A new variable-order fractional derivative with non-singular Mittag–Leffler kernel: application to variable-order fractional version of the 2D Richard equation

Abstract

In this study, to overcome the limitations of non-singular fractional derivatives in the Caputo–Fabrizio and Atangana–Baleanu senses (especially in dealing with the variable-order (VO) fractional calculus), we introduce a new non-singular VO fractional derivative with Mittag–Leffler function as its kernel. Some useful results are derived from this fractional derivative. Moreover, this fractional derivative is used for introducing the VO fractional version of the 2D Richard equation. A meshless scheme based on the thin plate spline radial basis functions (RBFs) is developed for solving this equation. The validity of the formulated method is investigated through three numerical examples.