Leibniz-type rule of variable-order fractional derivative and application to build Lie symmetry framework

In this paper, we first study some properties of the variable-order fractional derivative defined by the Caputo fractional derivative and particularly present a Leibniz-type rule, which makes the variable-order fractional derivative to be expressed as an infinite sum of integer-order derivatives. Then we use such properties to build a Lie symmetry framework for a class of scalar variable-order fractional partial differential equations and show that such type of equations has an elegant symmetry structure, which facilitates us to explore the symmetry properties. By means of the Lie symmetry framework and the symmetry structure, we perform a Lie symmetry classification of the variable-order fractional diffusion equations and construct the corresponding symmetry reductions and certain exact solutions. (C) 2022 Elsevier Inc. All rights reserved.