Search

This paper presents necessary and sufficient optimality conditions for variational problems involving distributed-order fractionals derivatives with arbitrary smooth kernels and generalized Lagrangians. Many important results from the fractional calculus of variations can be obtained from our main results. [ABSTRACT FROM AUTHOR]

The present paper deals with the applications of the half-derivative (derivative of order 1/2) and of the half-integral (integral of order 1/2) in control system design, with particular focus on sustainability. Fractional Calculus (FC) is the branch of mathematics which deals with derivatives and integrals whose order is non-integer. Nowadays FC is successfully applied in engineering applications, and in particular in control system design. The most widespread approach to apply FC to control systems is the so-called Fractional Order PID scheme (FOPID), which is a PID with non-integer order integrals and derivatives of the error. This scheme has two more control parameters, the orders of integration and derivation, thus improving the closed-loop behavior. An alternative approach to apply FC to control system is the Distributed Order PID (DOPID), in which the control output is the sum n differintegrators, with n odd and≥3, and with orders which are equally spaced in the interval [−1, +1]. For n=5, the differintegration orders are −1, −1/2, 0, +1/2, +1, adding to the PID the half-integral term (order −1/2) and the half-derivative term (order +1/2). The effectiveness of DOPID has been investigated in the scientific literature, and can lead to lower energy consumption with similar control readiness and error. Considering Multi-Input Multi-Output (MIMO) systems, and in particular robotic manipulators, a possible application of the half-derivative is the so-called KDHD impedance control. With respect to classical impedance control, KD, characterized by a stiffness and a damping term, a half-derivative damping term is added, proportional to the half-derivative of the end-effector error through the half-derivative damping matrix HD. Also in this case, additional control parameters, the elements of the matrix HD, allow a more efficient optimization of the closed-loop behavior. This kind of controllers represent an effective and nearly cost-free evolution of classical integer-order schemes, able to reduce energy consumption of mechatronic and automation devices. Another possible promising application of the half-derivative damping is the exploitation of renewable energies, for example for controlling the power take-off systems of Wave Energy Converters. In the paper, some possible future sustainability-oriented research directions involving the use of half-derivatives and half-integrals in control systems are outlined. [ABSTRACT FROM AUTHOR]

The optimum design of a controller plays a significant role in obtaining an adequate response from the process plant. However, in several areas of fractional order control system design, the coefficient diagram method (CDM) has not been thoroughly explored. The article proposes an optimal non-integer PIλDμ controller integrated with the CDM to accurately regulate the location of an unstable Magnetic Ball Levitation System (MLS). The controller maintains a predetermined force on the electromagnetic coil, allowing the ball to levitate. The controller's design entities are derived through the use of the CDM and a recently proposed Mayfly optimizer algorithm(MOA) that incorporates IAE, ITAE, ISE, and ITSE performance criteria, offering efficient, intelligent, and speedy solutions to complex design problems. The paper aims to leverage the powerful computational capabilities of fractional calculus (FC) in combination with the benefits of the CDM control strategy across various fields of control theory. The article also proposes a design methodology for the CDM-PIλDµ controller, which offers improved performance for systems containing non-linear and time-varying parameters. Simulation results demonstrate a comparativeanalysis of the transient response, including optimum overshoot, rise time, and settling time. [ABSTRACT FROM AUTHOR]

In the present paper, we explore some new formulae of fractional calculus, including the general class of polynomials given by Srivastava [5] and the generalized Hurwitz–Lerch zeta function given by Nadeem et al. [11]. The obtained results are in the form of hypergeometric function, which are made with the help of Hadamard product. We have also derived some important special cases from the main results. [ABSTRACT FROM AUTHOR]

