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Highlights • It presents a revision of previously proposed variable order fractional derivatives. • It introduces a new approach based on the inverse Laplace transform. • Variable order fractional linear systems are defined. • A variable order Mittag-Leffler function is introduced. • Variable order two-sided fractional derivatives. Abstract Fractional Calculus had a remarkable evolution during recent decades, and paved the way towards the definition of variable order derivatives. In the literature we find several different alternative definitions of such operators. This paper presents an overview of the fundamentals of this topic and addresses the questions of finding out which of them are reasonable according to simple criteria used for constant order fractional derivatives. This approach leads to the definitions of variable order fractional derivative based on the Grünwald–Letnikov and the Liouville formulations defined on R , as well as to the definition of a Mittag-Leffler function for variable orders, and to the application of these definitions to dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper we discuss a concept of dynamic memory and an application of fractional calculus to describe the dynamic memory. The concept of memory is considered from the standpoint of economic models in the framework of continuous time approach based on fractional calculus. We also describe some general restrictions that can be imposed on the structure and properties of dynamic memory. These restrictions include the following three principles: (a) the principle of fading memory; (b) the principle of memory homogeneity on time (the principle of non-aging memory); (c) the principle of memory reversibility (the principle of memory recovery). Examples of different memory functions are suggested by using the fractional calculus. To illustrate an application of the concept of dynamic memory in economics we consider a generalization of the Harrod–Domar model, where the power-law memory is taken into account. [ABSTRACT FROM AUTHOR]

A discretization of differential and integral operators of integer and non-integer orders is suggested. New type of differences, which are represented by infinite series, is proposed. A characteristic feature of the suggested differences is an implementation of the same algebraic properties that have the operator of differentiation (property of algebraic correspondence). Therefore the suggested differences are considered as an exact discretization of derivatives. These differences have a property of universality, which means that these operators do not depend on the form of differential equations and the parameters of these equations. The suggested differences operators allows us to have difference equations whose solutions are equal to the solutions of corresponding differential equations. The exact discretization of the derivatives of integer orders is given by the suggested differences of the same integer orders. Similarly, the exact discretization of the Riesz derivatives and integrals of integer and non-integer order is given by the proposed fractional differences of the same order. [ABSTRACT FROM AUTHOR]

In this paper, we give a general version of Chebyshev type inequality for pseudo-convolution integral on a semiring ( [ a , b ] , ⊕ , ⊙ ) . Our result is flexible enough to support both pseudo-integral and convolution integral, (e.g., fractional integral), thus closing the series of papers. It includes the corresponding results of Agahi et al. [1] as a special case. Finally, some concluding remarks are drawn and some open problems for further investigations are given. [ABSTRACT FROM AUTHOR]

We prove a discrete analogue for the composition of the fractional integral and Caputo derivative. This result is relevant in numerical analysis of fractional PDEs when one discretizes the Caputo derivative with the so-called L1 scheme. The proof is based on asymptotic evaluation of the discrete sums with the use of the Euler–Maclaurin summation formula. • The discrete version of the composition mimics the continuous case. • The remainder involves the fractional integral and the derivative of the function. • The remainder with a step h decays to zero at a rate h min (a l p h a , 1 − a l p h a) . [ABSTRACT FROM AUTHOR]

• Covariance of the fractional integral of the Brownian motion. • Covariance of the fractional integral of a Ornstein--Uhlenbeck process. • Numerical evaluations of above covariance functions. • Simulations of fractionally integrated Gaussian processes. • Fractional integration of Integrate--and--fire neuronal model with correlated inputs. We investigate the stochastic processes obtained as the fractional Riemann-Liouville integral of order α ∈ (0 , 1) of Gauss-Markov processes. The general expressions of the mean, variance and covariance functions are given. Due to the central role, for the fractional integral of standard Brownian motion and of the non-stationary/stationary Ornstein-Uhlenbeck processes, the covariance functions are carried out in closed-form. In order to clarify how the fractional order parameter α affects these functions, their numerical evaluations are shown and compared also with those of the corresponding processes obtained by ordinary Riemann integral. The results are useful for fractional neuronal models with long range memory dynamics and involving correlated input processes. The simulation of these fractionally integrated processes can be performed starting from the obtained covariance functions. A suitable neuronal model is proposed. Graphical comparisons are provided and discussed. [ABSTRACT FROM AUTHOR]

• We calculate calculus of hidden variable recurrent fractal interpolation function and give a theorem on the existence of Cr -HVRFIF. • We construct CSHVRFIFs of Type-I, Type-II and cardinal CSHVRFIFs and estimate error bound between the constructed CSHVRFIF and an original function. • We show that fractional calculus of cardinal CSHVRFIFs are also HVRFIFs. In this paper, we construct a cubic spline hidden variable recurrent fractal interpolation function(CSHVRFIF) and prove that its Riemann-Liouville fractional integral and derivative are also HVRFIFs. To do it, firstly, we calculate calculus of hidden variable recurrent fractal interpolation function(HVRFIF) and give a theorem on the existence of Cr -HVRFIF. Secondly, we construct CSHVRFIFs of Type-Ⅰ and Type-Ⅱ, cardinal CSHVRFIFs and estimate error bound between the constructed CSHVRFIF and an original function. Finally, we insist that fractional calculus of cardinal CSHVRFIFs are also HVRFIFs. [ABSTRACT FROM AUTHOR]

• We present a construction of HVFIF using functions of which Lipschitz exponents are in (0, 1], so that the Riemann Liuoville fractional integral of the HVFIF becomes a fractal interpolation function, and give an example where Lipschitz exponents of functions of IFS are in (0, 1]. • We prove that the Riemann Liuoville fractional integral is also a HVFIF with function vertical scaling factors defined newly. • We give the graphs of 0.8- and 0.2-order fractional integrals of the HVFIFs constructed in the above example. In this paper, we study Riemann Liouville fractional integral of hidden variable fractal interpolation function (HVFIF) constructed by functions whose Lipschitz exponents are in (0, 1]. Firstly, we present a construction of HVFIF using functions of which Lipschitz exponents are in (0, 1], so that the Riemann Liouville fractional integral of the HVFIF becomes a fractal interpolation function, and give an example where Lipschitz exponents of functions of IFS are in (0, 1]. Secondly, we prove that the Riemann Liouville fractional integral is also a HVFIF with function vertical scaling factors defined newly. Finally, we give the graphs of 0.8- and 0.2-order fractional integrals of the HVFIFs constructed in the above example. [ABSTRACT FROM AUTHOR]