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In his first paper on the generalization of the tautochrone problem, that was published in 1823, Niels Henrik Abel presented a complete framework for fractional-order calculus, and used the clear and appropriate notation for fractional-order integration and differentiation. [ABSTRACT FROM AUTHOR]

A new lumped capacitance model that employs fractional order operators is proposed for use on transient heat conduction problems. Details and implications of the fractional lumped capacitance model's development and application are discussed. The model is shown to agree with observed heating and cooling temperature profiles of laser aiming paper being heated by a laser under various conditions. [ABSTRACT FROM AUTHOR]

In this paper we provide a way to handle some symmetric fractional boundary-value problems. Indeed, first, we consider some system of fractional equations. We introduce the existence and uniqueness of solutions of the systems of equations and we show that they are entire functions of the spectral parameter. In particular, we show that the solutions are at most of order 1/2. Moreover we share the integration by parts rule for vector-valued functions that enables us to obtain some symmetric equations. These symmetries allow us to handle 2 - sequential and 4 - sequential fractional boundary-value problems. We provide some expansion formulas for the bilinear forms of the solutions of 2 - sequential and 4 - sequential fractional equations which admit us to impose some unusual boundary conditions for the solutions of fractional differential equations. We show that the systems of eigenfunctions of 2 - sequential and 4 - sequential fractional boundary value problems are complete in both energy and mean. Furthermore, we study on the zeros of solutions of 2 - sequential fractional differential equations. At the end of the paper we show that 6 - sequential fractional differential equation can also be handled as a system of equations and hence almost all the results obtained in the paper can be carried for such boundary-value problems. [ABSTRACT FROM AUTHOR]

We prove existence of solutions, and particularly positive solutions, of initial value problems (IVPs) for nonlinear fractional differential equations involving the Caputo differential operator of order α ∈ (0 , 1) . One novelty in this paper is that it is not assumed that f is continuous but that it satisfies an L p -Carathéodory condition for some p > 1 α (detailed definitions are given in the paper). We prove existence on an interval [0, T ] in cases where T can be arbitrarily large, called global solutions. The necessary a priori bounds are found using a new version of the Bihari inequality that we prove here. We show that global solutions exist when f (t , u) grows at most linearly in u , and also in some cases when the growth is faster than linear. We give examples of the new results for some fractional differential equations with nonlinearities related to some that occur in combustion theory. We also discuss in detail the often used alternative definition of Caputo fractional derivative and we show that it has severe disadvantages which restricts its use. In particular we prove that there is a necessary condition in order that solutions of the IVP can exist with this definition, which has often been overlooked in the literature. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new multiple-parameters (multi-index) extension of the Wright function that arises from an eigenvalue problem for a case of hyper-Bessel operator involving Caputo fractional derivatives. We show that by giving particular values to the parameters involved in this special function, this leads to some known special functions (as the classical Wright function, the α-Mittag-Leffler function, the Tricomi function, etc.) that on their turn appear as cases of the so-called multi-index Mittag-Leffler functions. As an application, we mention that this new generalization Wright function nis an isochronous solution of a nonlinear fractional partial differential equation. [ABSTRACT FROM AUTHOR]

The paper investigates fractional differential equations involving the Liouville derivative. Solution to these equations under a boundary condition inside the time interval are derived in explicit form, their uniqueness is established using integral transforms technique. [ABSTRACT FROM AUTHOR]

In this paper, the averaging principle for stochastic Caputo fractional differential equations with the nonlinear terms satisfying the non-Lipschitz condition is considered. The work in the article is roughly divided into three parts. Firstly, we establish a generalized Gronwall inequality with singular integral kernel which is a key part in our analysis. Secondly, we discuss the existence and uniqueness of solution. And finally, the averaging principle is considered. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the tempered subdiffusive Black–Scholes model. The main part of our work consists of the finite difference method as a numerical approach to option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme. The proposed method has an accuracy order 2 - α with respect to time, where α ∈ (0 , 1) is the subdiffusion parameter and 2 with respect to space. Furthermore, we provide stability and convergence analysis. Finally, we present some numerical results. [ABSTRACT FROM AUTHOR]

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 and the averaged equations has been proved. The main results of the paper are applied to the switched capacitor voltage inverter modeling problem which is described by the fractional differential equations. [ABSTRACT FROM AUTHOR]

