Related papers: Comment on "Some novel delta-function identities"

In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local…

We derive and prove a new formulation of the Lerch zeta function as a fractional derivative of an elementary function. We demonstrate how this formulation interacts very naturally with basic known properties of Lerch zeta, and use the…

Let $\Delta_\kappa$ be the Dunkl Laplacian on $\mathbb{R}^n$ and $\phi: \mathbb{R}^+ \to \mathbb{R}$ is a smooth function. The aim of this manuscript is twofold. First, we study the decay estimate for a class of dispersive semigroup of the…

Consider a normal function $f$ on the ordinals (i. e. a function $f$ that is strictly increasing and continuous at limit stages). By enumerating the fixed points of $f$ we obtain a faster normal function $f'$, called the derivative of $f$.…

We continue the development of the basic theory of generalized derivatives as introduced in \cite{JPA} and give some of their applications. In particular, we formulate versions of a weak maximum principle, Rolle's theorem, the Mean value…

We consider Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part. We consider abstract splitting methods associated with this decomposition where no discretization in space is made. We prove a…

In this paper, by making use of a certain family of fractional derivative operators in the complex domain, we introduce and investigate a new subclass $\mathcal{P}_{\tau,\mu}(k,\delta,\gamma)$ of analytic and univalent functions in the open…

We establish new integral inequalities of Hermite-Hadamard type for the recent class of $\eta$-convex functions. This is done via generalized $(k,r)$-Riemann-Liouville fractional integral operators. Our results generalize some known…

In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…

Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…

In this note we show that the general theory of vector valued singular integral operators of Calder\'on-Zygmund defined on general metric measure spaces, can be applied to obtain Sobolev type regularity properties for solutions of the…

It is proved that given $-1/2<s<1/2$, for any $f\in L^2(\mathbb{R})$, there is a unique $u\in \widehat{H}^{|s|}(\mathbb{R})$ such that $$ f=\boldsymbol{D}^{-s}u+\boldsymbol{D}^{s*}u\,, $$ where $\boldsymbol{D}^{-s}, \boldsymbol{D}^{s*}$ are…

In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and…

In this paper, a general integral identity for twice differentiable functions is derived. By using of this identity, the author establish some new estimates on Hermite-Hadamard type and Simpson type inequalities for s-convex via Riemann…

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…

In this article, we impose a new class of fractional analytic functions in the open unit disk. By considering this class, we define a fractional operator, which is generalized Salagean and Ruscheweyh differential operators. Moreover, by…

This is the revised version of the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory,…

In this paper, a general integral identity for convex functions is derived. Then, we establish new some inequalities of the Simpson and the Hermite-Hadamard's type for functions whose absolute values of derivatives are convex. Some…

We consider the problem of estimating an unknown function f* and its partial derivatives from a noisy data set of n observations, where we make no assumptions about f* except that it is smooth in the sense that it has square integrable…

In this paper, we consider the local smoothing estimate of fractional Schr\"{o}dinger operator $e^{it(-\Delta)^{\alpha/2}}$ with $\alpha>1$. Using the $k$-broad "norm" estimate developed by Guth, we improve the previous best results of…