Showing papers in "Fractional Calculus and Applied Analysis in 2014"
TL;DR: In this article, the existence and uniqueness of solutions for a coupled system of Hadamard type fractional differential equations and integral boundary conditions is studied. But the uniqueness of solution is not established by the contraction principle.
Abstract: This paper is concerned with the existence and uniqueness of solutions for a coupled system of Hadamard type fractional differential equations and integral boundary conditions. We emphasize that much work on fractional boundary value problems involves either Riemann-Liouville or Caputo type fractional differential equations. In the present work, we have considered a new problem which deals with a system of Hadamard differential equations and Hadamard type integral boundary conditions. The existence of solutions is derived from Leray-Schauder’s alternative, whereas the uniqueness of solution is established by Banach’s contraction principle. An illustrative example is also included.
124 citations
TL;DR: In this paper, the existence and multiplicity of solutions for an impulsive boundary value problem for fractional order differential equations are studied and the notions of classical and weak solutions are introduced.
Abstract: In this paper we study the existence and the multiplicity of solutions for an impulsive boundary value problem for fractional order differential equations. The notions of classical and weak solutions are introduced. Then, existence results of at least one and three solutions are proved.
94 citations
TL;DR: In this article, the authors investigated the asymptotic behavior of solutions to the initial-boundary-value problem for distributed order time-fractional diffusion equations in bounded multi-dimensional domains.
Abstract: This article deals with investigation of some important properties of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations in bounded multi-dimensional domains In particular, we investigate the asymptotic behavior of the solutions as the time variable t → 0 and t → +∞ By the Laplace transform method, we show that the solutions decay logarithmically as t → +∞ As t → 0, the decay rate of the solutions is dominated by the term (t log(1/t))−1 Thus the asymptotic behavior of solutions to the initial-boundary-value problem for the distributed order time-fractional diffusion equations is shown to be different compared to the case of the multi-term fractional diffusion equations
90 citations
TL;DR: In this article, a re-printed version of the paper, translated from Russian into English by Marina Shitikova, under the kind permission by the Editorial Board of the journal "Prikladnaya Matematika i Mekhanika", Institute of Mechanics, Russian Academy of Sciences, http://pmmnet.ipmnet.ru/
Abstract: Editorial Note: The original version of the paper is usually cited as Yu.N. Rabotnov, Equilibrium of an elastic medium with after-effect (in Russian). Prikladnaya Matematika i Mekhanika (J. Appl. Math. Mech.) 12, No 1 (1948), 53–62. This is a re-printed version of the paper, translated from Russian into English by Marina Shitikova, under the kind permission by the Editorial Board of the journal “Prikladnaya Matematika i Mekhanika”, Institute of Mechanics — Russian Academy of Sciences, http://pmm.ipmnet.ru/ru/
, a journal published with English translation since 1958, as “Journal of Applied Mathematics and Mechanics” by Elsevier http://www.journals.elsevier.com/journal-of-applied-mathematics-and-mechanics/
. Note that nowadays, the fractional exponential function ∋
α
(β, t) introduced by Rabotnov as (2.5), is known also as the Rabotnov function and as a special case of the Mittag-Leffler function widely used in fractional calculus. Rabotnov spoke also about differential equations with fractional derivatives (Sect. 3) but preferred to work with integral equations methods. Yury Nikolaevich Rabotnov (24 February 1914–15 May 1985) was a great Russian scientist in the field of Mechanics. More about his life and contributions, the readers can find in this journal (Fract. Calc. Appl. Anal.), in the articles:
90 citations
TL;DR: In this article, a fractional-order generalized Laguerre pseudo-spectral approximation method was proposed for solving nonlinear initial value problem of fractional order ν (0 < ν < 1).
Abstract: Fractional-order generalized Laguerre functions (FGLFs) are proposed depends on the definition of generalized Laguerre polynomials. In addition, we derive a new formula expressing explicitly any Caputo fractional-order derivatives of FGLFs in terms of FGLFs themselves. We also propose a fractional-order generalized Laguerre tau technique in conjunction with the derived fractional-order derivative formula of FGLFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). The fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on FGLFs and compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.
84 citations
TL;DR: In this paper, the initial-boundary-value problems for the one-dimensional linear and non-linear fractional diffusion equations with the Riemann-Liouville time-fractional derivative are analyzed.
Abstract: In this paper, the initial-boundary-value problems for the one-dimensional linear and non-linear fractional diffusion equations with the Riemann-Liouville time-fractional derivative are analyzed. First, a weak and a strong maximum principles for solutions of the linear problems are derived. These principles are employed to show uniqueness of solutions of the initial-boundary-value problems for the non-linear fractional diffusion equations under some standard assumptions posed on the non-linear part of the equations. In the linear case and under some additional conditions, these solutions can be represented in form of the Fourier series with respect to the eigenfunctions of the corresponding Sturm-Liouville eigenvalue problems.
