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 $E_{\alpha ,\beta }(-x)$ with $x\in [0,+\infty )$. 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 $E_{\alpha ,\beta }(-x)$, we work out the global Padé approximation [1/2] for the particular cases $\{0<\alpha <1,\beta >\alpha \}$, $\{0<\alpha =\beta <1\}$, and $\{\alpha =1,\beta >1\}$, respectively. Moreover, these approximations are inverted to yield a global Padé approximation of the inverse generalized Mittag-Leffler function $-L_{\alpha ,\beta }(x)$ with $x\in (0,1/\Gamma (\beta )]$. We also provide several examples with selected values $\alpha$ and $\beta$ 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.

15 pages, 4 figures, 1 table

References in corpus (3)

• Numerical evaluation of two and three parameter Mittag-Leffler functions
• Time-fractional derivatives in relaxation processes: a tutorial survey
• The Mittag-Leffler function

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