Fractional calculus appears to be a powerful tool of solid mechanics. In recent years several papers have been published including various forms of non-localities. Two basic fields can be distinguished, there are non-locality in space and in time. When term non-locality is used in its original meaning the value of some quantity in an internal point of the body is determined by the values of other quantities in a whole region around that location. The second one means to include hereditary effects like non-classical viscosity and study fractional visco-elasticity or visco-plasticity. In non-linear stability investigation the way of the loss of stability is studied and classified as a generic static or dynamic bifurcation. To do that step some kind of regular condition is necessary. This condition is connected with non-locality. Generally such behaviour is a result of viscous (time derivative) and gradient dependent terms in the constitutive equations. Such derivatives are not always of first order. There are materials where tests justify models with fractional order derivatives. Moreover, there are (for example Riemann-Liouville) type of fractional derivatives which are non-local itself. Thus by defining strain by fractional derivation of the displacement field a non-local quantity appears instead of the conventional (local) strain. In such a way various versions of non-localities are obtained by using various types of fractional derivatives. The paper studies how the selection of that fractional derivative effects the way of the loss of stability. Basically the study aims constitutive modelling via instability phenomena, that is, by observing the way of loss of stability of some material we can be informed about the form of fractional derivative in its mathematical model. [ABSTRACT FROM AUTHOR]

This paper is a continuation, as Part 2, of the recent author’s papers [18], [19], [20], where we have emphasized on our general and unified approach to evaluate operators of fractional calculus of a very general class of special functions. Namely, we have results for images of the Wright generalized hypergeometric functions (W. g.h.f.-s) under the operators of classical and generalized fractional calculus. Thus, great part of results published by other authors in numerous papers (for part of them - see references in the above mentioned 3 papers and also, herein) come as immediate particular cases. It is because the special functions considered there are all of them Wright g.h.f.-s, and the FC operators like the Riemann-Liouville (RL), Erdélyi-Kober (EK), Saigo, Marichev-Saigo-Maeda (MSM), etc., are all of them particular cases of the operators of Generalized Fractional Calculus (GFC), [11]. We gave previously long list of illustrative examples for the efficiency of the general approach, and now continue it. Recently, some authors repeated the job to evaluate FC operators of some special functions which they introduced and considered as “new” ones. Among them are some examples of the so-called k-analogues of the Bessel and Mittag-Leffler functions, generalized multi-index Bessel and Mittag-Leffler functions, K- and M-series and S -functions. In Section 5 we show that all these are just cases, again, of the Wright generalized hypergeometric function. Then, the results provided by the mentioned authors can come easily from our general ones. [ABSTRACT FROM AUTHOR]

In this paper, the homotopy perturbation method will be applied to find the approximate solution of fractional order random ordinary differential equations, in which the fractional order derivatives and integrals are defined using Caputo and Riemann-Liouville definitions of fractional derivatives and integrals, respectively. Also, the convergence of the approximated solution is stated and proved in this work. The work is verified for three different examples, which are simulated for different generations of Brownian motion. [ABSTRACT FROM AUTHOR]

Fractional differential equations with strong nonlinearity are important in modelling complex physical phenomena. Therefore, there is an urgent need to employ new technique to help researches and physicists to understand the physical problems better and to deal with such equations. In this paper, we develop and analyze a semi-analytical technique called the homotopy analysis method (HAM) which solve first order fuzzy fractional differential equation with strong nonlinearity under Caputo derivative sense by adjusting the convergence of the series solution. The paper also presents the fuzzy formulation of the proposed HAM using the concepts of fuzzy set theory combined with the properties of fractional calculus and analyzed it in order to obtain the semi-analytical solution. The feasibility of the technique is presented by solving an example of first order nonlinear fractional fuzzy initial value problem (NFFIVP). Finally, the numerical results show that the applied technique is accurate and reliable in solving first order NFFIVP. Thus, the proposed HAM can be extended to treat more general fuzzy fractional order models in physics and engineering. [ABSTRACT FROM AUTHOR]

The harmonic properties concerning the fractional derivative of Riemann zeta function are presented through the computation of the double one-sided Fourier transform. In this paper, it is computed both analytically and numerically. The symmetry of this integral transform is shown and discussed through the investigation of real and imagine parts. In addition, the link between the fractional derivative of Riemann zeta function and wavelet analysis is introduced. [ABSTRACT FROM AUTHOR]

In this paper, we define a subclass of p-valent (starlike of order α and type β) functions with negative coefficients using fractional calculus operator. We determine and obtain the coefficient estimate, distortion bounds and interesting properties for the subclass of p-valent functions. [ABSTRACT FROM AUTHOR]

