Two approaches for defining fractional derivatives of periodic distributions are presented. The first is a distributional version of the Weyl fractional derivative in which a derivative of arbitrary order of a periodic distribution is defined via Fourier series. The second is based on the Grünwald-Letnikov formula for defining a fractional derivative as a limit of a fractional difference quotie...

In this paper, biochemical reaction problem is given in the form of a system of non-linear differential equations involving Caputo fractional derivative. The aim is to suggest an instrumental scheme to approximate the solution of this problem. To achieve this goal, the fractional derivation terms are expanded as the elements of shifted Legendre scaling functions. Then, applying operational matr...

In this paper, we propose a numerical method to estimate the unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid. The implicit numerical method is employed to solve the direct problem. For the inverse problem, we first obtain the fractional sensitivity equation by means of the digamma function, and then we...

In this paper, the computation of a fractional derivative using the Fourier transform and a digital FIR di!erentiator is investigated. First, the Cauchy integral formula is generalized to de"ne the fractional derivative of functions. Then the fractional di!erentiation property of the Fourier transform of functions is presented. Using this property, the fractional derivative of a function can be...

Recently, an extended operator of fractional derivative related to a generalized Beta function was used in order to obtain some generating relations involving the extended hypergeometric functions [1]. The main object of this paper is to present a further generalization of the extended fractional derivative operator and apply the generalized extended fractional derivative operator to derive lin...

In this paper, a computational intelligence method is used for the solution of fractional optimal control problems (FOCP)'s with equality and inequality constraints. According to the Ponteryagin minimum principle (PMP) for FOCP with fractional derivative in the Riemann- Liouville sense and by constructing a suitable error function, we define an unconstrained minimization problem. In the optimiz...

Fractional Cauchy problems replace the usual first order time derivative by a fractional derivative. This paper develops classical solutions and stochas-tic analogues for fractional Cauchy problems in a bounded domain D ⊂ R d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional ...

Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain D ⊂ Rd with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional ti...

To accurately characterize the mechanical behavior of magnetorheological elastomer (MRE) under a wide range strain amplitude, excitation frequency, and magnetic field, viscoelastic fractional derivative was introduced, modified Bouc-Wen model based on for MRE in nonlinear region established. The can describe hysteretic characteristics region, but it cannot simulate magneto-viscoelasticity frequ...