Volume 18, Issue 6

We present non-existence results for systems of non-local in space hyperbolic equations, for systems of non-local in space parabolic equations, and for systems of non-local in space hyperbolic equations with linear damping terms. Our method of proof is based on the test function method with a help of a convexity inequality recently proved in [2].

Maxwell’s equations modified for dielectric permittivity lead to a modified wave equation similar to the viscoelastic wave equation for solid materials. In the process of the modification, certain terms arise that do not have traditional Laplace transforms. This paper addresses the necessary conditions needed to remove these problematic terms from the analysis.

In this paper Ionic Polymer-Metal Composites (IPMC) actuators motion control is addressed by using a new approach based on the modified fractional super twisting control strategy. We present the theoretical aspect of the proposed control strategy and then the performances of the controlling strategy are validated for an IPMC actuator by using an ad hoc developed experimental setup. The reported results show that the standard fractional super twisting control and the proposed modified fractional super twisting control over perform standard PI controllers.

In this paper we are concerned with the controllability of control systems governed by a fractional differential equation in Banach spaces. Using the properties of the Mittag-Leffler function we generalize to these systems a result of Korobov and Rabakh, which was established for first order systems. We apply our results to study the controllability of a system modeled by a fractional integral equation in a Hilbert space.

This paper concerns the dissipativity and stability of the Caputo nonlinear fractional functional differential equations (F-FDEs) with order 0 < α < 1. The fractional generalization of the Halanay-type inequality is proposed, which plays a central role in studies of stability and dissipativity of F-FDEs. Then the dissipativity and the absorbing set are derived under almost the same assumptions as the classical integer-order functional differential equations (FDEs). The asymptotic stability of F-FDEs are also proved under the one-sided Lipschitz conditions. Those extend the corresponding properties from integer-order FDEs to the Caputo fractional ones. The results can also be directly applied to some special cases of fractional nonlinear equations, such as the fractional delay differential equations (F-DDEs), fractional integro-differential equations (F-IDEs) and fractional delay integro-differential equations (F-DIDEs). The fractional Adams-Bashforth-Moulton algorithm is employed to simulate the F-FDEs, and several numerical examples are given to illustrate the theoretical results.

We present a numerical algorithm for solving nonlinear fractional boundary value problems of order n, n ∈ ℕ. The Bernstein polynomials (BPs) are redefined in a fractional form over an arbitrary interval [a, b]. Theoretical results related to the ractional Bernstein polynomials (FBPs) are proven. The well-known shooting technique is extended for the numerical treatment of nonlinear fractional boundary value problems of arbitrary order. The initial value problems were solved using a collocation method with collocation points at the location of the local maximum of the FBPs. Several examples are discussed to illustrate the efficiency and accuracy of the present scheme.

In this paper, a sliding mode control strategy is discussed for a class of nonlinear mechanical sub-systems with varying trajectory dynamics. The proposed class of sub-systems are represented in this simulation example by a two link robot actuator/manipulator. The fractional order is introduced in the setpoint definition as to represent changes in the desired trajectory of this sub-system. Furthermore, the same order is used to adapt the control law to the new dynamics. Uncertainties are introduced in the model used for the control law, hence robustness is intrinsic.

The space-time fractional advection diffusion equations are linear partial pseudo-differential equation with spatial fractional derivatives in time and in space and are used to model transport at the earth surface. We use the implicit difference scheme, the theta-method, to find the approximation solution of these equations in the long run. The proofs of stability of the difference scheme of each models are given. We compare the numerical results of these models for different values of the space and the time fractional orders and for different values of theta.

In this paper we study the existence of solutions for a fractional differential inclusion with impulsive effects. By using the nonsmooth critical point theory, we give some new criteria to guarantee that the hemivariational inequality problems produce at least two nontrivial solutions.

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.

The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions e inωx are replaced by the Mittag-Leffler functions E α (i(nωx) α ) , over the interval [0,M α /ω] where 0 < ω < ∞ and M α > 0 is the period of the function E α (ix α ) , i.e., E α (ix α ) = E α (i(x +M α ) α ) . He showed that any smooth periodic function f with period M α /ω can be expanded in a Fourier-type series. We will show that the only possible period of the function E α (ix α ) is M α = 0; hence the invalidity of any Fourier-type series expansion of f.

Atmospheric temperatures characterize Earth as a slow dynamics spatiotemporal system, revealing long-memory and complex behavior. Temperature time series of 54 worldwide geographic locations are considered as representative of the Earth weather dynamics. These data are then interpreted as the time evolution of a set of state space variables describing a complex system. The data are analyzed by means of multidimensional scaling (MDS), and the fractional state space portrait (fSSP). A centennial perspective covering the period from 1910 to 2012 allows MDS to identify similarities among different Earth’s locations. The multivariate mutual information is proposed to determine the “optimal” order of the time derivative for the fSSP representation. The fSSP emerges as a valuable alternative for visualizing system dynamics.