Related papers: Fractional calculus for power functions

We deal with the higher-order fractional Laplacians by two methods: the integral method and the system method. The former depends on the integral equation equivalent to the differential equation. The latter works directly on the…

In this note we present some of the most basic aspects of the fractional Laplacean with a self-contained and purely didactic intent, and with a somewhat different slant from the several excellent existing references. Given the interest that…

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.

We generalize the fractional Caputo derivative to the fractional derivative ${{^CD}^{\alpha,\beta}_{\gamma}}$, which is a convex combination of the left Caputo fractional derivative of order $\alpha$ and the right Caputo fractional…

We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and…

We establish fractional Leibniz rules for the Dunkl Laplacian $\Delta_k$ of the form $$\|(-\Delta_k)^s(fg)\|_{L^p(d\mu_k)} \lesssim \|(-\Delta_k)^s f\|_{L^{p_1}(d\mu_k)} \|g\|_{L^{p_2}(d\mu_k)} + \|f\|_{L^{p_1}(d\mu_k)} \|(-\Delta_k)^s…

Fragmentation functions for hadrons composed of heavy quarks are calculated directly from the definitions given by Collins and Soper and are compared with those calculated in another way. A new fragmentaion function for a P-wave meson is…

In this manuscript, we generalize F-calculus to apply it on fractal Tartan spaces. The generalized standard F-calculus is used to obtain the integral and derivative of the functions on the fractal Tartan with different dimensions. The…

We establish interior $C^{1,\alpha}$ regularity estimates for some $\alpha > 0$, for solutions of the fractional $p$-Laplace equation $(-\Delta_p)^s u = 0$ when $p$ is in the range $p \in [2,2/(1-s))$.

We use the polyharmonic extension approach to develop a numerical technique for discretizing higher-order powers of the spectral fractional Laplacian $(-\Delta)^s$ with $s \in (1,2)$.

The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the…

The technique of Caffarelli and Silvestre, characterizing the fractional Laplacian as the Dirichlet-to-Neumann map for a function U satisfying an elliptic equation in the upper half space with one extra spatial dimension, is shown to hold…

We study fractional differential equations of Riemann-Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on $\mathbb{R}$, we define fractional operators by means of a functional calculus…

A class of Laplace transforms is examined to show that particular cases of this class are associated with production-destruction and reaction-diffusion problems in physics, study of differences of independently distributed random variables…

We introduce a new fractional oscillator process which can be obtained as solution of a stochastic differential equation with two fractional orders. Basic properties such as fractal dimension and short range dependence of the process are…

Fractional variation is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. Fractional velocity can be suitable for characterizing singular behavior of derivatives…

There have been many proposed forms of fractional calculus, which can be grouped into a few broad classes of operators. By replacing the kernel of the power function with another kernel function, the traditional Riemann-Liouville formula…

The functional derivative of the effective action with respect to an external field is part of the equation of motion of this field if one-loop effects induced by quantum fluctuations or thermal fluctuations are included when minimizing the…

The traditional first approach to fractional calculus is via the Riemann-Liouville differintegral $_{a}D_{x}^{k}$. The intent of this paper will be to create a space $K$, pair of maps $g: C^{\omega}(\mathbb{R}) \to K$ and $g': K \to…

We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ \partial_t^\alpha u - Lu + \lambda u= f \quad \mathrm{in} \quad (0,T) \times \mathbb{R}^d,$$ where $\partial_t^\alpha u$ is the Caputo…