In the present paper, r-modified Crank Nicholson difference schemes are presented for a fractional Schro¨dinger differential equation. The stability theorem for r-modified Crank Nicholson difference schemes is established. Theoretical results are supported by a numerical experiment. [ABSTRACT FROM AUTHOR]

In this research paper, we extended an existing SIR epidemic integer model containing two strains and two vaccinations by using a system of fractional ordinary differential equations in the sense of Caputo derivative of order σ ∈ (0, 1]. Four equilibrium points were established which are disease free equilibrium, strain1 disease free equilibrium, strain2 disease free equilibrium and endemic equilibrium. Explicit analysis of the equilibrium points of the model was given by applying fractional calculus and Routh-Hurwitz criterion. Stability analysis of the equilibrium points was carried out by employing the Jacobian matrix. Numerical simulations were iterated to support the analytic results. It was shown that when both of the reproduction numbers R1 and R2 are less than one, the disease die out over time and while it persist in relation to the thriving strain when either of them is greater than one. We also studied the effect of vaccine. With the fractional order technique, the memory effect of the system is made visible and hence easier to predict. [ABSTRACT FROM AUTHOR]

Fractional calculus is a dynamic research field for mathematicians, engineers and physicists. The qualitative properties of fractional differential equations have significant growth due to their ability to model the real-world phenomena. In this research paper, Ulam-Hyers stability of nonlinear Pantograph fractional differential equation involving the Mittag–Leffler integral operator in the form Atangana – Baleanu derivative is analyzed. The existence and uniqueness of solutions are obtained by employing the fixed point theorems such as Arzela-Ascoli theorem, Schauders theorem and Banach contraction principle. Also using results of fixed points theorems and properties, adequate conditions for Ulam-Hyers(UH) stability and Generalized Ulam-Hyers stability are established. [ABSTRACT FROM AUTHOR]

Over the recent years, from among new mathematical methods the fractional differential calculus arouses particularly big hope. Some scientists expect that the fractional calculus will enable new discoveries and offer a new perspective on the old well-known problems. The equations of the harmonic oscillator is fundamental for many theories and models in physics and mechanics, and their solutions in the classical calculus are very well known. Some scientists attempt to look at them in a different way by replacing the classical derivatives with fractional ones and creating a set of new fractional equations. The aim of the paper is comparison of the classical equation for a damped harmonic oscillator with the fractional differential equation especially with fractional Riemann derivative. [ABSTRACT FROM AUTHOR]

Fractional calculus is widely used in formulation of constitutive relations for viscoelastic materials. Problems of vibration of continuous structures, when such constitutive relations are used, leads to fractional differential equations similar to the equation of forced, damped, harmonic oscillator. In this paper two fractional oscillators are investigated (corresponding to two constitutive relations of viscoelastic materials). Closed-form solutions are obtained using Laplace Transform method, resonance curves for these oscillators are calculated numerically, and influence on its shape of the model parameters are analyzed. [ABSTRACT FROM AUTHOR]

The Ward-Tordai integral equation governs the diffusion-controlled surfactant adsorption at air/liquid interfaces. In this paper the Ward-Tordai equation is generalized in two directions. First, the adsorption is assumed to take place at a liquid/liquid interface, where the surfactant is soluble in both liquid phases. Second, the diffusion in the bulk phases is anomalous and is governed by time-fractional diffusion equations. For the computation of the change of adsorption with time two numerical techniques are proposed and compared. Numerical results are presented. [ABSTRACT FROM AUTHOR]

In this paper, some elements of the theory of the Wright function φ are discussed. The Wright function-along with the Mittag-Leffler function-plays a prominent role in the theory of the partial differential equations of the fractional order that are actively used nowadays for modeling of many phenomena including e.g. the anomalous diffusion processes or in the theory of the complex systems. This function appears there simultaneously as a Green function in the initial-value problems for the model linear equations with the constant coefficients and as a special solution invariant under the groups of the scaling transformations of the fractional differential equations. In this paper, both of these applications are shortly introduced. Whereas the analytical theory of the Wright function is already more or less well developed, its numerical evaluation is still an area of the active research. In this paper, the numerical evaluation of the Wright function is discussed with a focus on the case of the real axis that is very important for applications. In particular, several approaches are presented including the method of series summation, integral representations, and asymptotical expansions. In different parts of the complex plane different numerical techniques are employed. In each case, estimates for accuracy of the computations are provided. [ABSTRACT FROM AUTHOR]

