Band 23, Heft 3

In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.

We propose a model for the transmission of perturbations across the amino acids of a protein represented as an interaction network. The dynamics consists of a Susceptible-Infected (SI) model based on the Caputo fractional-order derivative. We find an upper bound to the analytical solution of this model which represents the worse-case scenario on the propagation of perturbations across a protein residue network. This upper bound is expressed in terms of Mittag-Leffler functions of the adjacency matrix of the network of inter-amino acids interactions. We then apply this model to the analysis of the propagation of perturbations produced by inhibitors of the main protease of SARS CoV-2. We find that the perturbations produced by strong inhibitors of the protease are propagated far away from the binding site, confirming the long-range nature of intra-protein communication. On the contrary, the weakest inhibitors only transmit their perturbations across a close environment around the binding site. These findings may help to the design of drug candidates against this new coronavirus.

We survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

We analyze the inverse boundary value-problem to determine the fractional order ν of nonautonomous semilinear subdiffusion equations with memory terms from observations of their solutions during small time. We obtain an explicit formula reconstructing the order. Based on the Tikhonov regularization scheme and the quasi-optimality criterion, we construct the computational algorithm to find the order ν from noisy discrete measurements. We present several numerical tests illustrating the algorithm in action.

This paper deals with the existence of nontrivial solutions for critical possibly degenerate Kirchhoff fractional ( p , q ) systems. For clarity, the results are first presented in the scalar case, and then extended into the vectorial framework. The main features and novelty of the paper are the ( p , q ) growth of the fractional operator, the double lack of compactness as well as the fact that the systems can be degenerate. As far as we know the results are new even in the scalar case and when the Kirchhoff model considered is non–degenerate.

The paper deals with the initial value problem for linear systems of FDEs with variable coefficients involving Riemann–Liouville derivatives. The technique of the generalized Peano–Baker series is used to obtain the state-transition matrix. Explicit solutions are derived both in the homogeneous and inhomogeneous case. The theoretical results are supported by an example.

In this paper, based on the “fuzzy” calculus covering the continuous range of operations between two couples of arithmetic operations (+, –) and (×, :), a new form of the fractional integral is proposed occupying an intermediate position between the integral and derivative of the first order. This new form of the fractional integral satisfies the C 1 criterion according to the Ross classification. The new calculus is tightly related to the continuous values of the continuous spin S = 1 and can generalize the expression for the fractional values of the shifting discrete index. This calculus can be interpreted as the appearance of the hidden states corresponding to unobservable values of S = 1. Many well-known formulas can be generalized and receive a new extended interpretation. In particular, one can factorize any rectangle matrix and receive the “perfect” filtering formula that allows transforming any (deterministic or random) function to another arbitrary function and vice versa. This transformation can find unexpected applications in data transmission, cryptography and calibration of different gadgets and devices. One can also receive the hybrid (”centaur”) formula for the Fourier (F-) transformation unifying both expressions for the direct and inverse F-transformations in one mathematical unit. The generalized Dirichlet formula, which is obtained in the frame of the new calculus to allow selecting the desired resonance frequencies, will be useful in discrete signals processing, too. The basic formulas are tested numerically on mimic data.

A class of two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order α ∈ (1, 2] with a convection term is considered. Its boundary conditions are of Robin type including the Dirichlet boundary conditions as a special case. An explicit formula for the associated Green function is obtained in terms of two-parameter Mittag-Leffler functions. This work improves Meng and Stynes’ work [9] in three aspects. Firstly, the Green function is constructed by the use of the Laplace transform. This method may be more straightforward and easy to be generalized to solving other problems. Secondly, the monotonicity of a function is proven using monotone definition rather than using its derivative. Thirdly, a direct proof of the necessity of the positive property of Green function is given so that we can avoid the difficulty of construction any counter-example.

Inverse problem for a family of multi-term time fractional differential equation with non-local boundary conditions is studied. The spectral operator of the considered problem is non-self-adjoint and a bi-orthogonal set of functions is used to construct the solution. The operational calculus approach has been used to obtain the solution of the multi-term time fractional differential equations. Integral type over-determination condition is considered for unique solvability of the inverse problem. Different estimates of multinomial Mittag-Leffler functions alongside Banach fixed point theorem are used to prove the unique local existence of the solution of the inverse problem. Stability of the solution of the inverse problem on the given datum is established.

Two significant inequalities for generalized time fractional derivatives at extreme points are obtained. Then, we apply the inequalities to establish the maximum principles for multi-term time-space fractional variable-order operators. Finally, we employ the principles to investigate two kinds of diffusion equations involving generalized time-fractional Caputo derivatives and space-fractional Riesz-Caputo derivatives.

In this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent (P)(− Δ )su=λ uγ +|u|2α ∗ − 2u|x|α  in Ω ,u> -->0 in Ω ,u=0 in RN∖ Ω ,$$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^s u = \displaystyle{\frac{\lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \text{ in } \ \ \Omega, \\ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{array}$$ where Ω ⊂ ℝ N is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < α < 2 s < N , 0 < γ < 1 < 2 < 2s∗ $\begin{array}{} \displaystyle 2_s^* \end{array}$, where 2s∗ =2NN− 2s and 2α ∗ =2(N− α )N− 2s$\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and}~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (–Δ) s with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by (− Δ )su(x)=− 12∫ RNu(x+y)+u(x− y)− 2u(x)|y|N+2sdy, for all x∈ RN.$$\begin{array}{} \displaystyle (-\Delta)^s u(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ for all }\, x \in \mathbb{R}^N. \end{array}$$ By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P).

In this paper, we consider a regular Fractional Sturm–Liouville Problem (FSLP) of order μ (0 < μ < 1). We approximate the eigenvalues and eigenfunctions of the problem using a fractional variational approach. Recently, Klimek et al. [16] presented the variational approach for FSLPs defined in terms of Caputo derivatives and obtained eigenvalues, eigenfunctions for a special range of fractional order 1/2 < μ < 1. Here, we extend the variational approach for the FSLPs and approximate the eigenvalues and eigenfunctions of the FSLP for fractional-order μ (0 < μ < 1). We also prove that the FSLP has countably infinite eigenvalues and corresponding eigenfunctions.

In this paper, we consider a Takagi-like function on 2-series field and give its 2-adic derivatives by applying Vladimirov operator. The 2-adic derivatives of Takagi-like function with order 0 < α < 1 exist and show some fractal feature. Furthermore, both box dimension and Hausdorff dimension of the graph of its derivatives are obtained and equal to 1 + α .

In this article, we establish a technique for transforming arbitrary real order delta difference equations with impulses to corresponding summation equations. The technique is applied to non-integer order delta difference equation with some boundary conditions. Furthermore, the summation formulation for impulsive fractional difference equation is utilized to construct fixed point operator which in turn are used to discuss existence of solutions.

This paper presents an averaging principle for fractional stochastic differential equations in ℝ n with fractional order 0 < α < 1. We obtain a time-averaged equation under suitable conditions, such that the solutions to original fractional equation can be approximated by solutions to simpler averaged equation. By mathematical manipulations, we show that the mild solution of two equations before and after averaging are equivalent in the sense of mean square, which means the classical Khasminskii approach for the integer order systems can be extended to fractional systems.

In this work, we study a new type of linear partial differential equations – the variable-order subdiffusion equation. Here the Laplace operator in space acts on the Riemann-Liouville time derivative of space-dependent order. We construct a variable-order Sobolev and prove the weak solvability of the initial-boundary value problem for this equation, which confirms the well-posedness of the problem. Finally, we briefly discuss the application of the developed approach to the more general variable-order reaction-subdiffusion equation.