64 citations
TL;DR: In this paper, the multiplicity of solutions for fractional differential equations subject to boundary value conditions and impulses is studied, and the existence of at least three solutions to the impulsive problem considered is proved.
Abstract: We study the multiplicity of solutions for fractional differential equations subject to boundary value conditions and impulses. After introducing the notions of classical and weak solutions, we prove the existence of at least three solutions to the impulsive problem considered.
64 citations
TL;DR: In this article, the existence of a class of fractional initial value problems (FIVP) with fixed-point theorems and lower and upper solution methods was proved.
Abstract: In this paper, by using fixed-point theorems, and lower and upper solution method, the existence for a class of fractional initial value problem (FIVP) D αu(t )= f (t, u(t)) ,t ∈ (0 ,h ), t 2−α u(t) |t=0= b1 ,D α−1 0+ u(t) |t=0= b2,
62 citations
TL;DR: In this article, a distributed coordination of fractional-order multi-agent systems (FOMAS) with communication delays is studied, and a critical value of time delay is obtained to ensure the consensus of FOMAS.
Abstract: Because of the complexity of the practical environments, many distributed multi-agent systems can not be illustrated with the integer-order dynamics and can only be described with the fractional-order dynamics. Under the connected network with directed weighted topologies, the dynamical characteristics of agents with fractional-order derivative operator is analyzed in this paper. Applying the Laplace transform and frequency domain theory of the fractional-order operator, the distributed coordination of fractional-order multi-agent systems (FOMAS) with communication delays is studied, and a critical value of time delay is obtained to ensure the consensus of FOMAS. Since the integer-order model is a special case of fractional-order model, the extended results in this paper are in accordance with that of the integer-order model. Finally, numerical examples are provided to verify our results.
62 citations
TL;DR: In this paper, an initial-boundary value problem for the velocity distribution of a viscoelastic flow with generalized fractional Oldroyd-B constitutive model is studied.
Abstract: An initial-boundary value problem for the velocity distribution of a viscoelastic flow with generalized fractional Oldroyd-B constitutive model is studied. The model contains two Riemann-Liouville fractional derivatives in time. The eigenfunction expansion of the solution is constructed. The behavior of the time-dependent components of the solution is studied and the results are used to establish convergence of the series under some conditions. Further, applying the convolutional calculus approach proposed by Dimovski (I.H. Dimovski, Convolutional Calculus, Kluwer, Dordrecht (1990)), a Duhamel-type representation of the solution is found, containing two convolution products of particular solutions and the given initial and source functions. A non-classical convolution with respect to spatial variable is used. The obtained representation is applied for numerical computation of the solution in the case of a generalized second grade fluid. Numerical results for several one-dimensional examples are given and the present technique is compared to a finite difference method in terms of efficiency, accuracy, and CPU time.
60 citations
TL;DR: In this paper, the authors considered a nonlinear fractional boundary value problem defined on a star graph and used a transformation to obtain an equivalent system of boundary value problems with mixed boundary conditions, then the existence and uniqueness of solutions are investigated by fixed point theory.
Abstract: In this paper, the authors consider a nonlinear fractional boundary value problem defined on a star graph. By using a transformation, an equivalent system of fractional boundary value problems with mixed boundary conditions is obtained. Then the existence and uniqueness of solutions are investigated by fixed point theory.
TL;DR: In this paper, a particular hyper-Bessel operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient is considered.
Abstract: From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument due to the presence of Erdelyi-Kober fractional integrals in our operator. We present solutions, both singular and regular in the time origin, that are locally integrable and completely monotone functions in order to be consistent with the physical phenomena described by non-negative relaxation spectral distributions.
TL;DR: In this article, the authors study thermomechanical models with memory and compare them with the classical Volterra theory, showing that the fractional models involve significant differences in the type of kernels and predict important changes in the behavior of fluids and solids.
Abstract: Within the fractional derivative framework, we study thermomechanical models with memory and compare them with the classical Volterra theory. The fractional models involve significant differences in the type of kernels and predicts important changes in the behavior of fluids and solids. Moreover, an analysis of the thermodynamic restrictions provides compatibility conditions on the kernels and allows us to determine certain free energies, which in turn enables the definition of a topology on the history space. Finally, an analogous analysis is carried out for the phenomenon of heat propagation with memory.
TL;DR: In this paper, a fractional differential equation of the Euler-Lagrange/Sturm-Liouville type is considered, and a numerical scheme is presented.
