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We show that the Bergman, Szego, and Poisson kernels associated to a finitely connected domain in the plane are all composed of finitely many easily computed functions of one variable. The new formulas give rise to new methods for computing…
Fractional vector calculus is discussed in the spherical coordinate framework. A variation of the Legendre equation and fractional Bessel equation are solved by series expansion and numerically. Finally, we generalize the hypergeometric…
We determine all entire functions $f$ such that for nonzero complex values $a\neq b$ the implications $f=a \Rightarrow f' =a$ and $f' =b \Rightarrow f=b$ hold. This solves an open problem in uniqueness theory. In this context we give a…
The exact universal functional of integer charge leads to an extension to fractional charge asymptotically when it is applied to a system made of asymptotically separated densities. The extended functional is asymptotically local and is…
An algebraizable singularity is a germ of a singular holomorphic foliation which can be defined in some appropriate local chart by a differential equation with algebraic coefficients. We show that there exists at least countably many…
The two Fresnel Integrals are real and imaginary part of the integral over complex-valued exp(ix^2) as a function of the upper limit. They are special cases of the integrals over x^m*exp(i*x^n) for integer powers m and n, which are…
We review the recent generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives and study them using indirect methods. In particular, we provide necessary…
System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of…
An invex function generalizes a convex function in the sense that every stationary point is a global minimizer. Recently, invex functions and their subclasses have attracted attention in signal processing and machine learning. However,…
The counting function on binary values is extended to the signed case in order to count the number of transitions between contiguous locations. A generalized subdifferential for the sign change counting function is given where classical…
This article studies sufficient conditions on families of approximating kernels which provide $N$--term approximation errors from an associated nonlinear approximation space which match the best known orders of $N$--term wavelet expansion.…
This article addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalised functions. We employ the recently…
Usually such area of mathematics as differential equations acts as a consumer of results given by functional analysis. This article will give an example of the reverse interaction of these two fields of knowledge. Namely, the derivation and…
The general theme of this note is illustrated by the following theorem: Theorem 1. Suppose $K$ is a compact set in the complex plane and 0 belongs to the boundary $\partial K$. Let ${\cal A}(K)$ denote the space of all functions $f$ on $K$…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
A differential algebra of nonlinear generalized functions is presented as a tool for a wide range of nonsmooth nonlinear problems. The power of the differential algebra is used to do mathematical calculations or proofs; then the final…
In this paper, we study the existence of positive solutions for a class of conformable fractional differential equations with integral boundary conditions. By using the properties of the Green's function and the fixed point theorem in a…
Ordinary differential equation (ODE) is widely used in modeling biological and physical processes in science. In this article, we propose a new reproducing kernel-based approach for estimation and inference of ODE given noisy observations.…
We prove that a locally integrable function $f:(a,b) \to \mathbb R$ must be affine if its mean oscillation, considered as a function of intervals, can be extended to a locally finite Borel measure. In particular, we show that any function…
A further significant extension is presented of the infinitely large class of differential algebras of generalized functions which are the basic structures in the nonlinear algebraic theory listed under 46F30 in the AMS Mathematical Subject…