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Our aim in this paper is to investigate the asymptotic behavior of solutions of linear fractional differential equations. First, we show that the classical Lyapunov exponent of an arbitrary nontrivial solution of a bounded linear fractional differential equation is always nonnegative. Next, using the Mittag-Leffler function, we introduce an adequate notion of fractional Lyapunov exponent for an arbitrary function. We show that for a linear fractional differential equation, the fractional Lyapunov spectrum which consists of all possible fractional Lyapunov exponents of its solutions provides a good description of asymptotic behavior of this equation. Consequently, the stability of a linear fractional differential equation can be characterized by its fractional Lyapunov spectrum. Finally, to illustrate the theoretical results we compute explicitly the fractional Lyapunov exponent of an arbitrary solution of a planar time-invariant linear fractional differential equation.

In this paper a fractional differential equation of the Euler-Lagrange/Sturm-Liouville type is considered. The fractional equation with derivatives of order α ∈ (0, 1] in the finite time interval is transformed to the integral form. Next the numerical scheme is presented. In the final part of this paper examples of numerical solutions of this equation are shown. The convergence of the proposed method on the basis of numerical results is also discussed.

What is the most efficient search strategy for the random located target sites subject to the physical and biological constraints? Previous results suggested the Lévy flight is the best option to characterize this optimal problem, however, which ignores the understanding and learning abilities of the searcher agents. In this paper we propose the Continuous Time Random Walk (CTRW) optimal search framework and find the optimum for both of search length’s and waiting time’s distributions. Based on fractional calculus technique, we further derive its master equation to show the mechanism of such complex fractional dynamics. Numerous simulations are provided to illustrate the non-destructive and destructive cases.

In the present manuscript we analyze non-linear multi-order fractional differential equation $$L\left( D \right)u\left( t \right) = f\left( {t,u\left( t \right)} \right), t \in \left[ {0,T} \right], T > 0,$$ where $$L\left( D \right) = \lambda _n ^c D^{\alpha _n } + \lambda _{n - 1} ^c D^{\alpha _{n - 1} } + \cdots + \lambda _1 ^c D^{\alpha _1 } + \lambda _0 ^c D^{\alpha _0 } ,\lambda _i \in \mathbb{R}\left( {i = 0,1, \cdots ,n} \right),\lambda _n \ne 0, 0 \leqslant \alpha _0 < \alpha _1 < \cdots < \alpha _{n - 1} < \alpha _n < 1,$$ and c D α denotes the Caputo fractional derivative of order α. We find the Greens functions for this equation corresponding to periodic/anti-periodic boundary conditions in terms of the two-parametric functions of Mittag-Leffler type. Further we prove existence and uniqueness theorems for these fractional boundary value problems.

This paper is concerned with the existence and uniqueness of solutions for a coupled system of Hadamard type fractional differential equations and integral boundary conditions. We emphasize that much work on fractional boundary value problems involves either Riemann-Liouville or Caputo type fractional differential equations. In the present work, we have considered a new problem which deals with a system of Hadamard differential equations and Hadamard type integral boundary conditions. The existence of solutions is derived from Leray-Schauder’s alternative, whereas the uniqueness of solution is established by Banach’s contraction principle. An illustrative example is also included.

A family of generalized Erdélyi-Kober type fractional integrals is interpreted geometrically as a distortion of the rotationally invariant integral kernel of the Riesz fractional integral in terms of generalized Cassini ovaloids on R N. Based on this geometric point of view, several extensions are discussed.

A fractional Stefan’s problem with a boundary convective condition is solved, where the fractional derivative of order α ∈ (0, 1) is taken in the Caputo sense. Then an equivalence with other two fractional Stefan’s problems (the first one with a constant condition on x = 0 and the second with a flux condition) is proved and the convergence to the classical solutions is analyzed when α ↗ 1 recovering the heat equation with its respective Stefan’s condition.

In this paper, we extend a reliable modification of the Adomian decomposition method presented in [34] for solving initial value problem for fractional differential equations. In order to confirm the applicability and the advantages of our approach, we consider some illustrative examples.

In this paper we introduce and we study fractional Sobolev type spaces associated with a singular second order differential operator on (0, ∞) and propose several results. As applications we give certain properties including estimates for the solution of the generalized wave equation and generalized fractional operator.

From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument due to the presence of Erdélyi-Kober fractional integrals in our operator. We present solutions, both singular and regular in the time origin, that are locally integrable and completely monotone functions in order to be consistent with the physical phenomena described by non-negative relaxation spectral distributions.

The aim of this paper is to provide a survey on the recently obtained results which are useful in time response analysis of fractional-order control systems. In this survey, at first some results on error signal analysis in fractional-order control systems are presented. Then, some previously obtained results which are helpful for system output analysis in fractional-order control systems are summarized. In addition, some results on the analysis of the control signal and the system response to the load disturbances in fractional-order control systems are reviewed.

This paper deals with the application of adaptive fractional order control to networked control systems (NCSs) to compensate the effects of time-varying network-induced delays. In essence, it adapts both the gains and the orders of a local PIαDμ controller in accordance with the current network condition in order to avoid a decreased control performance. A frequency domain framework is provided to analyze the system stability on the basis of the switching systems theory. The velocity control of a servomotor through the Internet is given to show the effectiveness of the proposed adaptive controller, including a comparison with non- and gain scheduled controllers.

In this paper, the initial-boundary-value problems for the one-dimensional linear and non-linear fractional diffusion equations with the Riemann-Liouville time-fractional derivative are analyzed. First, a weak and a strong maximum principles for solutions of the linear problems are derived. These principles are employed to show uniqueness of solutions of the initial-boundary-value problems for the non-linear fractional diffusion equations under some standard assumptions posed on the non-linear part of the equations. In the linear case and under some additional conditions, these solutions can be represented in form of the Fourier series with respect to the eigenfunctions of the corresponding Sturm-Liouville eigenvalue problems.

In this paper, the authors consider a nonlinear fractional boundary value problem defined on a star graph. By using a transformation, an equivalent system of fractional boundary value problems with mixed boundary conditions is obtained. Then the existence and uniqueness of solutions are investigated by fixed point theory.

In this paper, we establish that the controllability and observability properties of fractional dynamical systems in a finite dimensional space are dual. Using this duality result and the Mittag-Leffler matrix function, we propose the stabilizability of fractional MIMO (Multiple-input Multipleoutput) systems. Some numerical examples are provided to show the effectiveness of the obtained results.

The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes.

In the last decades fractional calculus (FC) became an area of intensive research and development. This paper goes back and recalls important pioneers that started to apply FC to scientific and engineering problems during the nineteenth and twentieth centuries. Those we present are, in alphabetical order: Niels Abel, Kenneth and Robert Cole, Andrew Gemant, Andrey N. Gerasimov, Oliver Heaviside, Paul Lévy, Rashid Sh. Nigmatullin, Yuri N. Rabotnov, George Scott Blair.