Related papers: A Remark on the Memory Property of Fractional Diff…
This article is concerned with the existence and uniqueness of solutions to some fractional order boundary value problems. Our results are based on some fixed point theorems. For the applicability of our results, we provide an example.
We introduce a more general discrete fractional operator, given by convex linear combination of the delta and nabla fractional sums. Fundamental properties of the new fractional operator are proved. As particular cases, results on delta and…
Numerical solutions to fractional differential equations can be extremely computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current iteration. In finite difference…
This work establishes a comprehensive analytical framework for studying implicit fractional differential systems with distributed memory and time delays. We develop novel fractional integral inequalities of Gr\"onwall--Wendroff type that…
Recently, some authors have proved monotonicity results for delta and nabla fractional differences separately. In this article, we use dual identities relating delta and nabla fractional difference operators to prove shortly the…
We present an analytical investigation of the asymptotic behavior of non-resonance eigenvalues for the fractional Schr\"odinger operator under homogeneous Neumann boundary conditions. Our findings reveal an intriguing convergence: as the…
In this paper, we present the definitions and some properties of the general fractional integrals (GFIs) and general fractional derivatives (GFDs) of a function f(x) with respect to another function g(x). Examples of special cases of…
This article is a short review of the recent results on properties of nonlinear fractional maps which are maps with power- or asymptotically power-law memory. These maps demonstrate the new type of attractors - cascade of bifurcations type…
Derivatives of fractional order with respect to time describe long-term memory effects. Using nonlinear differential equation with Caputo fractional derivative of arbitrary order $\alpha>0$, we obtain discrete maps with power-law memory.…
We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the…
In this present paper, we introduce and study a dynamical systems involving fractional derivative operator and nonlocal condition, which is constituted of a fractional evolution equation and a time-dependent variational inequality, and is…
Fractional action-like variational problems have recently gained importance in studying dynamics of nonconservative systems. In this note we address multi-dimensional fractional action-like problems of the calculus of variations.
It is well known that martingale difference sequences are very useful in applications and theory. On the other hand, the operator fractional Brownian motion as an extension of the well-known fractional Brownian motion also plays important…
In this paper, we introduce the nabla fractional derivative and fractional integral on time scales in the Riemann-Liouville sense. We also introduce the nabla fractional derivative in Gr\"unwald-Letnikov sense. Some of the basic properties…
Fractionally integrated time series, exhibiting long memory with slowly decaying autocorrelations, are frequently encountered in economics, finance, and related fields. Since the seminal work of Robinson (1995), a variety of semiparametric…
This paper focuses on the fractional difference of Lyapunov functions related to Riemann-Liouville, Caputo and Grunwald-Letnikov definitions. A new way of building Lyapunov functions is introduced and then five inequalities are derived for…
The purpose of this article is to address the issues of dimensional consistency that arise in the process of replacing the ordinary time derivative operator by a fractional derivative operator in order to write a fractional differential…
Many natural and social systems possess power-law memory, and their mathematical modeling requires the application of discrete and continuous fractional calculus. Most of these systems are nonlinear and demonstrate regular and chaotic…
The present paper plans to examine the existence, uniqueness and data dependence of the solution of the fractional functional differential equation with the abstract operator of Volterra, in the context of the Picard operators. We present…
This work introduces and analyzes a finite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time we consider is employed…