By the Schauder fixed point theorem and generalized Ascoli-Arzela theorem, we present that the Riemann-Liouville fractional differential equation has at least one globally attractive solution and x (t) = x 0 t β - 1 + o (t β - γ 1 ) as t → + ∞ , where β < γ 1 < γ < 1 . The novelty in this paper is that the global solution of Riemann-Liouville fractional differential equation is made up of two functions. One function x 1 ∈ C 1 - β (0 , T ] is a solution of fractional differential equation, the other function x 2 ∈ C 0 [ T , + ∞) is a solution of fractional integral equation. Finally, several examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

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 Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms' theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian's results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, "Fractional derivatives and the Cauchy problem for differential equations of fractional order"), and were invited by the "FCAA" editors to publish its re-edited version in this same issue of the journal. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of the S-asymptotically ω -periodic mild solutions to a class of multi-term time-fractional measure differential equations with nonlocal conditions in an ordered Banach spaces. Firstly, we look for suitable concept of S-asymptotically ω -periodic mild solution to our concern problem, by means of Laplace transform and (β , γ k) -resolvent family { S β , γ k (t) } t ≥ 0 . Secondly, we construct monotone iterative method in the presence of the lower and upper solutions to the delayed fractional measure differential equations, and obtain the existence of maximal and minimal S-asymptotically ω -periodic mild solutions for the mentioned system. Finally, as the application of abstract results, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of stochastic Riemann-Liouville type fractional differential equations driven by Lévy noise. By using Itô formula for the considered equation, we attempt to explore the non-confluence property of solution for the considered equation under some appropriate conditions. Our approach is to construct some suitable Lyapunov functions which is novel in exploring the non-confluence property of differential equations. [ABSTRACT FROM AUTHOR]

In this paper, our focus is on exploring the relative controllability of systems governed by linear fractional differential equations incorporating state delay. We introduce a novel counterpart to the Cayley-Hamilton theorem. Leveraging a delayed perturbation of the Mittag-Leffler function, along with a determining function and an analog of the Cayley-Hamilton theorem, we establish an algebraic Kalman-type rank criterion for assessing the relative controllability of fractional differential equations with state delay. Moreover, we articulate necessary and sufficient conditions for relative controllability criteria concerning linear fractional time-delay systems, expressed in terms of a new α -Gramian matrix and define a control which transfer the system from any initial state to any final state within a given time. The theoretical findings are exemplified through the presentation of illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, a class of cyclic (noncyclic) operators of condensing nature are defined on Banach spaces via a pair of shifting distance functions. The best proximity point (pair) results are manifested using the concept of measure of noncompactness (MNC) for the said operators. The obtained best proximity point result is used to demonstrate existence of optimum solutions of a system of differential equations involving ψ -Hilfer fractional derivatives of complex order. [ABSTRACT FROM AUTHOR]

Nonstationary fractional-order systems represent a new class of dynamic systems characterized by time-varying parameters as well as memory effect and hereditary properties. Differential game described by system of linear nonstationary differential equations of fractional order is treated in the paper. The game involves two players, one of which tries to bring the system's trajectory to a terminal set, whereas the other strives to prevent it. Using the technique of set-valued maps and their selections, sufficient conditions for reaching the terminal set in a finite time are derived. Theoretical results are supported by a model example. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for a system of fractional differential equations at resonance set on the interval [0, 1]. We associate a Dirichlet condition at t = 0 and an integral boundary condition at t = 1 . Our main existence theorem relies on the coincidence degree theory. A detailed study of the linear operators involved in the fixed point formulation is carried out. Finally, an example is included to illustrate the applicability of theoretical result. [ABSTRACT FROM AUTHOR]

In statistical seismology, the Epidemic Type Aftershocks Sequence (ETAS) model is a branching process used world-wide to forecast earthquake intensity rates and reproduce many statistical features observed in seismicity catalogs. In this paper, we describe a fractional differential equation that governs the earthquake intensity rate of the pure temporal ETAS model by using the Caputo fractional derivative and we solve it analytically. We highlight that the tools and special functions of fractional calculus simplify the classical methods employed to obtain the intensity rate and let us describe the change of solution decay for large times. We also apply and discuss the theoretical results to the Japanese catalog in the period 1965-2003. [ABSTRACT FROM AUTHOR]

