This paper discuss the longstanding problems of fractional calculus such as too many definitions while lacking physical or geometrical meanings, and try to extend fractional calculus to any dimension. First, some different definitions of fractional derivatives, such as the Riemann-Liouville derivative, the Caputo derivative, Kolwankar’s local derivative and Jumarie’s modified Riemann-Liouville ...

In this work, we study chaos control and synchronization of the commensurate fractional order Liu system. Based on the stability theory of fractional order systems, the conditions of local stability of nonlinear three-dimensional commensurate fractional order systems are discussed. The existence and uniqueness of solutions for a class of commensurate fractional order Liu systems are investigate...

In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian (−Δ)s with s∈(0,1) in two dimensions. We theoretically prove and numerically verify numerical stability convergence rate no worse than O(hk+12) k≥1.

The purpose of this paper is to solve fractional calculus of variational Herglotz problem depending on an Atangana-Baleanu fractional derivative. Since the new Atangana-Baleanu fractional derivative is non-singular and non-local, the Euler-Lagrange equations are proposed for the problems of Herglotz. Fractional variational Herglotz problems of variable order are considered and two cases are sho...

A fractional order HIV model with both virus-tocell and cell-to-cell transmissions and therapy effect is investigated. The conditions for the existence of the equilibria are determined. Local stability analysis of the HIV model is studied by using the fractional Routh-Hurwitz stability conditions. We have generalized the integer LaSalle’s invariant theorem into fractional system and given some ...

In this paper we introduce the notion of α-martingale as the fractional derivative of order α of a continuous local martingale, where α ∈ (−12 , 1 2), and we show that it has a nonzero finite variation of order 2 1+2α , under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of Lévy’s characterization theorem for the f...

This article concerns a general fractional differential equation of order between 1 and 2. We consider the cases where the nonlinear term contains or does not contain other (lower order) fractional derivatives (of RiemannLiouville type). Moreover, the nonlinearity involves also a nonlinear non-local in time term. The case where this non-local term has a singular kernel is treated as well. It is...

In this paper, the local fractional Laplace decomposition method is implemented to obtain approximate analytical solution of the telegraph and Laplace equations on Cantor sets. This method is a combination of the Yang-Laplace transform and the Adomian decomposition method. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to ...

In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst index 1/2 < H < 1, which is analogous to the linear self-attracting diffusion. For 1-dimensional process we study its convergence and the corresponding weighted local time. For 2-dimensional process, as a related problem, we show that the renormalized selfintersection lo...