Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the same idea for fractional variational problems. We consider a minimization problem with a Lagrangian that depends on the left Riemann– Liouville fractional deri...

Fractional centred differences and derivatives definitions are proposed generalising to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalisations of the Cauchy integral derivative. To compute this integral, a special two straight line path was use...

An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Grünwald-Letnikov formula, and the spatial derivative by a three-point centered formula. The accura...

Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the same idea for fractional variational problems. We consider a minimization problem with a Lagrangian that depends only on the left Riemann–Liouville fractional ...

The relation showing that the Grünwald-Letnikov and generalised Cauchy derivatives are equal is presented. This establishes a bridge between two different formulations and simultaneously between the classic integer order derivatives and the fractional ones. Starting from the generalised Cauchy derivative formula, new relations are obtained, namely a regularised version that makes the concept of...

Abstract. This paper presents a step-by-step procedure for the numerical integration of fractional differential equation governing response single-degree-of-freedom (SDOF) system with derivative damping. The is developed by extending improved pseudo-force method proposed second author classical equations. To this aim, Grünwald–Letnikov approximation adopted. exploited to compute statistics SDOF...

Abstract We study distributed optimal control problems, governed by space-time fractional parabolic equations (STFPEs) involving time-fractional Caputo derivatives and spatial of Sturm-Liouville type. first prove existence uniqueness solutions STFPEs on an open bounded interval their regularity. Then we show to a quadratic problem. derive adjoint problem using the right-Caputo derivative in tim...

Abstract This paper deals with the fractional calculus of zeta functions. In particular, study is focused on Hurwitz ? \zeta function. All results are based complex generalization Grünwald-Letnikov derivative. We state and prove functional equation together an integral representation by Bernoulli numbers. Moreover, we ...