Related papers: Generalized Fractional Order Derivatives, Its Prop…
Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type…
We present exact analytical results for the Caputo fractional derivative of a wide class of elementary functions, including trigonometric and inverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian, quartic Gaussian, and…
The fractional order system, which is described by the fractional order derivative and integral, has been studied in many engineering areas. Recently, the concept of fractional order has been generalized to the distributed order concept,…
We introduce a new fractional derivative which obeys classical properties including: linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, the Rolle's Theorem and the Mean Value…
We introduce a new derivative, the so-called truncated $\mathcal{V}$-fractional derivative for $\alpha$-differentiable functions through the six parameters truncated Mittag-Leffler function, which generalizes different fractional…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
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…
We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When…
In the present paper, a generalized local Taylor formula with the local fractional derivatives (LFDs) is proposed based on the local fractional calculus (LFC). From the fractal geometry point of view, the theory of local fractional…
To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation. In this paper, we develop Laguerre spectral collocation methods for solving…
The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the…
The paper discusses the characteristic properties of fractional derivatives of non-integer order. It is known that derivatives of integer orders are determined by properties of differentiable functions only in an infinitely small…
In this paper we find fractional Riemann-Liouville derivatives for the Takagi-Landsberg functions. Moreover, we introduce their generalizations called weighted Takagi-Landsberg functions which have arbitrary bounded coefficients in the…
Several approaches to the formulation of a fractional theory of calculus of "variable order" have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an…
A generalization of the Gr\"{u}nwald difference approximation for fractional derivatives in terms of a real sequence and its generating function is presented. Properties of the generating function are derived for consistency and order of…
Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law non-locality, power-law long-term memory or fractal properties by using integrations and…
In this work, we rigorously investigate the robustness of graph neural fractional-order differential equation (FDE) models. This framework extends beyond traditional graph neural (integer-order) ordinary differential equation (ODE) models…
We study Bohr type inequalities within the framework of fractional calculus. Using Riemann Liouville fractional differential and integral operators, we establish generalized Bohr radii for analytic functions in the unit disk, including the…
Fractional partial differential equations (FDEs) are used to describe phenomena that involve a "non-local" or "long-range" interaction of some kind. Accurate and practical numerical approximation of their solutions is challenging due to the…
Derivative-free optimization (DFO) is the mathematical study of the optimization algorithms that do not use derivatives. One branch of DFO focuses on model-based DFO methods, where an approximation of the objective function is used to guide…