Volume 18, Issue 3

Our aim is to study a new class of differential variational inequalities involving fractional derivatives. Using the fixed point approach, the existence of decay solutions to the mentioned problem is proved.

We perform a biomathematical analysis of a model of cellulose degradation with derivative of fractional order γ. In the theory of biopolymers division, the phenomenon of shattering remains partially unexplained by classical models of clusters’ fragmentation. Thus, we first examine the case where the breakup rate H is independent of the size of the cellulose chain breaking up, following by the case where H is proportional to the size of the cellulose chain. Both cases show that the evolution of the biopolymer sizes distribution is governed by a combination of higher transcendental functions, namely the Mittag-Leffler function, the further generalized G-function and the Pochhammer polynomial. In particular, this shows existence of an eigen-property, that is, the system describing fractional cellulose degradation contains replicated and partially replicated fractional poles, whose effects are given by these functions.

This paper is devoted to the study of some spreading properties of a prey-predator reaction-diffusion system where the diffusion term is replaced by the fractional Laplacian. We focus on the invasion of the introduced predator in some environment which is initially well-populated of prey. In contrast with the case of the standard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Lévy process, the front position is exponential in time.

The paper demonstrates the basic properties of the local fractional variation operators (termed fractal variation operators). The action of the operators is demonstrated for local characterization of Hölderian functions. In particular, it is established that a class of such functions exhibits singular behavior under the action of fractal variation operators in infinitesimal limit. The link between the limit of the fractal variation of a function and its derivative is demonstrated. The paper presents a number of examples, including the calculation of the fractional variation of Cauchy sequences leading to the Dirac’s delta-function.

This paper proposes a simple fractional-order derivative controller, based on an adaptive orthogonal signals generator, which permits both to reconstruct an unknown multi-sinusoidal disturbance and cancel its effect on the system output. An interesting feature is that the disturbance is removed by the generated internal signals with no additional dynamics in the cancellation algorithm. An opportune choice of the fractional-order controller guarantees the closed-loop stability of the system if the location of the plant frequency response at the estimated frequencies belongs to a halfplane passing through the origin of the complex plane, i.e. no information about the order of the system to be controlled, the relative degree, the nature of its poles and zeros, is required. The case of multi-sinusoidal disturbance is also analyzed. Simulations are presented that highlight the performances of the proposed method.

This paper outlines a reliable strategy to approximate the local stable manifold near a hyperbolic equilibrium point for nonlinear systems of differential equations of fractional order. Furthermore, the local behavior of these systems near a hyperbolic equilibrium point is investigated based on the fractional Hartman-Grobman theorem. The fractional derivative is described in the Caputo sense. The solution existence, uniqueness, stability and convergence of the proposed scheme is discussed. Finally, the validity and applicability of our approach is examined with the use of a solvable model method.

When do fractional differential equations have solutions on the half line that are bounded by the Mittag-Leffler function? This work answers the above question through fixed-point methods, providing a deeper understanding of the long term growth behaviour of solutions, in addition to advancing our knowledge on the existence and uniqueness of solutions. A YouTube video by the Author that complements this work can be found at http://tinyurl.com/FDE-ML.

The paper describes the stability area for the difference system (Δ α y)(n + 1 − α) = Ay(n), n= 0, 1, . . . , with the Caputo forward difference operator Δ α of a real order α ∈ (0, 1) and a real constant matrix A. Contrary to the existing result on this topic, our stability conditions are fully explicit and involve the decay rate of the solutions. Some comparisons with a difference system of the Riemann- Liouville type are discussed as well, including related consequences and illustrating examples.

In this paper we study fractional differential inclusions in the sense of Almgren. We begin with a discussion of multiple-valued functions in the Almgren sense and include the basic results needed to make the paper selfcontained. Sufficient background on the fractional calculus is provided to make the material accessible also to the non-specialist readers. Our main result gives sufficient conditions for the existence of at least one solution to the problem under investigation. In addition, we show that the solution set to the problem is compact.

