Article Frontmatter was published on April 1, 2021 in the journal Fractional Calculus and Applied Analysis (volume 24, issue 2).

Article Anniversary of Prof. S.G. Samko, FC Events (FCAA–Volume 24–2–2021) was published on April 1, 2021 in the journal Fractional Calculus and Applied Analysis (volume 24, issue 2).

In this paper, we first address the general fractional integrals and derivatives with the Sonine kernels that possess the integrable singularities of power function type at the point zero. Both particular cases and compositions of these operators are discussed. Then we proceed with a construction of an operational calculus of the Mikusiński type for the general fractional derivatives with the Sonine

We establish intertwining relations between Riesz potentials associated with fractional powers of minus-Laplacian and orthogonal Radon transforms 𝓡 j , k of the Gonzalez-Strichartz type. The latter take functions on the Grassmannian of j -dimensional affine planes in ℝ n to functions on a similar manifold of k -dimensional planes by integration over the set of all j -planes that meet a given k -plane

We introduce a new type of dyadic maximal operators and prove that under the log-Hölder continuity condition of the variable exponent p (⋅), it is bounded on L p (⋅) if 1 < p − ≤ p + ≤ ∞. Moreover, the space generated by the L p (⋅) -norm (resp. the L p (⋅), q -norm) of the maximal operator is equivalent to the Hardy space H p (⋅) (resp. to the Hardy-Lorentz space H p (⋅), q ). As special cases, our

Obstacle avoidance is one of the main interests regarding path planning. In many situations (mostly those regarding applications in urban environments), the obstacles to be avoided are dynamical and unpredictable. This lack of certainty regarding the environment introduces the need to use local path planning techniques rather than global ones. A well-known method uses artificial potential fields introduced

By observing that the fractional Caputo derivative of order α ∈ (0, 1) can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of

The asymptotic behaviour of solutions in an appropriate space is discussed for a fractional problem involving Hadamard left-sided fractional derivatives of different orders. Reasonable sufficient conditions are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach a logarithmic function as time goes to infinity. This generalizes and extends

In this paper, we investigate a nonlocal boundary value problem for an equation of special type. For y > 0 it is a fractional diffusion equation, which contains the Riemann-Liouville fractional derivative. For y < 0 it is a generalized equation of moisture transfer. A unique solvability of the considered problem is proved.

Mikusiński’s operational calculus is a formalism for understanding integral and derivative operators and solving differential equations, which has been applied to several types of fractional-calculus operators by Y. Luchko and collaborators, such as for example [26], etc. In this paper, we consider the operators of Riemann–Liouville fractional differentiation of a function with respect to another function

We study the duality theory for fractional resolvents, extending and improving some corresponding theorems on semigroups. As applications, we develop the variational technique to analyze the finite-approximate controllability of a backward fractional control system with a right-sided Riemann-Liouville fractional derivative. Moreover, validity of our theoretical findings is given by a fractional diffusion

This work is devoted to examining the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal super-critical diffusion operator and a multiplicative noise. Our proof for uniqueness is based upon the analysis on double variables method and the existence is enabled by a parabolic approximation.

In this work, we consider the n -dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem. More precisely, we show that the eigenfunctions are orthogonal and the

The theory of averaging is a classical component of applied mathematics and has been applied to solve some engineering problems, such as in the filed of control engineering. In this paper, we develop a theory of averaging on both finite and infinite time intervals for fractional non-autonomous differential equations. The closeness of the solutions of fractional no-autonomous differential equations

Article Frontmatter was published on February 1, 2021 in the journal Fractional Calculus and Applied Analysis (volume 24, issue 1).

Article FCAA related news, events and books (FCAA–volume 24–1–2021) was published on February 1, 2021 in the journal Fractional Calculus and Applied Analysis (volume 24, issue 1).

The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity

The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book Fractional Calculus and Waves in Linear Viscoelasticity (2010)], Fσ(x)=∑n=0∞(−x)nn!Γ(−nσ) , Mσ(x)=∑n=0∞(−x)nn!Γ(−nσ+1−σ) (0<σ<1)$$F_\sigma(x)=\sum\limits_{n=0}^\infty \frac{(-x)^n}{n! {\mathrm{\Gamma}}(-n\sigma)}~,\quad M_\sigma(x)=\sum\limits_{n=0}^\infty \frac{(-x)^n}{n! {\mat

In this paper, the long-time behavior of the Cesaro mean of the fundamental solution for fractional Heat equation corresponding to random time changes in the Brownian motion is studied. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order

The multinomial Mittag-Leffler function plays a crucial role in the study of multi-term time-fractional evolution equations. In this work we establish basic properties of the Prabhakar type generalization of this function with the main emphasis on complete monotonicity. As particular examples, the relaxation functions for equations with multiple time-derivatives in the so-called “natural” and “modified”

This paper is devoted to a fractional generalization of the Dirichlet distribution. The form of the multivariate distribution is derived assuming that the n partitions of the interval [0, W n ] are independent and identically distributed random variables following the generalized Mittag-Leffler distribution. The expected value and variance of the one-dimensional marginal are derived as well as the

