Volume 22, Issue 5

This survey applies the fractional asymptotic expansion to analyze some differential equations with boundary value problem. The method leads to the approximate solution in a wide range of applications, and avoids the limitations of algorithms based on Taylor expansion and the perturbation technique. The new method gives approximation series efficiently and overcomes the problems revealed by other analytical schemes that were proposed in the literature.

This paper presents how to design as simply as possible a robust MIMO controller for a refrigeration system with a 2 inputs, 2 outputs and a nonlinear model. Even if the proposed method starts from the CRONE MIMO approach, the analysis of the linear MIMO behavior of the system model shows that its MIMO controller can be decentralized and designed in two simple steps based on the SISO CRONE methodology. Simulation results show the good control performance and robust stability for a wide set of operating points.

We study embeddings of weighted local and consequently global generalized Morrey spaces defined on a quasi-metric measure set ( X , d , μ ) of general nature which may be unbounded, into Lebesgue spaces L s ( X ), 1 ≤ s ≤ p < ∞. The main motivation for obtaining such an embedding is to have an embedding of non-separable Morrey space into a separable space. In the general setting of quasi-metric measure spaces and arbitrary weights we give a sufficient condition for such an embedding. In the case of radial weights related to the center of local Morrey space, we obtain an effective sufficient condition in terms of (fractional in general) upper Ahlfors dimensions of the set X . In the case of radial weights we also obtain necessary conditions for such embeddings of local and global Morrey spaces, with the use of (fractional in general) lower and upper Ahlfors dimensions. In the case of power-logarithmic-type weights we obtain a criterion for such embeddings when these dimensions coincide.

Weighted spaces of continuous functions are introduced such that Weyl fractional integrals with orders from any finite nonnegative interval define equicontinuous sets of continuous linear endomorphisms for which the semigroup law of fractional orders is valid. The result is obtained from studying continuity and boundedness of convolution as a bilinear operation on general weighted spaces of continuous functions and measures.

We study fractional smoothness of measures on ℝ k , that are images of a Gaussian measure under mappings from Gaussian Sobolev classes. As a consequence we obtain Nikolskii–Besov fractional regularity of these distributions under some weak nondegeneracy assumption.

We establish necessary and sufficient conditions for the compactness of fractional integral operators from L p ( X , μ ) to L q ( X , μ ) with 1 < p < q < ∞, where μ is a measure on a quasi-metric measure space X . As an application we obtain criteria for the compactness of fractional integral operators defined in weighted Lebesgue spaces over bounded domains of the Euclidean space ℝ n with the Lebesgue measure, and also for the fractional integral operator associated to rectifiable curves of the complex plane.

The paper [5] by R. Garrappa, S. Rogosin, and F. Mainardi, entitled “On a generalized three-parameter Wright function of the Le Roy type” and published in Fract. Calc. Appl. Anal . 20 (2017), 1196–1215, ends up leaving the open question concerning the range of the parameters α , β and γ for which Mittag-Leffler functions of Le Roy type Fα,β(γ)$\begin{array}{} F_{\alpha, \beta}^{(\gamma)} \end{array}$ are completely monotonic. Inspired by the 1948 seminal H. Pollard’s paper which provides the proof of the complete monotonicity of the one-parameter Mittag-Leffler function, the Pollard approach is used to find the Laplace transform representation of Fα,β(γ)$\begin{array}{} F_{\alpha, \beta}^{(\gamma)} \end{array}$ for integer γ = n and rational 0 < α ≤ 1/ n . In this way it is possible to show that the Mittag-Leffler functions of Le Roy type are completely monotone for α = 1/ n and β ≥ ( n + 1)/(2 n ) as well as for rational 0 < α ≤ 1/2, β = 1 and n = 2. For further integer values of n the complete monotonicity is tested numerically for rational 0 < α < 1/ n and various choices of β . The obtained results suggest that for the complete monotonicity the condition β ≥ ( n + 1)/(2 n ) holds for any value of n .

In this paper, we extend the applications of the method of upper and lower solutions for a class of nonlinear nabla fractional difference equations involving Caputo derivative. We obtain the existence of coupled minimal and maximal solutions which constructed by two monotone sequences. In order to illustrate our main results, we present two numerical examples in the end.

