In this paper, a numerical method for nonlinear fractional-order differential equations with constant or time-varying delay is devised. The order here is an arbitrary positive real number, and the differential operator is the Grünwald-Letnikov derivative. The detailed error analysis for this algorithm is given, meanwhile, the convergence of the iteration algorithm is proved. Compared with the e...

In this work, a second-order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial Grünwald formula and its asymptotic expansion. Moreover, the proposed approximation is applied to a fractional diffusion equation with fractional substantial derivative in time. With the use of the fourth-order compact scheme in space, we give a fully di...

In  this  paper a  non-linear  model  with  fractional  order  is  presented  for  analyzing  and  controlling the  spread  of  HIV virus.  Both  the  disease-free  equilibrium and the endemic equilibrium are  found  and  their  stability is  discussed. The basic reproduction number , which is a function of the constant parameters in the model, plays an essential  role in the stability of  the ...

and Applied Analysis 3 Definition 2. The Grünwald-Letnikov fractional derivative with order α of function x(t) is defined below:

The work in this paper extends a memristive chaotic system with transcendental nonlinearities to the fractional-order domain. extended system’s properties were validated through bifurcation analysis and spectral entropy. presented was employed substitution stage of an image encryption algorithm, including generalized Arnold map for permutation. scheme demonstrated its efficiency statistical tes...

In this paper, we approximate the solution to time-fractional telegraph equation by two kinds of difference methods: the Grünwald formula and Caputo fractional difference.

The formulations of Riemann-Liouville and Caputo derivatives in the complex plane are presented. Two versions corresponding to the whole or half plane. It is shown that they can be obtained from the Grünwald-Letnikov derivative.