This paper studies a class of the fractional stochastic evolution equation with delay. With the aid of the compact semigroup theory and Schauder fixed point theorem, the existence of square-mean S-asymptotically periodic mild solutions is obtained under the situation that the nonlinear functions satisfy certain growth conditions. Moreover, by establishing a new Grönwall integral inequality corresponding to fractional differential equation with delay, the global asymptotic stability of the square-mean S-asymptotically periodic mild solutions are discussed. Finally, an example is given to illustrate our abstract results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to deal with a new class of fractional evolutionary equations involving the time-fractional order integral and nonlinear weakly continuous operators. Exploiting the Rothe method and using a surjectivity result for weakly continuous operators, the solvability for the problem is established. The result is applied to prove the existence of solutions to time-fractional nonstationary incompressible Navier–Stokes equation and Navier–Stokes–Voigt equation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the exact solutions of linear homogeneous and nonhomogeneous fractional differential equations with double delays. Firstly, a new concept of double-delayed Mittag-Leffler type matrix function is introduced, which is the promotion of the double-delayed matrix exponential. Secondly, we apply the double-delayed Mittag-Leffler type matrix function and Laplace transform approach to obtain the exact solutions of fractional differential equations with double delays. Furthermore, the solution is used to investigate the Hyers-Ulam stability of the system. Lastly, we illustrate our techniques by an example. [ABSTRACT FROM AUTHOR]

In this paper, we study the parameter sensitivity analysis a class of fractional impulsive stochastic differential equation with S p -weighted pseudo almost periodic coefficients and infinite delay in Hilbert spaces. Firstly, a more appropriate concept of piecewise S p -weighted pseudo almost periodic in distribution for stochastic processes of class ∞ is introduced. The existence of piecewise weighted pseudo almost periodic mild solutions in distribution is presented by using α -order fractional resolvent operator theory, the stochastic analysis techniques and a fixed-point theorem. Secondly, the sensitivity properties of these infinite delay systems under non-Lipschitz conditions is also obtained. Finally, two examples are given for demonstration. [ABSTRACT FROM AUTHOR]

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, and discover that the approach of Luchko can be followed, with small modifications, in this more general setting too. The Mikusiński's operational calculus approach is used to obtain exact solutions of fractional differential equations with constant coefficients and with this type of fractional derivatives. These solutions can be expressed in terms of Mittag-Leffler type functions. [ABSTRACT FROM AUTHOR]

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. Dubatovskaya to prepare the survey paper in this same issue of "FCAA" and also to translate and edit the present paper in English. On behalf of Editorial Board and fractional calculus' community, we express to them our thanks for this hard work, including also retyping, mentioning some typos, etc. Authors' Summary: The concept of fractional integro-differentiation has found a number of applications in earlier papers of the present authors. With this paper we begin the publication of our results in the field of boundary problems for differential operators of fractional order. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of fractional semilinear integro-differential equations of order β ∈ (1,2] with nonlocal conditions. By using the solution operator, measure of noncompactness and some fixed point theorems, we obtain the existence of local and global mild solutions for the problem. The results presented in this paper improve and generalize many classical results. An example about fractional partial differential equations is given to show the application of our theory. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a Cauchy problem for a nonlinear fractional differential equation with the Caputo derivative of order α ∈ (0 , 1) . As initial data, we consider a pair consisting of an initial point, which does not necessarily coincide with the inferior limit of the fractional derivative, and a function that determines the values of a solution on the interval from this inferior limit to the initial point. We study differentiability properties of the functional associating initial data with the endpoint of the corresponding solution of the Cauchy problem. Stimulated by recent results on the dynamic programming principle and Hamilton–Jacobi–Bellman equations for fractional optimal control problems, we examine so-called fractional coinvariant derivatives of order α of this functional. We prove that these derivatives exist and give formulas for their calculation. [ABSTRACT FROM AUTHOR]

