The Mittag–Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.

Publicly Available February 27, 2020

Crossover dynamics from superdiffusion to subdiffusion: Models and solutions

Emad Awad, Ralf Metzler

Page range: 55-102

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Abstract

The Cattaneo or telegrapher’s equation describes the crossover from initial ballistic to normal diffusion. Here we study and survey time-fractional generalisations of this equation that are shown to produce the crossover of the mean squared displacement from superdiffusion to subdiffusion. Conditional solutions are derived in terms of Fox H -functions and the δ th-order moments as well as the diffusive flux of the different models are derived. Moreover, the concept of the distribution-like is proposed as an alternative to the probability density function.

Research Paper

Publicly Available February 27, 2020

Operational method for solving fractional differential equations with the left-and right-hand sided Erdélyi-Kober fractional derivatives

Latif A-M. Hanna, Maryam Al-Kandari, Yuri Luchko

Page range: 103-125

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Abstract

In this paper, we first provide a survey of some basic properties of the left-and right-hand sided Erdélyi-Kober fractional integrals and derivatives and introduce their compositions in form of the composed Erdélyi-Kober operators. Then we derive a convolutional representation for the composed Erdélyi-Kober fractional integral in terms of its convolution in the Dimovski sense. For this convolution, we also determine the divisors of zero. These both results are then used for construction of an operational method for solving an initial value problem for a fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives defined on the positive semi-axis. Its solution is obtained in terms of the four-parameters Wright function of the second kind. The same operational method can be employed for other fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives.

February 27, 2020

Well-posedness of the fractional Zener wave equation for heterogeneous viscoelastic materials

Ljubica Oparnica, Endre Süli

Page range: 126-166

Abstract

Zener’s model for viscoelastic solids replaces Hooke’s law σ = 2 μ ε ( u ) + λ tr( ε ( u )) I , relating the stress tensor σ to the strain tensor ε ( u ), where u is the displacement vector, μ > 0 is the shear modulus, and λ ≥ 0 is the first Lamé coefficient, with the constitutive law (1 + τ D t ) σ = (1 + ρ D t )[2 μ ε ( u ) + λ tr( ε ( u )) I ], where τ > 0 is the characteristic relaxation time and ρ ≥ τ is the characteristic retardation time. It is the simplest model that predicts creep/recovery and stress relaxation phenomena. We explore the well-posedness of the fractional version of the model, where the first-order time-derivative D t in the constitutive law is replaced by the Caputo time-derivative Dtα $\begin{array}{} D_t^\alpha \end{array} $ with α ∈ (0, 1), μ , λ belong to L ∞ (Ω), μ is bounded below by a positive constant and λ is nonnegative. We show that, when coupled with the equation of motion ϱ ü = Div σ + f , considered in a bounded open Lipschitz domain Ω in ℝ 3 and over a time interval (0, T ], where ϱ ∈ L ∞ (Ω) is the density of the material, assumed to be bounded below by a positive constant, and f is a specified load vector, the resulting model is well-posed in the sense that the associated initial-boundary-value problem, with initial conditions u (0, x ) = g ( x ), u̇ (0, x ) = h ( x ), σ (0, x ) = S ( x ), for x ∈ Ω, and a homogeneous Dirichlet boundary condition, possesses a unique weak solution for any choice of g ∈ [H01$\begin{array}{} \mathrm{H}^1_0 \end{array} $(Ω)] 3 , h ∈ [L 2 (Ω)] 3 , and S = S T ∈ [L 2 (Ω)] 3×3 , and any load vector f ∈ L 2 (0, T ; [L 2 (Ω)] 3 ), and that this unique weak solution depends continuously on the initial data and the load vector.

February 27, 2020

A note on models for anomalous phase-change processes

Andrea N. Ceretani

Page range: 167-182

Abstract

We review some fractional free boundary problems that were recently considered for modeling anomalous phase-transitions. All problems are of Stefan type and involve fractional derivatives in time according to Caputo’s definition. We survey the assumptions from which they are obtained and observe that the problems are nonequivalent though all of them reduce to a classical Stefan problem when the order of the fractional derivatives is replaced by one. We further show that a simple heuristic approach built upon a fractional version of the energy balance and the classical Fourier’s law leads to a natural generalization of the classical Stefan problem in which time derivatives are replaced by fractional ones.

