arXiv subjects

Dejenie A. Lakew

Publications and source records attributed to Dejenie A. Lakew.

10 recordsLinked to original sources

arXiv2026

The Fractional Half Dirac Operator Over Sobolev Spaces

Let a bounded and smooth domain in the n dimensional Euclidean space be given and consider a boundary value problem of the half order Dirac operator and its power, with f be the value of the half derivative of the solution in the domain and, g be the trace value of the solution over the boundary. We will show that the solution will be a generalized function u in the half order Hilbert space which is the orthogonal sum of the solution that emerged as a domain integral from the domain value of the fractional half Dirac operator and the one that is the boundary integrals that emerges from the boundary or trace value of the solution. We use two different symbols to designate orthogonal sums for subspaces and functions in orthogonal subspaces of the Sobolev space.

math.AP↗

arXiv2016

On Orthogonal Decomposition of a Sobolev Space

The theme of this short article is to investigate an orthogonal decomposition of a Sobolev space and look at some properties of the inner product therein and the distance defined from the inner product. We also determine the dimension of the orthogonal difference space and show the expansion of spaces as their regularity increases.

math.FA↗

arXiv2015

New Proofs on Properties of an Orthogonal Decomposition of a Hilbert Space

We establish new and different kinds of proofs of properties that arise due to the orthogonal decomposition of the Hilbert space, including projections, over the unit interval of one dimension. We also see angles between functions, particularly between those which are non zero constant multiples of each other and between functions from the kernel space and the derivative image of the trace less Sobolev space.

math.FA↗

arXiv2015

On Orthogonal Decomposition of a Hilbert Space

In this short article we show an orthogonal decomposition of a Hilbert space as a sum of null solutions of the first derivative and the first derivative of a traceless higher order Hilbert/Sobolev space. We define orthogonal projections and see some of its properties and show case some decomposition of elementary functions as corollaries.

math.FA↗

arXiv2014

Norm Estimates for Solutions of Elliptic BVPs of the Dirac Operator

We present norm estimates for solutions of first and second order elliptic BVPs of the Dirac operator considered over a bounded and smooth domain of the n-dimension Euclidean space. The solutions whose norms to be estimated are in some Sobolev spaces and the boundary conditions as traces of solutions and their derivatives are in some Slobodeckji spaces.

math.FA↗

arXiv2014

The Fibonacci Sequence via the sigma transform

In this short article, we study different problems described as initial value problems of discrete differential equations and develop a a transform method called the sigma transform, a discrete version of the continuous Laplace transform to generate solutions as rational functions of integers to these initial value problems. Particularly we look how the method generates the traditionally known numbers called Fibonacci sequence as a solution to an initial value problem of a discrete differential equation.

math.GM↗

arXiv2009

On Hyper Singular Integral Operators over Weighted Sobolev Spaces

In this paper we study singular integral operators which are hyper or weak over Lipschitz or Holder spaces and over weghted Sobolev spaces defined on unbounded domains in the standard $n$-D space $R^n$ for $n>0$. The $π$-operator in this case is one of the hyper integral operators which has been studied extensively than other hyper singular integral operators. It will be shown the control of singularity of such integral operators that are defined interms of Cauchy generating kernels by working on weghted Sobolev spaces $W^{p,k}(Ω,|x|^{ζ+epsilon}dx)$ for some $ε>0$ and $ζ$ some positive integer.

math.FA↗

arXiv2008

The Spherical $π$-Operator

In this article, we define the spherical $π$-operator over domains in the $(n-1)$-D unit sphere $S^n$ of $R^n$ and develop new and analogous results on the operator it self and its mapping properties. We introduce the spherical Dirac operator $Γ_α$ as an $α$- shift of of $Γ_omega$, where $Γ_omega$ is the negative of the wedge (or Grassmann) product of $ω$ with that of the Dirac operator $D_ω$. A gegenbauer polynomial is used as a Cauchy kernel for the spherical Dirac operator $Γ_alpha$.

math.CA↗

arXiv2008

On Some Discrete Differential Equations

In this short note, we present few results on the use of the discrete Laplace transform in solving first and second order initial value problems of discrete differential equations.

math.GM↗

arXiv2008

Mollifiers in Clifford Analysis

We introduce mollifiers in Clifford analysis setting and construct a sequence of $\C^{\infinity}$-functions that approximate a $γ$-regular function and a solution to a non homogeneous BVP of an in homogeneous Dirac like operator in certain Sobolev spaces over bounded domains whose boundary is not that wild. One can extend the smooth functions upto the boundary if the domain has a $\C^1$-boundary and this is the case in the paper as we consider a domain whose boundary is a $\C^2$-hyper surface.

math.AP↗