Fractional differential equations: Change of variables and nonlocal symmetries

Abstract

The paper considers point changes of variables in integrals and fractional derivatives of various types. In the general case, such changes lead to the appearance of operators of fractional integro-differentiation of a function with respect to another function. The problem of extending the action of a group of point transformations to a given type of operators is solved, and the corresponding formulas for the continuation of the infinitesimal operator of the group are presented and proved. Using a simple example of an ordinary differential equation with a fractional derivative, we illustrate the application of continuation formulas to find some of its nonlocal symmetries and check them admitted by the equation.