We study Markovian continuos-time random walk models for Lévy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i ) with regard to the probabilistic derivation of the fractional diffusion equation and also ii ) with regard to the

We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary

In this paper, a weighted and shifted Grünwald-Letnikov difference (WSGD) Legendre spectral method is proposed to solve the two-dimensional nonlinear time fractional mobile/immobile advection-dispersion equation. We introduce the correction method to deal with the singularity in time, and the stability and convergence analysis are proven. In the numerical implementation, a fast method is applied based

Necessary and sufficient conditions are explored for the asymptotic stability and instability of linear two-dimensional autonomous systems of fractional-order differential equations with Caputo derivatives. Fractional-order-dependent and fractional-order-independent stability and instability properties are fully characterised, in terms of the main diagonal elements of the systems’ matrix, as well as

This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential

The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of errors between the exact solution and the Monte Carlo approximation, the estimate of the fluctuation via the appropriate central limit theorem (CLT) and the construction

In this paper, we consider a discrete-time fractional model of abstract form involving the Riemann-Liouville-like difference operator. On account of the C 0 -semigroups generated by a closed linear operator A and based on a distinguished class of sequences of operators, we show the existence of stable solutions for the nonlinear Cauchy problem by means of fixed point technique and the compact method

For 1 < ν ≤ 2 a real number and T ≥ 3 a natural number, conditions are given for the existence of solutions of the ν th order Atıcı-Eloe fractional difference equation, Δ ν y ( t ) + f ( t + ν − 1, y ( t + ν − 1)) = 0, t ∈ {0, 1, …, T }, and satisfying the left focal boundary conditions Δ y ( ν − 2) = y ( ν + T ) = 0.

Editorial Note: This is a paper by M.M. Djrbashian and A.B. Nersesian of 1968, that was published in Russian.

This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the

In this article, we aim to analyze a mathematical model of tumor growth as a problem of fractional optimal control. The considered fractional-order model describes the interaction of effector-immune cells and tumor cells, including combined chemo-immunotherapy. We deduce the necessary optimality conditions together with implementing the Adomian decomposition method on the suggested fractional-order

We consider a nonlinear stochastic heat equation with Riesz space-fractional derivative and variable thermal conductivity, on infinite domain. First we approximate the original problem by regularizing the Riesz space-fractional derivative. Then we prove that the approximate problem has almost surely a unique solution within a Colombeau generalized stochastic process space. In our solving procedure

The present paper deals with initial value problem (IVP) for semilinear fractional Schrödinger integro-differential equation

We investigate extreme properties of a class of integro-differential operators. We prove an assertion that extends the Nakhushev extremum principle, known for fractional Riemann-Liouville derivatives, to integro-differential operators with kernels of a general form. We establish the weighted extremum principle for convolution operators and the Riemann-Liouville fractional derivative. In addition, as

We study uniqueness of a solution for an inverse source problem arising in linear time-fractional diffusion equations with time-dependent coefficients. We consider source term in a separated form h(t)f (x). The unknown source f (x) is recovered from the final time measurement u (x, T). A new uniqueness result is formulated in under the assumption that h ∈ C ([0, T]) and 0 ≢ h ≥ 0. No monotonicity in

Two inverse problems with final overdetermination for diffusion and wave equations containing the Caputo fractional time derivative and a fractional Laplacian of distributed order are considered. They are: 1) the problem to reconstruct a time-dependent source term; 2) the problem to recover simultaneously the source term, the order of the time derivative and the fractional Laplacian. Uniqueness of

The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with

The identification of the right order of the equation in applied fractional modeling plays an important role. In this paper we consider an inverse problem for determining the order of time fractional derivative in a subdiffusion equation with an arbitrary second order elliptic differential operator. We prove that the additional information about the solution at a fixed time instant at a monitoring

The survey is devoted to numerical solution of the equation Aαu=f, 0 < α<1, where A is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in ℝd. The fractional power Aα is a non-local operator and is defined though the spectrum of A. Due to growing interest and demand in applications of sub-diffusion models to physics and engineering

Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to the Karhunen-Loéve expansion theorem are presented

Journal Name: Fractional Calculus and Applied Analysis Volume: 23 Issue: 6 Pages: 1561-1569

Journal Name: Fractional Calculus and Applied Analysis Volume: 23 Issue: 6 Pages: i-v

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 []. 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

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

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

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

In this study, we consider a class of nonlinear fractional difference equations following form:

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

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 [] and in contrast to

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

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

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

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

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

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.

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

Fractional relaxation equations, as well as relaxation functions time-changed by independent stochastic processes have been widely studied (see, for example, [], [] and []). 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

Journal Name: Fractional Calculus and Applied Analysis Volume: 23 Issue: 5 Pages: 1241-1247

Journal Name: Fractional Calculus and Applied Analysis Volume: 23 Issue: 5 Pages: i-v