Accurate Padé Global Approximations for the Mittag-Leffler Function, Its Inverse, and Its Partial Derivatives to Efficiently Compute Convergent Power Series

Abstract

The fractional derivative operator provides a mathematical means to concisely describe a heterogeneous and relatively complex system that exhibits non-local, power-law behavior. Discretization of a fractional partial differential equation is non-trivial and computationally intensive. Furthermore, the closed form solution, particularly in the case of the fractional time derivative, comes in the form of a convergent power series of the Mittag-Leffler type, which involves considerable computational burden for accurate and efficient calculation. Here, we extend the Pade global approximation technique in order to accurately compute the Mittag-Leffler function and its inverse operation. Furthermore, we introduce the use of the Pade technique for the partial derivatives of the Mittag-Leffler function, which holistically eliminates the need for the calculation of any convergent power series when fitting to data sets. The Pade approximations provide great utility, particularly in the field of diffusion-weighted magnetic resonance imaging, in which Mittag-Leffler function and its partial derivatives are computed on a voxel-wise basis, with each data set containing on the order of \(10^7\) voxels. Finally, the Pade approximation codes are provided freely for those interested in modeling and fitting data for processes encompassed by this convergent power series.