February 27, 2020

Stability and resonance analysis of a general non-commensurate elementary fractional-order system

Shuo Zhang, Lu Liu, Dingyu Xue, YangQuan Chen

Page range: 183-210

Abstract

The elementary fractional-order models are the extension of first and second order models which have been widely used in various engineering fields. Some important properties of commensurate or a few particular kinds of non-commensurate elementary fractional-order transfer functions have already been discussed in the existing studies. However, most of them are only available for one particular kind elementary fractional-order system. In this paper, the stability and resonance analysis of a general kind non-commensurate elementary fractional-order system is presented. The commensurate-order restriction is fully released. Firstly, based on Nyquist’s Theorem, the stability conditions are explored in details under different conditions, namely different combinations of pseudo-damping ( ζ ) factor values and order parameters. Then, resonance conditions are established in terms of frequency behaviors. At last, an example is given to show the stable and resonant regions of the studied systems.

February 27, 2020

A comment on a controversial issue: A generalized fractional derivative cannot have a regular kernel

Andrzej Hanyga

Page range: 211-223

Abstract

The problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation. It is shown for some selected classes of functions that a necessary condition for a function to be the kernel of a fractional derivative is an integrable singularity at 0. It is shown that locally integrable completely monotone functions satisfy the Sonine equation if and only if they are singular at 0.

February 27, 2020

Space-time fractional stochastic partial differential equations with Lévy noise

Xiangqian Meng, Erkan Nane

Page range: 224-249

Abstract

We consider non-linear time-fractional stochastic heat type equation ∂ β u∂ tβ +ν (− Δ )α /2u=It1− β [∫ Rdσ (u(t,x),h)N~ ⋅ (t,x,h)]$$\begin{array}{} \displaystyle \frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u = I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg] \end{array} $$ and ∂ β u∂ tβ +ν (− Δ )α /2u=It1− β [∫ Rdσ (u(t,x),h)N⋅ (t,x,h)]$$\begin{array}{} \displaystyle \frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u = I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h)\stackrel{\cdot}{N }(t,x,h)\bigg] \end{array} $$ in ( d + 1) dimensions, where α ∈ (0, 2] and d < min{2, β −1 } α , ν > 0, ∂ tβ $\begin{array}{} \partial^\beta_t \end{array} $ is the Caputo fractional derivative, −(−Δ) α /2 is the generator of an isotropic stable process, It1− β $\begin{array}{} I^{1-\beta}_t \end{array} $ is the fractional integral operator, N ( t , x ) are Poisson random measure with Ñ ( t , x ) being the compensated Poisson random measure. σ : ℝ → ℝ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in [16, 33]. Under the linear growth of σ , we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when σ grows faster than linear.

February 27, 2020

Stability of scalar nonlinear fractional differential equations with linearly dominated delay

Hoang The Tuan, Stefan Siegmund

Page range: 250-267

Abstract

In this paper, we study the asymptotic behavior of solutions to a scalar fractional delay differential equations around the equilibrium points. More precise, we provide conditions on the coefficients under which a linear fractional delay equation is asymptotically stable and show that the asymptotic stability of the trivial solution is preserved under a small nonlinear Lipschitz perturbation of the fractional delay differential equation.

February 27, 2020

Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators

Pengyu Chen, Xuping Zhang, Yongxiang Li

Page range: 268-291

Abstract

In this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of fractional evolution equations with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green’s function and constructing a control function involving Gramian controllability operator. The discussions are based on Schauder’s fixed point theorem as well as the theory of α -order solution operator and α -order resolvent operator. An example is given to illustrate the feasibility of our theoretical results.

Short Paper

February 27, 2020

Comments on: “The failure of certain fractional calculus operators in two physical models” by M. Ortigueira, V. Martynyuk, M. Fedula and J.A.T. Machado

Dumitru Baleanu

Page range: 292-297

Abstract

In these comments, I analyse the results reported by Ortigueira et al. [18] regarding the potential applications of non-singular fractional operators suggested by Caputo-Fabrizio and Atangana-Baleanu. My purpose is to show that the opinions of [18] are not consistent.

February 27, 2020

A note on vanishing Morrey → VMO result for fractional integrals of variable order

Humberto Rafeiro, Stefan Samko

Page range: 298-302

Abstract

In the limiting case of Sobolev-Adams theorem for Morrey spaces of variable order we prove that the fractional operator of variable order maps the corresponding vanishing Morrey space into VMO.

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Please note that, from the 2022 volume onwards, Fractional Calculus and Applied Analysis is published by Springer Nature. For more information, please see the new journal homepage here. (please put under ‘here’ the link to Springer's website: https://www.springer.com/journal/13540.