Volume 23, Issue 5

Fractional relaxation equations, as well as relaxation functions time-changed by independent stochastic processes have been widely studied (see, for example, [21], [33] and [11]). We start here by proving that the upper-incomplete Gamma function satisfies the tempered-relaxation equation (of index ρ ∈ (0, 1)); thanks to this explicit form of the solution, we can then derive its spectral distribution, which extends the stable law. Accordingly, we define a new class of selfsimilar processes (by means of the n -times Laplace transform of its density) which is indexed by the parameter ρ : in the special case where ρ = 1, it reduces to the stable subordinator. Therefore the parameter ρ can be seen as a measure of the local deviation from the temporal dependence structure displayed in the standard stable case.

We show that integrability properties of integral transforms with kernel depending on the product of arguments (which include in particular, popular Laplace, Hankel, Mittag-Leffler transforms and various others) are better described in terms of Morrey spaces than in terms of Lebesgue spaces. Mapping properties of integral transforms of such a type in Lebesgue spaces, including weight setting, are known. We discover that local weighted Morrey and complementary Morrey spaces are very appropriate spaces for describing integrability properties of such transforms. More precisely, we show that under certain natural assumptions on the kernel, transforms under consideration act from local weighted Morrey space to a weighted complementary Morrey space and vice versa, where an interplay between behavior of functions and their transforms at the origin and infinity is transparent. In case of multidimensional integral transforms, for this goal we introduce and use anisotropic mixed norm Morrey and complementary Morrey spaces.

In this paper we study the global solvability of several ordinary and partial fractional integro-differential equations in the Wiener space of functions with bounded square averages.

This paper introduces the notion of “fractional fractals”. The main idea is to establish a connection between the classical iterated function system and the first order truncation of the Gründwald-Letnikov fractional derivative. This allows us to consider higher order truncations, and also to study the limit sets for these higher order systems. We prove several results involving the existence and dimension of such limit sets, that will be called “fractional fractals”. Some numerical calculations and representations illustrate relevant examples.

Two new high-order time discretization schemes for solving subdiffusion problems with nonsmooth data are developed based on the corrections of the existing time discretization schemes in literature. Without the corrections, the schemes have only a first order of accuracy for both smooth and nonsmooth data. After correcting some starting steps and some weights of the schemes, the optimal convergence orders O ( k 3– α ) and O ( k 4– α ) with 0 < α < 1 can be restored for any fixed time t for both smooth and nonsmooth data, respectively. The error estimates for these two new high-order schemes are proved by using Laplace transform method for both homogeneous and inhomogeneous problem. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

In this paper we consider Cauchy problem of time-fractional Tricomi-Keldysh type equation. Based on the theory of a Erdélyi-Kober fractional integral operator, the formal solution of the inhomogeneous differential equation involving hyper-Bessel operator is presented with Mittag-Leffler function, then nonlinear equations are considered by applying Gronwall-type inequalities. At last, we establish the existence and uniqueness of L p -solution of time-fractional Tricomi-Keldysh type equation by use of Mikhlin multiplier theorem.

In this article, we study the existence and uniqueness of solutions for nonlinear fractional integro-differential equations with nonlocal Erdélyi-Kober type integral boundary conditions. The existence results are based on Krasnoselskii’s and Schaefer’s fixed point theorems, whereas the uniqueness result is based on the contraction mapping principle. Examples illustrating the applicability of our main results are also constructed.

We consider parabolic nonlocal Venttsel’ problems in polygonal and piecewise smooth two-dimensional domains and study existence, uniqueness and regularity in (anisotropic) weighted Sobolev spaces of the solution. The nonlocal term can be regarded as a regional fractional Laplacian on the boundary. The regularity results deeply rely on a priori estimates, obtained via the so-called Munchhausen trick, and sophisticated extension theorem for anisotropic weighted Sobolev spaces.

We extend the Kulkarni class of multivariate phase–type distributions in a natural time–fractional way to construct a new class of multivariate distributions with heavy-tailed Mittag-Leffler(ML)-distributed marginals. The approach relies on assigning rewards to a non–Markovian jump process with ML sojourn times. This new class complements an earlier multivariate ML construction [2] and in contrast to the former also allows for tail dependence. We derive properties and characterizations of this class, and work out some special cases that lead to explicit density representations.

D. Adams type trace inequalities for multiple fractional integral operators in grand Lebesgue spaces with mixed norms are established. Operators under consideration contain multiple fractional integrals defined on the product of quasi-metric measure spaces, and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously necessary and sufficient for appropriate trace inequalities. The derived results are new even for multiple Riesz potential operators defined on the product of Euclidean spaces.

In this study, we consider a class of nonlinear fractional difference equations following form: Δ atgΔ α xt+Ft,∑ s=t0t− 1+α t− s− 1− α xs=0,$$\begin{array}{} \displaystyle \Delta \left( a\left( t\right) g\left( \Delta ^{\alpha }x\left( t\right) \right) \right) +F\left( t,\sum_{s=t_{0}}^{t-1+\alpha }\left( t-s-1\right) ^{\left( -\alpha \right) }x\left( s\right) \right) =0\text{,} \end{array}$$ where t ∈ ℕ t 0 +1– α and Δ α denotes the Riemann-Liouville fractional difference operator of order α . Using the generalized Riccati function, we obtain some oscillation criteria. Finally, we give some illustrative examples.

In this paper, we introduce and study two counting processes by considering state dependency on the order of fractional derivative as well as on the exponent of backward shift operator involved in the governing difference-differential equations of the state probabilities of space-time fractional Poisson process. The Adomian decomposition method is employed to obtain their state probabilities and then their Laplace transforms are evaluated. Also, the compound versions of these state dependent models are studied and the corresponding governing fractional integral equations of their state probabilities are obtained.

This paper is mainly concerned with stochastic fractional hemivariational inequalities of degenerate (or Sobolev) type in Caputo and Riemann-Liouville derivatives with order (1, 2), respectively. Based upon some properties of fractional resolvent family and generalized directional derivative of a locally Lipschitz function, some sufficient conditions are established for the existence and approximate controllability of the aforementioned systems. Particularly, the uniform boundedness for some nonlinear terms, the existence and compactness of certain inverse operator are not necessarily needed in obtained approximate controllability results.

An experimental investigation is presented of an oscillating disk submerged in water. The system is adapted from one modeled by the Bagley-Torvik equation, being modified to more closely approximate idealizations in the derivation. The modified system eliminates alignment problems that would violate the assumption of fluid forces being due only to shear in the fluid. An implicit finite difference model based on the Riemann-Liouville fractional derivative was used to model the observed oscillatory free response resulting from a non-homogeneous initial condition. The numerical model accounted for motion well until the amplitude was small.

In the present work, we consider the Cauchy problem for the time fractional diffusion equation involving the general Caputo-type differential operator proposed by Kochubei [11]. First, the existence, the positivity and the long time behavior of solutions of the diffusion equation without source term are established by using the Fourier analysis technique. Then, based on the representation of the solution of the inhomogenous linear ordinary differential equation with the general Caputo-type operator, the general diffusion equation with source term is studied.