In this paper we study the exact solutions of a class of fractional delay differential equations. We consider the fractional derivative of the order between 1 and 2 in the sense of Caputo. In the first part, we introduce two novel matrix functions, namely, the generalized cosine-type and sine-type delay Mittag-Leffler matrix functions. Then we obtain the explicit solutions for the linear homogeneous equations subjecting to corresponding initial conditions, by means of undetermined coefficients. In the second part, we first obtain a particular solution by means of the Laplace transform for the inhomogeneous equations with null initial conditions. Then we give an analytical representation of the general solution of the inhomogeneous equations through the sum of its particular solution and the general solution of the corresponding homogeneous equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two types of problems (the initial-value problem and nonlocal Cauchy problem) for fractional differential equations involving ψ-Hilfer derivative in multivariable case (ψ-m-Hilfer derivative). First we propose and discuss ψ-fractional integral, ψ-fractional derivative and ψ-Hilfer type fractional derivative of a multivariable function f : ℝm → ℝ (m is a positive integer). Then, using the properties of the ψ-m-Hilfer fractional derivative with m = 1 (the ψ-Hilfer derivative), we derive an equivalent relationship between solutions to the initial-value (Cauchy) problem and solutions to some integral equations, and also present an existence and uniqueness theorem. Based on the equivalency relationship, we establish new and general existence results for the nonlocal Cauchy problem of fractional differential equations involving ψ-Hilfer multivariable operators in the space of weighted continuous functions. Moreover, we obtain a new Gronwall-type inequality with singular kernel, and derive the dependence of the solution on the order and the initial condition for the fractional Cauchy problem with the help of this Gronwall-type inequality. Finally, some examples are given to illustrate our results. Compared with the recent paper [2] and other previous works, the novelties in this paper are in treating the multivariable case of operators (f : ℝm → ℝ, m is a positive integer). [ABSTRACT FROM AUTHOR]

Hadamard type fractional calculus involves logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenge in numerical treatment. In this paper we present a spectral collocation method with mapped Jacobi log orthogonal functions (MJLOFs) as basis functions and obtain an efficient algorithm to solve Hadamard type fractional differential equations. We develop basic approximation theory for the MJLOFs and derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e, the fractional Helmholtz equation, and the fractional Burgers equation. [ABSTRACT FROM AUTHOR]

This paper concentrates on the stability and stabilization of short memory fractional differential equations with delayed impulses. The sufficient conditions for asymptotic stability of short memory fractional differential equations with two kinds of delayed impulses are derived, respectively. The results show that the delayed impulses in short memory fractional differential equations exhibit double effects on system performance. For an unstable system, one can stabilize the system by inputting delays in impulses; for a stable system, the stability would be destroyed if the delays were too long. Further, a class of fractional chaotic systems is presented to test the validity of the established theoretical results, some criteria for impulsive synchronization of fractional chaotic systems are derived, and the corresponding impulsive controllers are designed. Finally, a fractional Chua chaotic oscillator is presented to illustrate the practicability of the established impulsive controllers. [ABSTRACT FROM AUTHOR]

In this paper we introduce fractional quasi-Bessel equations ∑ i = 1 m d i x ξ i D α i u (x) + (x β - ν 2) u (x) = 0 and construct their existence theory in the class of fractional series solutions. In order to find the parameters of the series, we derive the characteristic equation, which is surprisingly independent of the terms with non-matching parameters ξ i ≠ α i . Our methodology allows us to obtain new results for a broad class of fractional differential equations including quasi-Euler equations. As a particular example, we demonstrate how our approach works for the constant-coefficient equations. The theoretical results are justified computationally. [ABSTRACT FROM AUTHOR]

This paper is devoted to the fractional generalization of the Fokker-Planck equation associated with a nonlinear stochastic differential equation on a bounded domain. The driving process of the stochastic differential equation is a Lévy process subordinated to the inverse of Lévy's mixed stable subordinators. The Fokker-Planck equation is given through the general Waldenfels operator, while the boundary condition is given through the general Wentcel's boundary condition. As a fractional operator a distributed order differential operator with a Borel mixing measure is considered. In the paper fractional generalizations of the Fokker-Planck equation are derived and the existence of a unique solution of the corresponding initial-boundary value problems is proved. [ABSTRACT FROM AUTHOR]

In this paper the accurate relationships between the averaging procedure of a smooth function over 1D-fractal sets and the fractional integral of the RL-type are established. The numerical verifications are realized for confirmation of the analytical results and the physical meaning of these obtained formulas is discussed. Besides, the generalizations of the results for a combination of fractal circuits having a discrete set of fractal dimensions were obtained. We suppose that these new results help understand deeper the intimate links between fractals and fractional integrals of different types. These results can be used in different branches of the interdisciplinary physics, where the different equations describing the different physical phenomena and containing the fractional derivatives and integrals are used. [ABSTRACT FROM AUTHOR]

Considerable attention has been recently given to the existence of solutions of initial or boundary value problems for fractional differential equations and inclusions with Hilfer fractional derivative. Motivated by these results, in this paper we will present existence, data dependence and Ulam stability results for some differential inclusions with Hilfer fractional derivative. The results follow as applications of the multi-valued weakly Picard operator theory. An example illustrates the main result of the paper. [ABSTRACT FROM AUTHOR]