In this paper, we propose a novel approach for the numerical solution of fractional-order ordinary differential equations. The method is based on the infinite state representation of the Caputo fractional differential operator, in which the entire history of the state of the system is considered for correct initialization. The infinite state representation contains an improper integral with respect to frequency, expressing the history dependence of the fractional derivative. The integral generally has a weakly singular kernel, which may lead to problems in numerical computations. A reformulation of the integral generates a kernel that decays to zero at both ends of the integration interval leading to better convergence properties of the related numerical scheme. We compare our method to other schemes by considering several benchmark problems.

In this paper we deal with the following fractional Kirchhoff problem M∬Rn×Rn|u(x)−u(y)|p|x−y|n+spdxdyp−1(−Δ)psu=f(x,u)+λ|u|r−2u in Ω,u=0  in Rn∖Ω.$$\begin{array}{} \displaystyle \left\{ {\begin{array}{l} \left[M\left(\displaystyle \iint_{\mathbb R^n\times \mathbb R^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}} dx dy\right)\right]^{p-1}(-\Delta)^{s}_{p}u = f(x, u)+\lambda |u|^{r-2}u \\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \mbox{ in } \, \Omega, \\ \\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad u=0 \, \, ~\mbox{ in } \, \mathbb R^n\setminus \Omega. \end{array}} \right. \end{array}$$ Here Ω ⊂ ℝ n is a smooth bounded open set with continuous boundary ∂ Ω, p ∈ (1, +∞), s ∈ (0, 1), n > sp , (−Δ)ps$\begin{array}{} (-\Delta)^{s}_{p} \end{array}$ is the fractional p -Laplacian, M is a Kirchhoff function, f is a continuous function with subcritical growth, λ is a nonnegative parameter and r > ps∗$\begin{array}{} p^*_s \end{array}$, where ps∗=npn−sp$\begin{array}{} p^*_s=\frac{np}{n-sp} \end{array}$ is the fractional critical Sobolev exponent. By combining variational techniques and a truncation argument, we prove two existence results for this problem, provided that the parameter λ is sufficiently small.

This paper proposes a framework for modelling velocity profiles and suspended objects in non-Newtonian fluid environment. A setup is proposed to allow mimicking blood properties and arterial to venous dynamic flow changes. Navier-Stokes relations are employed followed by fractional constitutive equations for velocity profiles and flow. The theoretical analysis is performed under assumptions of steady and pulsatile flow conditions, with incompressible properties. The fractional derivative model for velocity and friction drag effect upon a suspended object are determined. Experimental data from such an object is then recorded in real-time and identification of a fractional order model performed. The model is determined from step input changes during pulsatile flow for velocity in the direction of the flow. Further on, this model can be employed for controller design purposes for velocity and position in pulsatile non-Newtonian fluid flow.

The aim of this paper is to design an algebraic fractional order differentiator for a class of commensurate fractional order linear systems modeled by the pseudo-state space representation. For this purpose, a new algebraic method is introduced by designing an operator which can transform the considered system into a fractional order integral equation by eliminating unknown initial conditions. Based on the obtained equation, the desired fractional derivative is exactly given by a new algebraic formula using a recursive way. Then, a digital fractional order differentiator is introduced in discrete noisy cases. Finally, numerical results are given to illustrate the accuracy and the robustness of the proposed method.

In this paper we study the Dirichlet eigenvalue problem −Δpu−ΔJ,pu=λ|u|p−2u in Ω,u=0 in Ωc=RN∖Ω.$$\begin{array}{} \displaystyle -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \text{ in } \Omega,\quad u=0 \, \text{ in } \, \Omega^c=\mathbb{R}^N\setminus\Omega. \end{array}$$ Here Ω is a bounded domain in ℝ N , Δ p u is the standard local p -Laplacian and Δ J , p u is a nonlocal p -homogeneous operator of order zero. We show that the first eigenvalue (that is isolated and simple) satisfies limp→∞$\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$(λ 1 ) 1/ p = Λ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u ∞ = limp→∞$\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$ u p , and find the limit problem that is satisfied in the limit.