The purpose of this presentation is to describe a recent family of basis functions—the fractional B-splines—which appear to be intimately connected to fractional calculus. Among other properties, we show that they are the convolution kernels that link the discrete (finite differences) and continuous (derivatives) fractional differentiation operators. We also provide simple closed forms for the ...

The fractional calculus in the neuronal models provides memory properties. In fractional-order model, dynamics of neuron depends on derivative order, which can produce various types memory-dependent dynamics. this paper, behaviors coupled FitzHugh–Nagumo neurons are investigated. effects coupling strength and order under consideration. It is revealed that level synchronization decreased by decr...

This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of t...

In this paper, we prove the existence of mild solutions for the semilinear fractional order functional of Volterra-Fredholm type differential equations with impulses in a Banach space. The results are obtained by using the theory of fractional calculus, the analytic semigroup theory of linear operators and the fixed point techniques. ifx

This paper investigates the existence of solutions for fractional-order neutral impulsive differential inclusions with nonlocal conditions. Utilizing the fractional calculus and fixed point theorem for multivalued maps, new sufficient conditions are derived for ensuring the existence of solutions. The obtained results improve and generalize some existed results. Finally, an illustrative example...

This paper proposes a novel method for controlling the convergence rate of a particle swarm optimization algorithm using fractional calculus (FC) concepts. The optimization is tested for several wellknown functions and the relationship between the fractional order velocity and the convergence of the algorithm is observed. The FC demonstrates a potential for interpreting evolution of the algorit...

‎‎in this paper‎, ‎the existence and uniqueness of positive solutions for a class of nonlinear initial value problem for a finite fractional difference equation obtained by constructing the upper and lower control functions of nonlinear term without any monotone requirement‎ .‎the solutions of fractional difference equation are the size of tumor in model tumor growth described by the gompertz f...

Differential and integral calculus belongs to basic courses of mathematics and everybody understands geometrical and physical meaning of derivative or integral. But, hardly anybody can imagine for example derivative of order 1/2 or even of non-rational order. The branch of mathematics which generalizes calculus to non-integer case is known under the term “fractional calculs” although this name ...

Significant progress has been made in incorporating fractional calculus into the projection and lag synchronization of complex networks. However, real-world networks are highly complex, making derivative used dynamics more susceptible to changes over time. Therefore, it is essential incorporate variable-order asymptotic hybrid Firstly, this approach considers nonidentical models with characteri...

A special function is a that typically entitled after an early scientist who studied its features and has specific application in mathematical physics or another area of mathematics. There are few significant examples, including the hypergeometric unique species. These types functions generalized by fractional calculus, fractal, q-calculus, (q,p)-calculus k-calculus. By engaging notion q-fracti...