Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse

Abstract

This paper proposes a global Padé approximation of the generalized Mittag-Leffler function E α,β (−x) with x ∈ [0, +∞). This uniform approximation can account for both the Taylor series for small arguments and asymptotic series for large arguments. Based on the complete monotonicity of the function E α,β (−x), we work out the global Padé approximation [1/2] for the particular cases {0 < α < 1, β > α}, {0 < α = β < 1}, and {α = 1, β > 1}, respectively. Moreover, these approximations are inverted to yield a global Padé approximation of the inverse generalized Mittag-Leffler function −L α,β (x) with x ∈ (0, 1/Γ(β)]. We also provide several examples with selected values α and β to compute the relative error from the approximations. Finally, we point out the possible applications using our established approximations in the ordinary and partial time-fractional differential equations in the sense of Riemann-Liouville.MSC 2010 : Primary 26A33; Secondary 33E12, 35S10, 45K05

Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches

“…The speed improvement is because in the direct integration method, each S(t j ) in Equation (32) Additionally, from Equations (33) and (34), the direct integration method can be used to calculate Mittag-Leffler-type functions and their derivatives. Figure 3 shows that the MLF calculated from the direct integration method agrees with that calculated by other methods in [49,50]. The Pade approximation method in [50] is only for subdiffusion.…”

“…Figure 3 shows that the MLF calculated from the direct integration method agrees with that calculated by other methods in [49,50]. The Pade approximation method in [50] is only for subdiffusion. Because the direct integration method does not cause overflow, it can be a useful method for calculating Mittag-Leffler-type functions.…”

“…(33) from the direct integration method, the Pade approximation method [50], and the method used in [49].…”

On the Analysis of Mixed-Index Time Fractional Differential Equation Systems

“…We find an analytical expression for the solution of the linear system that generalises the Mittag-Leffler expansion of a matrix and the solution of linear sequential fractional differential equations. We can use this result to derive new numerical methods that generalise the concept of exponential methods used in the approximation of the Mittag-Leffler matrix function (see [37][38][39], for example) and exponential integrators [40,41]. The second element would deal with developing numerical techniques for the integration component that incorporates the integral of a function times a Green function.…”

“…define the asymptotic stability boundary; see also (38). In order to make this more specific, let m 1 = m 2 = 1 and:…”