation is approximated by an integro-differential equation with a small parameter for which approximate symmetries can be found. A theorem is proved about the form of prolongation of one-parameter point transformations group to a new variable represented by a nonlocal operator included in the expansion of the fractional derivative. Knowing such a prolongation allows us to apply an approximate invariance criterion to the equation under consideration.
The proposed algorithm is illustrated by the problem of finding approximate symmetries for a nonlinear fractional filtration equation of subdiffusion type. It is shown that the dimension of approximate symmetries algebra for such an equation is significantly larger than the dimension of the algebra of exact symmetries. This fact opens the possibility of constructing a large number of approximately invariant solutions. Also, it is shown that the algorithm makes it possible to find nonlocal approximate symmetries of a certain type. This possibility is illustrated on a linear fractional differential subdiffusion equation.