As it is known various dynamical processes can be modeled through systems of time-fractional order pseudo-differential equations. In the modeling process one frequently faces with the problem of determination of adequate orders of time-fractional derivatives in the sense of Riemann–Liouville or Caputo. This problem is qualified as an inverse problem. The correct (vector) order can be found utilizing the available data. In this paper we offer an new method of solution of this inverse problem for linear systems of fractional order pseudo-differential equations. We prove that the Fourier transform of the vector-solution U ^ (t , ξ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{U}(t, \xi)$$\end{document} evaluated at a fixed time instance, which becomes possible due to the available data, recovers uniquely the unknown vector-order of a system of governing pseudo-differential equations. [ABSTRACT FROM AUTHOR]

I Aims i : The workshop follows the great success of Special Session "Fractional Diffusion Problems: Numerical Methods, Algorithms and Applications" organized within the scientific program of the 12th International Conference on Large-Scale Scientific Computations (LSSC'19), June 10-14, 2019, Sozopol, Bulgaria. Historically, fractional calculus has been recognized since the inception of regular calculus, with the first written reference dated in September 1695 in a letter from Leibniz to L'Hospital. B Description b : Due to its ubiquity across a variety of fields in science and engineering, fractional calculus has gained momentum in industry and academia. While a number of books and papers introduce either fractional calculus or numerical approximations, no current literature provides a comprehensive collection of both topics. [Extracted from the article]

In the paper we investigate the fractional-order evaluation of the fractional-order discrete integration element. We assume that the input and output signals are known. The main problem is to calculate fractional-order value. From a theoretical point of view there is no mathematical problem of the solution. One should solve linear algebraic equation or find roots of a polynomial in a variable ν. The problem arises when the measured output signal contains a noise. Then, the solution is unsettled because the polynomial roots are very sensitive to coefficients variability. In the paper we propose a method of evaluating of the discrete integrator fractional-order. The investigations are supported by numerical examples. [ABSTRACT FROM AUTHOR]

The accurate modelling of an electrical coil over a wide range of frequency is the keystone for a precise modelling of an electrical machine. As a consequence of copper losses, eddy-current losses and hysteresis losses; electrical coils with conductive ferromagnetic core show different behaviour from that of an ideal coil. Throughout this paper, dynamic modelling and performance analysis of conventional as well as fractional-order models of an electrical coil with an interchangeable core are achieved. Measurement results are acquired through an integration between Matlab and the high-speed measurement system LTT24. In order to assess the accuracy of these models, simulation results are compared with experimental results whereas unknown parameters are identified through an optimization process that is based on the method of least squares. It is known that the parameters of fractional-order model (Lα, α, Cβ, β) can not be measured directly. Therefore, the paper proposes a possibility based on system analysis to derive these parameters (indirect measurement) from the parameters of the classical model. A frequency band beyond the self-resonant frequency of the electrical coil is explored, thus the parasitic capacitance between coils windings must be considered as an important part of the equivalent circuit. The dependency of model parameters on frequency due to skin-effect is also examined. [ABSTRACT FROM AUTHOR]

It is a fundamental problem to determine a starting point in fractional differential equations which reveals the memory length in real life modeling. This paper describes it by an inverse problem. Fixed point theorems such as Krasnoselskii's and Schauder type's and nonlinear alternative for single–valued mappings are presented. Through existence analysis of the inverse problem, the range of the initial value points and the memory length of fractional differential equations are obtained. Finally, three examples are demonstrated to support the theoretical results and numerical solutions are provided. [ABSTRACT FROM AUTHOR]

In this paper we introduce a discrete fractional resolvent family { S α , β n } n ∈ N 0 $\begin{array}{} \displaystyle \{S_{\alpha,\beta}^n\}_{n\in\mathbb{N}_0} \end{array}$ generated by a closed linear operator in a Banach space X for a given α, β > 0. Moreover, we study its main properties and, as a consequence, we obtain a method to study the existence and uniqueness of the solutions to discrete fractional difference equations in a Banach space. [ABSTRACT FROM AUTHOR]

