Volume 21, Issue 5

Discussions under this title were held during a round table in frames of the International Conference “Fractional Differentiation and Its Applications” (ICFDA’18) held in 16-18 July 2018, Amman, The Hashemite Kingdom of Jordan, see details at http://conferences.ju.edu.jo/en/icfda2018/Home.aspx. The intention of this round table was to continue the useful traditions from the first conferences on the Fractional Calculus topics, to pose open problems, challenging hypotheses and questions “where to go”, to discuss them and try to find ways to resolve.

In this paper we survey the properties of the Schelkunoff modification of the Exponential integral and we generalize it with the Mittag-Leffler function. So doing we get a new special function (as far as we know) that may be relevant in linear viscoelasticity because of its complete monotonicity properties in the time domain. We also consider the generalized sine and cosine integral functions.

In this paper we investigate the existence and uniqueness of semilinear stochastic Volterra equations driven by multiplicative Lévy noise of pure jump type. In particular, we consider the equation du(t)=A∫0tb(t−s)u(s)dsdt+F(t,u(t))dt+∫ZG(t,u(t),z)η~(dz,dt)+∫ZLGL(t,u(t),z)ηL(dz,dt),t∈(0,T],u(0)=u0,$$\begin{array}{} \left\{ \begin{aligned} du(t) & = \left( A\int_0 ^t b(t-s) u(s)\,ds\right) \, dt + F(t,u(t))\,dt \\ & {} + \int_ZG(t,u(t), z) \tilde \eta(dz,dt) + \int_{Z_L}G_L(t,u(t), z) \eta_L(dz,dt),\, t\in (0,T],\\ u(0)&=u_0, \end{aligned} \right. \end{array} $$ where Z and Z L are Banach spaces, η̃ is a time-homogeneous compensated Poisson random measure on Z with intensity measure ν (capturing the small jumps), and η L is a time-homogeneous Poisson random measure on Z L independent to η̃ with finite intensity measure ν L (capturing the large jumps). Here, A is a selfadjoint operator on a Hilbert space H , b is a scalar memory function and F , G and G L are nonlinear mappings. We provide conditions on b , F G and G L under which a unique global solution exists. We also present an example from the theory of linear viscoelasticity where our result is applicable. The specific kernel b ( t ) = c ρ t ρ −2 , 1 < ρ < 2, corresponds to a fractional-in-time stochastic equation and the nonlinear maps F and G can include fractional powers of A .

We consider operator semigroups generated by Feller processes killed upon leaving a given domain. These semigroups correspond to Cauchy–Dirichlet type initial-exterior value problems in this domain for a class of evolution equations with (possibly non-local) operators. The considered semigroups are approximated by means of the Chernoff theorem. For a class of killed Feller processes, the constructed Chernoff approximation leads to a representation of the solution of the corresponding Cauchy–Dirichlet type problem by a Feynman formula, i.e. by a limit of n -fold iterated integrals of certain functions as n → ∞. Feynman formulae can be used for direct calculations, modelling of underlying dynamics, simulation of underlying stochastic processes. Further, a method to approximate solutions of time-fractional evolution equations is suggested. The method is based on connections between time-fractional and time-non-fractional evolution equations as well as on Chernoff approximations for the latter ones. This method leads to Feynman formulae for solutions of time-fractional evolution equations. A class of distributed order time-fractional equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy–Dirichlet type problems are obtained.

The paper is devoted to the development of control procedures with a guide for fractional order dynamical systems controlled under conditions of disturbances, uncertainties or counteractions. We consider a dynamical system which motion is described by ordinary fractional differential equations with the Caputo derivative of an order α ∈ (0, 1). For the case when the guide is, in a certain sense, a copy of the system, we propose a mutual aiming procedure between the original system and guide. The proof of proximity between motions of the systems is based on the estimate of the fractional derivative of the superposition of a convex Lyapunov function and a function represented by the fractional integral of an essentially bounded measurable function. This estimate can be considered as a generalization of the known estimates of such type. We give an example that illustrates the workability of the proposed control procedures with a guide.