In this paper, the anomalous diffusion processes are modeled with partial differential equations of the fractional order that are then discussed in details. The anomalous diffusion can be characterized by the property that it no longer follows the Gaussian statistics and in particular one observes a deviation from the linear time dependence of the mean squared displacement. This is the case for many different phenomena including, e.g., the translocation dynamics of a polymer chain through a nanopore, charge carrier transport in amorphous semiconductors, laser cooling in quantum optical systems to mention only few of them. In this paper, we consider the case of the anomalous diffusion that shows a power-low growth of the mean squared displacement in time. Our starting point is a stochastic formulation of the model in terms of the random walk processes. Following this line, a continuous time random walk model in form of a system of the integral equations of convolution type for the corresponding probability density functions is introduced. These so called master equations can be explicitly solved in the Fourier-Laplace domain. The time-fractional differential equation is then derived asymptotically from the master equations for the special classes of the probability density functions with the infinite first moment. For the obtained model equation and its generalizations the initial-boundary-value problems in the bounded domains are discussed. A special focus is on the initial-boundary-value problems for the generalized time-fractional diffusion equation. For this equation, the maximum principle well known for the elliptic and parabolic type PDEs is presented and applied both for the a priori estimates of the solution and for the proof of its uniqueness. Finally, first the existence of the generalized solution and then the existence of the solution under some restrictions are shown. [ABSTRACT FROM AUTHOR]

In this paper we give a generalization of the classical Fourier transformation, known as fractional Fourier transformation (FRFT). We also prove main properties of the FRFT and discuss some of its applications. [ABSTRACT FROM AUTHOR]

In this paper, we shall construct a new numerical scheme for fourth-order fractional diffusion wave model. The solvability, stability and convergence of proposed method are established in l2 norm and it is shown that the numerical scheme improves the earlier work done. Simulation is carried out to verity the accuracy and efficiency of the numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a numerical scheme for a fuzzy time fractional diffusion equation. Two finite difference schemes, that is the forward time centre space (FTCS) and the Crank-Nicholson methods, are studied. The time fractional derivative is defined using the Caputo formula. A numerical example is presented to illustrate the feasibility of the proposed methods. The obtained results show that the Crank-Nicholson method achieves more accurate solution compared with the FTCS method [ABSTRACT FROM AUTHOR]

In this paper we discuss a fractional order predator-prey model with ratio-dependent functional response. The dynamical properties of this model is analyzed. Here we determine all equilibrium points of this model including their existence conditions and their stability properties. It is found that the model has two type of equilibria, namely the predator-free point and the co-existence point. If there is no co-existence equilibrium, i.e. when the coefficient of conversion from the functional response into the growth rate of predator is less than the death rate of predator, then the predator-free point is asymptotically stable. On the other hand, if the co-existence point exists then this equilibrium is conditionally stable. We also construct a nonstandard Grnwald-Letnikov (NSGL) numerical scheme for the propose model. This scheme is a combination of the Grnwald-Letnikov approximation and the nonstandard finite difference scheme. This scheme is implemented in MATLAB and used to perform some simulations. It is shown that our numerical solutions are consistent with the dynamical properties of our fractional predator-prey model. [ABSTRACT FROM AUTHOR]

In this paper we point out on some author's ideas and developments, combined with few basic classical results, that show how one can do the task posed in the title at once, and in rather general case: for both operators of generalized fractional calculus and generalized hypergeometric functions. In this way, the greater part of the results in the publications mentioned in References (and many others not in this limited list) are well predicted and come just as very special cases of the discussed general scheme. [ABSTRACT FROM AUTHOR]

In this paper, the case of an equation involving fractional derivatives with respect to a single independent variable has been analyzed. Our aim is to determine its Lie's symmetry, and by using them, obtain analytical and numerical solutions. [ABSTRACT FROM AUTHOR]

