Related papers: Fractional differential and integral operations an…
We discuss the formation of stochastic fractals and multifractals using the kinetic equation of fragmentation approach. We also discuss the potential application of this sequential breaking and attempt to explain how nature creats fractals.
Using fractional calculus we define integrals of the form $% \int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued H\"{o}lder continuous functions of order $\displaystyle \beta \in (\frac13, \frac12)$ and $f$ is a continuously…
In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on…
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional…
We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…
We present a new formula for umbral operators that yields three main insights. First, it makes explicit a connection between umbral calculus and iteration theory. Second, it leads naturally to a definition of fractional exponents of umbral…
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
Duhamel's principle reduces the Cauchy problem for an inhomogeneous partial differential equation to the corresponding homogeneous problem. In the fractional-order setting, the classical principle does not apply directly because fractional…
We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is…
This paper discusses and summarizes some results on complex variables that are very useful in fractional-order systems analysis and design, specifically when the system is analyzed in the frequency domain. The author hopes that this…
Fractional calculus represents a natural tool for describing relativistic phenomena in pseudo-Euclidean space-time. In this study, Fractional modified special relativity is presented. We obtain fractional generalized relation for the time…
Fractional calculus is an effective tool in incorporating the effects of non-locality and memory into physical models. In this regard, successful applications exist rang- ing from signal processing to anomalous diffusion and quantum…
Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In…
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to…
A fractal is in essence a hierarchy with cascade structure, which can be described with a set of exponential functions. From these exponential functions, a set of power laws indicative of scaling can be derived. Hierarchy structure and…
This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of…
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions.…
This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant…
In this paper we introduce a new mathematical tool to solve fractional equations representing models of fractional systems : The Ultradistributions. Ultradistributions permit us to unify the notion of integral and derivative in one only…
We consider the integral and derivative operators of tempered fractional calculus, and examine their analytic properties. We discover connections with the classical Riemann-Liouville fractional calculus and demonstrate how the operators may…