We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic.

In this paper, we prove that Caputo type linear fractional evolution equations do not have nonconstant periodic solutions. Then, we study asymptotically periodic solutions of semilinear fractional evolution equations and establish existence and uniqueness results by using theory of semigroup and fixed point theorems. Finally, two examples are given to illustrate the theoretical results.

The classical wave equation is generalized within fractional framework, by using fractional derivatives of real and complex order in the constitutive equation, so that it describes wave propagation in one dimensional infinite viscoelastic rod. We analyze existence, uniqueness and properties of solutions to the corresponding initial-boundary value problem for generalized wave equation. Also, we provide a comparative analysis with the case of the same equation but considered on a bounded or half-bounded spatial domain. We conclude our investigation with a numerical example that illustrates obtained results.

We show a strong maximum principle and an Alexandrov-Bakelman-Pucci estimate for the weak solutions of a Cauchy problem featuring Caputo time-derivatives and non-local operators in space variables given in terms of Bernstein functions of the Laplacian. To achieve this, first we propose a suitable meaning of a weak solution, show their existence and uniqueness, and establish a probabilistic representation in terms of time-changed Brownian motion. As an application, we also discuss an inverse source problem.

In this paper we investigate the extension of the multiple Erdélyi-Kober fractional integral operator of Kiryakova to arbitrary complex values of parameters by the way of regularization. The regularization involves derivatives of the function in question and the integration with respect to a kernel expressed in terms of special case of Meijer’s G -function. An action of the regularized multiple Erdélyi-Kober operator on some simple kernels leads to decomposition formulas for the generalized hypergeometric functions. In the ultimate section, we define an alternative regularization better suited for representing the Bessel type generalized hypergeometric function p −1 F p . A particular case of this regularization is then used to identify some new facts about the positivity and reality of zeros of this function.

In this paper an extended generalized Mittag-Leffler function Eρ,σ,τδ,r,q,c(z;p)$\begin{array}{} \displaystyle E_{\rho,\sigma,\tau}^{\delta,r,q,c}(z;p) \end{array}$ and the corresponding fractional integral operator εa+,ρ,σ,τw,δ,q,r,cf$\begin{array}{} \displaystyle \varepsilon_{a^{+},\rho,\sigma,\tau}^{w,\delta,q,r,c}f \end{array}$ are defined and used to obtain generalizations of Opial-type inequalities due to Mitrinović and Pečarić. Also, some interesting properties of this function and its integral operator are discussed. Several known results are deduced.

The present work extends and generalizes the notion of frequency-distributed (FD) representation to a broad class of linear, stationary, continuous-time systems. On one hand, the proposed FD representation can be seen as a generalization of the diffusive representation, which is primarily utilized in the context of fractional order systems. Alternatively, it can also be seen as an extension to the Jordan canonical form, which is used as one of the main theoretical tools when analyzing finite-dimensional systems. Sufficient conditions under which FD representation can be achieved are derived. The proposed approach ensures real-valued state functions and output weights even when applied to oscillatory systems, and in a wast majority of cases manages to avoid utilization of generalized functions. Potential applications include simulation, representation theory and stability analysis, control synthesis, etc. All considerations have been illustrated by numerical examples.

We consider an ensemble of Ornstein–Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass like variable corresponding to this ensemble is statistically equivalent to a process driven by a non-autonomous stochastic differential equation with time-dependent drift and a white noise. In particular, the time scaling and the density function of such variable are driven by the population of timescales and of noise amplitudes, respectively. Moreover, we show that this variable is equivalent in distribution to a randomly-scaled Gaussian process, i.e., a process built by the product of a Gaussian process times a non-negative independent random variable. This last result establishes a connection with the so-called generalized grey Brownian motion and suggests application to model fractional anomalous diffusion in biological systems.