Volume 19, Issue 4

In this survey we consider two fractional-order discrete state-space models of linear systems. In both cases the crucial elements are the fundamental matrices. The difference between them is analyzed. A fundamental condition for the first state-space model is given. The investigations are illustrated by the numerical examples.

Recently, in series of papers we have proposed different concepts of solutions of impulsive fractional differential equations (IFDE). This paper is a survey of our main results about IFDE. We present several types of such equations with various boundary value conditions as well. Concept of solutions, existence results and examples are presented. Proofs are only sketched.

A generalization of the Poisson distribution based on the generalized Mittag-Leffler function E α,β (λ) is proposed and the raw moments are calculated algebraically in terms of Bell polynomials. It is demonstrated, that the proposed distribution function contains the standard fractional Poisson distribution as a subset. A possible interpretation of the additional parameter β is suggested.

This paper deals with the estimates of the convergence rates for various problems associated with diffusivity identification in a time-fractional nonlinear diffusion equation. We find the convergence rate of the corresponding Erd´elyi-Kober type operator as the anomalous parameter approaches the classical limit. Further, we use this result in proving an estimate of the difference between the identified diffusivity and its approximate value which was derived by the use of the previously developed framework. The exact formula for the diffusivity is computationally expensive thus having an accurate and easily calculable approximation is very relevant. In the last part of the paper we take up the problem of regularization strategy for solving the inverse problem. Calculation of the diffusivity requires a computation of the derivative which is a unstable operation and can amplify the measurement noise. We discuss how this ill-possedness can be mollified and prove some corresponding estimates on the convergence rates of the regularization

We prove that the (nonlocal) Marchaud fractional derivative in ℝ can be obtained from a parabolic extension problem with an extra (positive) variable as the operator that maps the heat conduction equation to the Neumann condition. Some properties of the fractional derivative are deduced from those of the local operator. In particular, we prove a Harnack inequality for Marchaud-stationary functions.

The strong maximum principle is a remarkable property of parabolic equations, which is expected to be partly inherited by fractional diffusion equations. Based on the corresponding weak maximum principle, in this paper we establish a strong maximum principle for time-fractional diffusion equations with Caputo derivatives, which is slightly weaker than that for the parabolic case. As a direct application, we give a uniqueness result for a related inverse source problem on the determination of the temporal component of the inhomogeneous term.

We discuss the existence and multiplicity of solutions to the generalized Bagley-Torvik fractional differential equation u″ = A c D α u + f ( t, u, u′ ) satisfying the Neumann boundary conditions u′ (0) = u′ ( T ) = 0. The solvability of the problem is proved by the combination of the Leray-Schauder degree method with the extremal principle.

In order to develop certain fractional probabilistic analogues of Taylor's theorem and mean value theorem, we introduce the n th-order fractional equilibrium distribution in terms of the Weyl fractional integral and investigate its main properties. Specifically, we show a characterization result by which the n th-order fractional equilibrium distribution is identical to the starting distribution if and only if it is exponential. The n th-order fractional equilibrium density is then used to prove a fractional probabilistic Taylor's theorem based on derivatives of Riemann-Liouville type. A fractional analogue of the probabilistic mean value theorem is thus developed for pairs of nonnegative random variables ordered according to the survival bounded stochastic order. We also provide some related results, both involving the normalized moments and a fractional extension of the variance, and a formula of interest to actuarial science. In conclusion, we discuss the probabilistic Taylor's theorem based on fractional Caputo derivatives.

The heat conduction equation is considered in a composite body consisting of two regions: 0 < x < L and − L < x < 0. Heat conduction in one region is described by the equation with the Caputo fractional derivative of order α , whereas in another region by the equation with the Caputo fractional derivative of order β . The integral transforms technique is used. The approximate solution valid for small values of time is analyzed in detail.

We prove the boundedness of the Riesz fractional integration operator from a generalized Morrey space L p, φ to a certain Orlicz-Morrey space L ϕ, φ which covers the Adams result for Morrey spaces. We also give a generalization to the case of weighted Riesz fractional integration operators for some class of weights.

This paper is about the fractional Schrödinger equation expressed in terms of the Caputo time-fractional and quantum Riesz-Feller space fractional derivatives for particle moving in a potential field. The cases of free particle (zero potential) and a linear potential are considered. For free particle, the solution is obtained in terms of the Fox H -function. For the case of a linear potential, the separation of variables method allows the fractional Schrödinger equation to be split into space fractional and time fractional ones. By using the Fourier and Mellin transforms for the space equation and the contour integrals technique for the time equation, the solutions are obtained also in terms of the Fox H -function. Moreover, some recent results related to the time fractional equation have been revised and reconsidered. The results obtained in this paper contain as particular cases already known results for the fractional Schrödinger equation in terms of the quantum Riesz space-fractional derivative and standard Laplace operator.

In this article, we consider a nonlinear higher-order m -point fractional boundary-value problem with integral boundary conditions. We establish criteria for the existence of at least one, two and three positive solutions for higher order nonlinear m -point fractional boundary-value problems with integral boundary conditions by using the four functionals fixed point theorem, the Avery-Henderson fixed point theorem and the Legget-Williams fixed point theorem, respectively.

Let T be a bounded linear operator defined on a Banach space X . We investigate the existence of solutions for a class of nonlinear fractional equation in the form (∗)Δαu(n)=Tu(n)+f(n,u(n)),n∈N0,0<α≤1;u(0)=x,$$ \begin{equation*} (*) \left\{ % \begin{array}{rll} \Delta^{\alpha} u(n) &= Tu(n) +f(n, u(n)), \quad n \in \mathbb{N}_0, \quad 0<\alpha \leq 1;\\ u(0) &= x, \\ \end{array} % \right. \end{equation*}$$ on the vector-valued weighted sequence space lf∞(N;X)=x:N→X/supn∈N∥x(n)∥nn!<∞.$$l_f^\infty(\mathbb{N};X)=\left\{x:\mathbb{N}\to X \,\, /\,\,\sup_{n\in\mathbb{N}}\frac{\|x(n)\|}{nn!}<\infty \right\}. $$ Our analysis relies on the fixed point theory and operator-theoretical methods.

Using estimates on the Green function and a perturbation argument, we prove the existence and uniqueness of a positive continuous solution to problem: {Dαu(t)=u(t)φ(t,u(t)),  t∈(0,∞), 1<α≤2,limt→0t2−αu(t)=a, limt→∞t1−αu(t)=b,$$\begin{equation*} \left\{ \begin{array}{l} D^{\alpha} u(t)=u(t)\varphi (t, u(t)),\ \ t\in (0,\infty),\ 1\lt \alpha \leq 2, \\ \underset{t\rightarrow 0}{\lim} t^{2-\alpha} u(t)=a,\ \underset{ t\rightarrow \infty} {\lim} t^{1-\alpha} u(t)=b, \end{array} \right. \end{equation*}$$ where D α is the standard Riemann-Liouville fractional derivative, a , b are nonnegative constants such that a + b > 0 and φ ( t, s ) is a nonnegative continuous function that is required to satisfy an appropriate condition related to a class 𝜅 α satisfying suitable integrability condition. We also give a global behavior of such solution.

In this paper we discuss functional delay fractional equations. A Banach fixed point theorem is applied to obtain the existence (uniqueness) theorem. We also discuss such problems when a delay argument has a form α ( t ) = αt , 0 < α < 1, by using the method of successive approximations. Some existence results are also formulated in this case. An example illustrates the main result.