In this paper, we introduce a short-time coupled fractional Fourier transform ( scfrft ) using the kernel of the coupled fractional Fourier transform ( cfrft ). We then prove that it satisfies Parseval’s relation, derive its inversion and addition formulas, and characterize its range on ℒ 2 (ℝ 2 ). We also study its time delay and frequency shift properties and conclude the article by a derivation

The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different route, namely, construction of explicit asymptotic approximations for the solutions. As a study case, we consider a problem with left and right Riemann-Liouville

A specific formulation of the “classical” problem of mathematical analysis is considered. This is the problem of calculating the derivative of a function. The purpose of this work is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on the methods of control theory. The input data of the algorithm is represented by inaccurate measured function

In this paper, we develop a sliding method for the higher order fractional Laplacians. We first obtain the key ingredients to obtain monotonicity of solutions, such as narrow region maximum principles in bounded or unbounded domains. Then we introduce a new idea of estimating the singular integrals defining the fractional Laplacian along a sequence of approximate maximum points and illustrate how this

Article Frontmatter was published on June 1, 2021 in the journal Fractional Calculus and Applied Analysis (volume 24, issue 3).

In this editorial paper, we start by surveying of the main milestones in the organization, foundation, and development of the journal Fractional Calculus and Applied Analysis ( FCAA ). The main potential of FCAA is in its readers, authors, and editors who contribute to the scientific advance and promote the progress of the journal. Among the editors, a special role of the honorary editors who contributed

The formula of the all-pole low-pass frequency filter transfer function of the fractional order ( N + α ) designated for implementation by non-cascade multiple-feedback analogue structures is presented. The aim is to determine the coefficients of this transfer function and its possible variants depending on the filter order and the distribution of the fractional-order terms in the transfer function

In this paper we characterize the Laplace transform of functions with power growth square averages and study several multi-term Caputo and Riemann-Liouville fractional integro-differential equations in this space of functions.

This paper gathers the tools for solving Riemann-Liouville time fractional non-linear PDE’s by using a Galerkin method. This method has the advantage of not being more complicated than the one used to solve the same PDE with first order time derivative. As a model problem, existence and uniqueness is proved for semilinear heat equations with polynomial growth at infinity.

This paper presents two new classes of Müntz functions which are called Jacobi-Müntz functions of the first and second types. These newly generated functions satisfy in two self-adjoint fractional Sturm-Liouville problems and thus they have some spectral properties such as: orthogonality, completeness, three-term recurrence relations and so on. With respect to these functions two new orthogonal projections

In this article, we consider the space-time fractional (nonlocal) equation characterizing the so-called “double-scale” anomalous diffusion ∂ tβ u(t,x)=− (− Δ )α /2u(t,x)− (− Δ )γ /2u(t,x),t> -->0,− 1< x< 1,$$\begin{array}{} \displaystyle \partial_t^\beta u(t, x) = -(-\Delta)^{\alpha/2}u(t,x) - (-\Delta)^{\gamma/2}u(t,x), \, \, t \gt 0, \, -1 \lt x \lt 1, \end{array}$$ where ∂ tβ $\begin{array}{} \displaystyle

For an integro-differential equation of the convolution type defined on the half-line [0, ∞) with a power nonlinearity and variable coefficient, we use the weight metrics method to prove a global theorem on the existence and uniqueness of a solution in the cone of nonnegative functions in the space C [0, ∞). It is shown that the solution can be found by a successive approximation method of the Picard

This paper introduces an efficient collocation solver, the generalized finite difference method (GFDM) combined with the recent-developed scale-dependent time stepping method (SD-TSM), to predict the anomalous diffusion behavior on surfaces governed by surface time-fractional diffusion equations. In the proposed solver, the GFDM is used in spatial discretization and SD-TSM is used in temporal discretization

The global stability of continuous-time fractional orders nonlinear feedback systems with positive linear parts and interval state matrices is investigated. New sufficient conditions for the global stability of this class of positive feedback nonlinear systems are established. The effectiveness of these new stability conditions is demonstrated on simple example.

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.

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

Abstract Editorial Note: This is a paper by M.M. Djrbashian and A.B. Nersesian of 1968, that was published in Russian. There is a constant interest to Djrbashian’s contributions to the topic of fractional calculus and theory of Mittag-Leffler function. Unfortunately, his works were published in Russian and thus, are not easy accessible and not enough popular. Therefore, we invited hS. Rogosin and M

Abstract 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

Abstract 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

Abstract 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

Abstract The present paper deals with initial value problem (IVP) for semilinear fractional Schrödinger integro-differential equation idudt+Au=∫0tfs,Dsαu(s)ds,0

Abstract 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

Abstract 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 Theorem 3.1 under the assumption that h ∈ C ([0, T]) and 0 ≢ h ≥

Abstract 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

Abstract 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

Abstract 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

Abstract The survey is devoted to numerical solution of the equation Aαu=f $ {\mathcal A}^\alpha u=f $ , 0 < α<1, where A $ {\mathcal 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 $ {\mathcal A}^\alpha $ is a non-local operator and is defined though the spectrum of A $ {\mathcal A}

Abstract 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

In 2020 some international conferences on Fractional Calculus (FC) and related topics have been held in a hybrid way mainly online via popular electronic platforms and by physical presence with reduced number of local participants. Despite of the pandemic situation, the FC community showed as usual great activities. In this special issue, we collected some of the plenary presentations and invited selected

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

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

Abstract 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