In this paper we establish the existence of solutions for the nonlinear abstract Cauchy problem of order α ∈ (1, 2), where the fractional derivative is considered in the sense of Caputo. The autonomous and nonautonomous cases are studied. We assume the existence of an α-resolvent family for the homogeneous linear problem. By using this α-resolvent family and appropriate conditions on the forcing function, we study the existence of classical solutions of the nonhomogeneus semilinear problem. The non-autonomous problem is discussed as a perturbation of the autonomous case. We establish a variation of the constants formula for the nonautonomous and nonhomogeneous equation. [ABSTRACT FROM AUTHOR]

This paper investigates the structure of solutions to the BVP of a class of fractional ordinary differential equations involving both fractional derivatives (R-L or Caputo) and fractional Laplacian with variable coefficients. This family of equations generalize the usual fractional diffusion equation and fractional Laplace equation. We provide a deep insight to the structure of the solutions shared by this family of equations. The specific decomposition of the solution is obtained, which consists of the "good" part and the "bad" part that precisely control the regularity and singularity, respectively. Other associated properties of the solution will be characterized as well. [ABSTRACT FROM AUTHOR]

In this paper, we formulate and prove two maximum principles to nonlinear fractional differential equations. We consider a fractional derivative operator with Mittag-Leffler function of two parameters in the kernel. These maximum principles are used to establish a pre-norm estimate of solutions, and to derive certain uniqueness and positivity results to related linear and nonlinear fractional initial value problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear fractional impulsive differential equation involving Sturm-Liouville boundary-value conditions and p-Laplacian operator. By making use of critical point theorem and variational methods, some new criteria are given to guarantee that the considered problem has infinitely many solutions. Our results extend some recent results and the conditions of assumptions are easily verified. Finally, an example is given as an application of our fundamental results. [ABSTRACT FROM AUTHOR]

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 kernels. This operational calculus is applied for analytical treatment of some initial value problems for the fractional differential equations with the general fractional derivatives. The solutions are expressed in form of the convolution series that generalize the power series for the exponential and the Mittag-Leffler functions. [ABSTRACT FROM AUTHOR]

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 of confidence intervals. Moreover, we provide rates of convergence in the CLT via Berry-Esseen type bounds. Concrete numerical computations and illustrations are included. [ABSTRACT FROM AUTHOR]

The present paper deals with initial value problem (IVP) for semilinear fractional Schrödinger integro-differential equation i du/dt + Au = ∫0t ƒ(s, Dsαu(s)ds, 0 < t < T, u(0) = 0 in a Hilbert space H with a self-adjoint positive definite (SAPD) operator A. Stable difference schemes (DSs) have significant interest in investigations of fractional partial differential equations. The main theorem concerns the existence and uniqueness of the uniformly bounded solutions (UBSs) with respect to step time of second order of accuracy DSs for this semilinear fractional Schrödinger differential problem. In practice, existence and uniqueness theorems for a UBS of the one-dimensional initial boundary value problem (BVP) with nonlocal condition and multi-dimensional problem with local condition on the boundary are proved. Numerical results and explanatory illustrations are presented to show the validation of the theoretical results. [ABSTRACT FROM AUTHOR]

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. MSC 2010: Primary 93B05; Secondary 34A08, 93E03 [ABSTRACT FROM AUTHOR]

In this paper, we discuss the maximum principle for a time-fractional diffusion equation ∂αt u(x, t)= ∑ni,j=1 ∂i(aij(x)∂ju(x, t))+c(x)u(x, t)+F(x, t), t>0, x ∈ Ω ⊂ ℝn, with the Caputo time-derivative of the order α ∈ (0, 1) in the case of the homogeneous Dirichlet boundary condition. Compared to the already published results, our findings have two important special features. First, we derive a maximum principle for a suitably defined weak solution in the fractional Sobolev spaces, not for the strong solution. Second, for the nonnegative source functions F = F(x, t) we prove the non-negativity of the weak solution to the problem under consideration without any restrictions on the sign of the coefficient c = c(x) by the derivative of order zero in the spatial differential operator. Moreover, we prove the monotonicity of the solution with respect to the coefficient c = c(x). [ABSTRACT FROM AUTHOR]

Series defined by means of the three-parametric Mittag-Leffler functions, called also the Prabhakar functions, are considered in this paper. Their behaviour is investigated on the boundaries of the convergence domains. Necessary and sufficient conditions for their overconvergence are proposed. The corresponding results for series in Mittag-Leffler functions are discussed as a particular case. Such kind of results are motivated by the fact that the solutions of some fractional order differential and integral equations can be written in terms of series (or series of integrals) of Mittag-Leffler type functions. [ABSTRACT FROM AUTHOR]