In this paper, operational matrices of Riemann-Liouville fractional integration and Caputo fractional differentiation for shifted Jacobi polynomials are considered. Using the given initial conditions, we transform the fractional differential equation (FDE) into a modified fractional differential equation with zero initial conditions. Next, all the existing functions in modified differential equ...

The concept of differentiation and integration to non-integer order has its origins in the seventeen century. However, only in the second-half of the twenty century appeared the first applications related to the area of control theory. In this paper we consider the study of a heat diffusion system based on the application of the fractional calculus concepts. In this perspective, several control...

This work explores different particle-based approaches to the simulation of one-dimensional fractional sub-diffusion equations in unbounded domains. We rely on smooth particle approximations, and consider four methods for estimating the fractional diffusion term. The first method is based on a direct differentiation of the particle representation; it follows the Riesz definition of the fraction...

This paper studies a new class of instantaneous and non-instantaneous impulsive boundary value problem involving the generalized ?-Caputo fractional derivative with weight. Depending on critical point theorems some properties ?-Caputo-type integration differentiation, variational construction multiplicity result solutions are established.

Fractional-order calculus is about the differentiation and integration of non-integer orders. Fractional (FC) based on fractional-order thinking (FOT) has been shown to help us understand complex systems better, improve processing signals, enhance control systems, increase performance optimization, even extend enabling potential for creativity. In this article, authors discuss fractional dynami...

in this paper, we prove the existence and uniqueness results to the random fractional functional differential equations under assumptions more general than the lipschitz type condition. moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.

The term fractional calculus is more than 300 years old. It is a generalization of the ordinary differentiation and integration to non-integer (arbitrary) order. The subject is as old as the calculus of differentiation and goes back to times when Leibniz, Gauss, and Newton invented this kind of calculation. In a letter to L’Hospital in 1695 Leibniz raised the following question (Miller and Ross...

This paper addresses issues surrounding the concept of fractional quantum mechanics, related to lights propagation in inhomogeneous nonlinear media, specifically restricted a so-called gravitational optics. Besides Schrödinger–Newton equation, we have also concerned with linear and Airy beam accelerations flat curved spaces fractal photonics, Schrödinger where impact Laplacian is discussed. Ano...