Abstract: In this paper a fractional differential equation of the Euler-Lagrange/Sturm-Liouville type is considered. The fractional equation with derivatives of order α ∈ (0, 1] in the finite time interval is transformed to the integral form. Next the numerical scheme is presented. In the final part of this paper examples of numerical solutions of this equation are shown. The convergence of the proposed method on the basis of numerical results is also discussed.
TL;DR: In this paper, the theory for u0-positive operators was applied to obtain eigenvalue comparison results for a fractional boundary value problem with the Caputo derivative, and the results were used to obtain the eigenvalues of the u 0-positive operator.
Abstract: We apply the theory for u0-positive operators to obtain eigenvalue comparison results for a fractional boundary value problem with the Caputo derivative.
TL;DR: In this paper, the authors define fractional Skellam processes via the time changes in Poisson and Skekam processes by an inverse of a standard stable subordinator.
Abstract: The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes.
Abstract: In the past decade, researchers working on fractional-order systems modeling and control have been considering working on the design and development of analog and digital fractional-order differentiators, i.e. circuits that can perform non-integer-order differentiation. It has been one of the major research areas under such field due to proven advantages over its integer-order counterparts. In particular, traditional integer-order proportional-integral-derivative (PID) controllers seem to be outperformed by fractional-order PID (FOPID or PIλDμ) controllers. Many researches have emerged presenting the possibility of designing analog and digital fractional-order differentiators, but only restricted to a fixed order. In this paper, we present the conceptual design of a variable fractional-order differentiator in which the order can be selected from 0 to 1 with an increment of 0.05. The analog conceptual design utilizes operational amplifiers and resistor-capacitor ladders as main components, while a generic microcontroller is introduced for switching purposes. Simulation results through Matlab and LTSpiceIV show that the designed resistor-capacitor ladders can perform as analog fractional-order differentiation.
TL;DR: In this paper, Dimovski introduced and started detailed studies on the Bessel type differential operators B of arbitrary (integer) order m ≥ 1, and also suggested a variant of the Obrechkoff integral transform (arising in a paper of 1958 by another Bulgarian mathematician Nikola Obrechtkoff) as a Laplace-type transform basis of a corresponding operational calculus for B and for its linear right inverse integral operator L. Later, the developments on these linear singular differential operators appearing in many problems of mathematical physics, have been continued by the author of this survey
Abstract: In 1966 Ivan Dimovski introduced and started detailed studies on the Bessel type differential operators B of arbitrary (integer) order m ≥ 1. He also suggested a variant of the Obrechkoff integral transform (arising in a paper of 1958 by another Bulgarian mathematician Nikola Obrechkoff) as a Laplace-type transform basis of a corresponding operational calculus for B and for its linear right inverse integral operator L. Later, the developments on these linear singular differential operators appearing in many problems of mathematical physics, have been continued by the author of this survey who called them hyper-Bessel differential operators, in relation to the notion of hyper-Bessel functions of Delerue (1953), shown to form a fundamental system of solutions of the IVPs for By(t )= λy(t). We have been able to extend Dimovski’s results on the hyper-Bessel operators and on the Obrechkoff transform due to the happy hint to attract the tools of the special functions as Meijer’s G-function and Fox’s H-function to handle successfully these matters. These author’s studies have lead to the
TL;DR: In this article, the authors prove that the initial value problem has the solution if and only if some initial values are zero, which is the case for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients.
Abstract: This paper provides results on the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski’s type. We prove that the initial value problem has the solution if and only if some initial values are zero.
TL;DR: In this article, the authors provide a critical assessment of numerical solutions of the space-fractional diffusion-advection equation, which is of high significance for applications in various natural sciences.
Abstract: The present work provides a critical assessment of numerical solutions of the space-fractional diffusion-advection equation, which is of high significance for applications in various natural sciences. In view of the fact that, in contrast to the case of normal (Gaussian) diffusion, no standard methods and corresponding numerical codes for anomalous diffusion problems have been established yet, it is of importance to critically assess the accuracy and practicability of existing approaches. Three numerical methods, namely a finite-difference method, the so-called matrix transfer technique, and a Monte-Carlo method based on the solution of stochastic differential equations, are analyzed and compared by applying them to three selected test problems for which analytical or semi-analytical solutions were known or are newly derived. The differences in accuracy and practicability are critically discussed with the result that the use of stochastic differential equations appears to be advantageous.
TL;DR: In this paper, the concept of fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities and fractional delay discrete-time linear systems are introduced.
Abstract: This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform
TL;DR: In this paper, the authors provide a survey on the recently obtained results which are useful in time response analysis of fractional-order control systems, including error signal analysis and the analysis of the control signal and the system response to the load disturbances.
