Review of approximation formulas for fractional derivatives of variable order

Fractional differential equations of variable order, depending on a time or space variable, are successfully used to study time and/or spatial dynamics. The purpose of this review is to consider the latest research and results related to the basic definitions and approximation formulas for fractional derivatives of variable order. The review begins with an examination of existing definitions based on various physical and practical knowledge. The following are formulas for discretizing fractional derivatives, since they play a key role in numerical modeling, providing an effective means of describing complex systems with long-term memory, heterogeneous parameters and non-local interactions. Their use not only improves the accuracy of models, but also facilitates the numerical solution of differential equations, which is essential for analyzing and predicting the behavior of real processes in various fields of science and engineering. This review is intended to help readers select appropriate definitions and discretization formulas to effectively solve specific physical and engineering problems.