The problem of time-optimal control of linear systems with fractional dynamics is treated in the paper from the convex-analytic standpoint. A linear system of fractional differential equations involving Riemann- Liouville derivatives is considered. A method to construct a control function that brings trajectory of the system to the terminal state in the shortest time is proposed in terms of attainability sets and their support functions

We consider the analytical solutions of multi-term time-space fractional reaction-diffusion equations on an infinite domain. The results are presented in a compact and elegant form in terms of the Mittag-Leffler functions. The importance of the derived results lies in the fact that numerous results on fractional reaction, fractional diffusion, fractional wave problems, and fractional telegraph equations scattered in the literature can be derived as special cases of the results presented in this paper.

We establish a nonexistence result of global solutions to the nonlinear evolution equation (|u| β ) tt − |ϑ| α H Δ H (|u| β ) + h(ϑ)(|u| q ) t = f(t, ϑ)|u| p + w(t, ϑ), ϑ ∈ H, where ΔH is the Kohn-Laplace operator on the (2N+1)-dimensional Heisenberg group H, |ϑ| H is the distance from ϑ to the origin, β, p, q > 0, α ≥ 0, f(t, v) ≥ 0, h(ϑ) and w(t, ϑ) are given functions. Next, we extend this result to the case of systems. Our technique of proof is based on Pohozaev’s nonlinear capacity method.

In this paper, we first establish a high-order numerical algorithm for α-th (0 < α < 1) order Caputo derivative of a given function f(t), where the convergence rate is (4 − α)-th order. Then by using this new formula, an improved difference scheme with high order accuracy in time to solve Caputo-type fractional advection-diffusion equation with Dirichlet boundary conditions is constructed. Finally, numerical examples are carried out to confirm the efficiency of the constructed algorithm.

In this paper we solve the initial value problem for the nonlocal diffusion generated by the space fractional derivative induced by the dyadic tilings of M. Christ on a space of homogeneous type. We consider the problems of pointwise and norm convergence to the initial data. The main tool is the use of the Haar system induced by a dyadic tiling, which is actually the set of eigenfunctions for the fractional derivative operator.

This study suggests that the power law decay of prime number distribution can be considered a sub-diffusion process, one of typical anomalous diffusions, and could be described by the fractional derivative equation model, whose solution is the statistical density function of Mittag-Leffler distribution. It is observed that the Mittag-Leffler distribution of the fractional derivative diffusion equation agrees well with the prime number distribution and performs far better than the prime number theory. Compared with the Riemann’s method, the fractional diffusion model is less accurate but has clear physical significance and appears more stable. We also find that the Shannon entropies of the Riemann’s description and the fractional diffusion models are both very close to the original entropy of prime numbers. The proposed model appears an attractive physical description of the power law decay of prime number distribution and opens a new methodology in this field.

The Caputo time-derivative is usually defined pointwise for well-behaved functions, say, for the continuously differentiable functions. Accordingly, in the publications devoted to the theory of the partial fractional differential equations with the Caputo derivatives, the functional spaces where the solutions are looked for are often the spaces of smooth functions that appear to be too narrow for several important applications. In this paper, we propose a definition of the Caputo derivative on a finite interval in the fractional Sobolev spaces and investigate it from the operator theoretic viewpoint. In particular, some important equivalences of the norms related to the fractional integration and differentiation operators in the fractional Sobolev spaces are given. These results are then applied for proving the maximal regularity of the solutions to some initial-boundary-value problems for the time-fractional diffusion equation with the Caputo derivative in the fractional Sobolev spaces.

In this paper continuous time and discrete random walk models approximating diffusion processes associated with time-fractional and spacedistributed order differential equations are studied. Stochastic processes associated with the considered equations represent time-changed processes, where the time-change process is the inverse to a Levy’s stable subordinator with the stability index β ∈ (0, 1). In the paper the convergence of modeled continuous time and discrete random walks to time-changed processes associated with distributed order fractional diffusion equations are proved using an analytic method.