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
This paper presents some sufficient and necessary conditions for reducing the nonlinear stochastic differential equations (SDEs) with fractional Brownian motion (fBm) to the linear SDEs. The explicit solution of the reduced equation is computed by its integral equation or the variation of parameters technique. Two illustrative examples are provided to demonstrate the applicability of the proposed approach.
This paper reports a new fractional-order Lorenz-like system with one saddle and two stable node-foci. First, some sufficient conditions for local stability of equilibria are given. Also, this system has a double-scroll chaotic attractor with effective dimension being less than three. The minimum effective dimension for this system is estimated as 2.967. It should be emphasized that the linear differential equation in fractional-order Lorenz-like system seems to be less "sensitive" to the damping, introduced by a fractional derivative, than two other nonlinear equations. Furthermore, mixed synchronization of this system is analyzed with the help of nonlinear feedback control method. The first two pairs of state variables between the interactive systems are anti-phase synchronous, while the third pair of state variables is complete synchronous. Numerical simulations are performed to verify the theoretical results.
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