Abstract: The aim of this paper is to provide a survey on the recently obtained results which are useful in time response analysis of fractional-order control systems. In this survey, at first some results on error signal analysis in fractional-order control systems are presented. Then, some previously obtained results which are helpful for system output analysis in fractional-order control systems are summarized. In addition, some results on the analysis of the control signal and the system response to the load disturbances in fractional-order control systems are reviewed.
TL;DR: In this paper, the authors extend a reliable modification of the Adomian decomposition method presented in [34] for solving the initial value problem for fractional differential equations (FDEs).
Abstract: In this paper, we extend a reliable modification of the Adomian decomposition method presented in [34] for solving initial value problem for fractional differential equations.
TL;DR: Using the memory conception developed in the framework of the Mori-Zwanzig formalism, the kinetic equations for relaxation functions that correspond to the previously suggested empirical functions (Cole-Davidson and Havriliak-Negami) are derived as discussed by the authors.
Abstract: Using the memory conception developed in the framework of the Mori-Zwanzig formalism, the kinetic equations for relaxation functions that correspond to the previously suggested empirical functions (Cole-Davidson and Havriliak-Negami) are derived. The obtained kinetic equations contain differential operators of non-integer order and have clear physical meaning and interpretation. The derivation of the memory function corresponding to the Havriliak-Negami relaxation law in the frame of Mori-Zwanzig formalism is given. A physical interpretation of the power-law exponents involved in the Havriliak-Negami empirical expression is provided too.
TL;DR: In this paper, the authors investigated the asymptotic behavior of solutions of linear fractional differential equations, and showed that the spectrum of the fractional Lyapunov exponent of a solution of a bounded linear FDE is always nonnegative.
Abstract: Our aim in this paper is to investigate the asymptotic behavior of solutions of linear fractional differential equations. First, we show that the classical Lyapunov exponent of an arbitrary nontrivial solution of a bounded linear fractional differential equation is always nonnegative. Next, using the Mittag-Leffler function, we introduce an adequate notion of fractional Lyapunov exponent for an arbitrary function. We show that for a linear fractional differential equation, the fractional Lyapunov spectrum which consists of all possible fractional Lyapunov exponents of its solutions provides a good description of asymptotic behavior of this equation. Consequently, the stability of a linear fractional differential equation can be characterized by its fractional Lyapunov spectrum. Finally, to illustrate the theoretical results we compute explicitly the fractional Lyapunov exponent of an arbitrary solution of a planar time-invariant linear fractional differential equation.
TL;DR: In this article, the existence of decay integral solutions to a class of retarded fractional differential equations involving impulsive effects was established by using the fixed point approach and fractional calculus tools in Banach spaces.
Abstract: The aim of this paper is to establish the existence of decay integral solutions to a class of retarded fractional differential equations involving impulsive effects. The results are obtained by using the fixed point approach and fractional calculus tools in Banach spaces. Applications to both ordinary and partial differential equations are presented.
TL;DR: In this article, a family of generalized Erdelyi-Kober type fractional integrals is interpreted geometrically as a distortion of the rotationally invariant integral kernel of the Riesz fractional integral in terms of generalized Cassini ovaloids on RN.
Abstract: A family of generalized Erdelyi-Kober type fractional integrals is interpreted geometrically as a distortion of the rotationally invariant integral kernel of the Riesz fractional integral in terms of generalized Cassini ovaloids on RN. Based on this geometric point of view, several extensions are discussed.
TL;DR: In this article, the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces were proved and generalized to the fractional order case.
Abstract: In this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.
TL;DR: For the fractional diffusion-wave equation with the Caputo-Djrbashian fractional derivative of order α ∈ (1, 2) with respect to the time variable, this paper proved an analog of the principle of limiting amplitude and a pointwise stabilization property of solutions.
Abstract: For the fractional diffusion-wave equation with the Caputo-Djrbashian fractional derivative of order α ∈ (1, 2) with respect to the time variable, we prove an analog of the principle of limiting amplitude (well-known for the wave equation and some other hyperbolic equations) and a pointwise stabilization property of solutions (similar to a well-known property of the heat equation and some other parabolic equations).
TL;DR: In this paper, the robust stability bound problem of uncertain fractional-order systems is considered and robust stability bounds on the uncertainties are derived. But the robustness of these bounds depends on the assumption that the system is already asymptotically stable.
Abstract: This paper considers the robust stability bound problem of uncertain fractional-order systems. The system considered is subject either to a two-norm bounded uncertainty or to a infinity-norm bounded uncertainty. The robust stability bounds on the uncertainties are derived. The fact that these bounds are not exceeded guarantees that the asymptotical stability of the uncertain fractional-order systems is preserved when the nominal fractional-order systems are already asymptotically stable. Simulation examples are given to demonstrate the effectiveness of the proposed theoretical results.