In this paper we present some useful properties of Katugampola fractional derivative which generalizes the Riemann-Liouville and the Hadamard fractional derivatives. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method is proposed to solve the Fredholm-Volterra fractional integro-differential equation with nonlocal boundary conditions by using Haar wavelets. A collocation based Galerkin's method is applied by using Haar wavelets as basis functions over the interval [0, 1). It converts the Fredholm-Volterra fractional integro-differential equation into a system of m linear equations. On incorporating q nonlocal boundary conditions, it leads to further q equations. All together it will give a system of (m + q) linear equations in (m + q) variables which can be solved. A variety of test examples are considered to illustrate the proposed method. The actual error is also measured with respect to a norm and the results are validated through error bounds. [ABSTRACT FROM AUTHOR]

Fractional calculus, which deals with the integration and differentiation of non-integer order, is widely being used these days to mathematically model many engineering and scientific systems ranging from microbiological processes to astronomical images. The efficient and accurate analysis of such systems, usually referred to as the fractional order systems, has become very important and a number of analysis techniques have been recently proposed. This paper provides a brief survey about these fractional order system analysis techniques while highlighting upon their key strengths and drawbacks. In particular, we evaluate each analysis technique with respect to its expressiveness, accuracy and scalability. We also propose a hybrid analysis technique, which is more efficient and reliable, for the analysis of fractional order systems. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence, uniqueness and controllability results for fractional impulsive neutral functional integro-differential evolution equation in Banach spaces. The main techniques depend on the fractional calculus properties of characteristic solution operators and sectorial operators. Particulary, we do not consider that the system produces a compact semi-group. So we claim that phase space for infinite delay with impulse. Finally an example is given to illustrate for our required results. [ABSTRACT FROM AUTHOR]

In the paper Euler-Lagrange equations of fractional mechanics are studied. The characteristic feature of such equations is mixing of left- and right-sided fractional derivatives due to the rules of integration by parts in fractional calculus. We propose to solve a class of equations of this type using transformation to equivalent fractional integral equations and then applying Banach theorem on fixed point of a contractive mapping. The method is explained in detail for nonlinear fractional oscillator equation in two versions, then the results for a wide class of fractional differential equations are reported. [ABSTRACT FROM AUTHOR]

In this paper, the necessary and sufficient conditions of the boundedness of a fractional integration type operator with variable upper limit are obtained under different parameters in weighted Lebesgue spaces. [ABSTRACT FROM AUTHOR]

The paper deals with the evolution of the parameters of a fractional model for different values of temperature. An experimental campaign has been performed on epoxy resin at different levels of temperature. It is shown that epoxy resin is very sensitive to the temperature. [ABSTRACT FROM AUTHOR]

In this paper, numerical algorithms for the approximate solution of a fractional inverse problem with an unknown source term are given using stable first order of accuracy difference schemes. The algorithm is tested in a one-dimensional fractional inverse problem. [ABSTRACT FROM AUTHOR]

In this paper a novel control strategy implying a Smith Predictor structure is proposed for which the primary controller is tuned based on the IMC method, but altered such that the final primary controller is a fractional order controller. A first order time delay process is used as a case study. The simulation results show that the proposed fractional order IMC controller ensures improved closed loop performance compared to the classical IMC, especially in terms of modeling uncertainties. [ABSTRACT FROM AUTHOR]

In this paper we survey some author’s results and developments relating the so-called Gel’fond-Leont’ev (G-L) operators of generalized integration and differentiation, classes of special functions (SF) of generalized hypergeometric type and the operators of generalized fractional calculus (GFC). The G-L operators have been introduced by Gel’fond-Leont’ev [9] in the classes of analytic functions in disks ΔR = {|z| < R}, by means of of multipliers’ sequences composed by the coefficients of suitable entire (generating) functions. Introducing classes of SF related to Fractional Calculus (FC), as the Mittag-Leffler (ML) function, the multi-index Mittag-Leffler (multi-ML) function and its various particular cases ([16]–[18]), we specify the G-L operators generated by these entire functions. It is shown that in these cases, the G-L operators can be extended to analytic functions in wider complex domains Ω starlike with respect to the origin z = 0 and represented by operators of the Generalized Fractional Calculus (GFC), Kiryakova [14], i.e. operators of generalized integration and differentiation of arbitrary fractional multi-order. Illustrative examples and some open problems are proposed. [ABSTRACT FROM AUTHOR]