scholarly journals The Fourier extension operator of distributions in Sobolev spaces of the sphere and the Helmholtz equation

2020 ◽  
pp. 1-22
Author(s):  
J. A. Barceló ◽  
M. Folch-Gabayet ◽  
T. Luque ◽  
S. Pérez-Esteva ◽  
M. C. Vilela
Keyword(s):  
Helmholtz Equation ◽  
Sobolev Spaces ◽  
Fractional Integration ◽  
Extension Operator ◽  
Wave Functions ◽  
Square Function ◽  
Entire Solutions ◽  
Fourier Extension ◽  
Fractional Integration Operator ◽  
Weighted Norms

The purpose of this paper is to characterize the entire solutions of the homogeneous Helmholtz equation (solutions in ℝ d ) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^\alpha (\mathbb {S}^{d-1}),$ with α ∈ ℝ. We present two characterizations. The first one is written in terms of certain L2-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For α > 0 this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for α < 0 it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.

CONTINUOUS-TIME FRACTIONAL ORDER LINEAR SYSTEMS IDENTIFICATION USING CHEBYSHEV WAVELET

2020 ◽  
Vol 6 (8(77)) ◽  
pp. 23-28
Author(s):  
Shuen Wang ◽  
Ying Wang ◽  
Yinggan Tang
Keyword(s):  
Linear Systems ◽  
Fractional Order ◽  
Continuous Time ◽  
Fractional Integration ◽  
Operational Matrices ◽  
Computation Complexity ◽  
Order Linear ◽  
Fractional Integration Operator ◽  
Systems Identification ◽  
Chebyshev Wavelet

In this paper, the identification of continuous-time fractional order linear systems (FOLS) is investigated. In order to identify the differentiation or- ders as well as parameters and reduce the computation complexity, a novel identification method based on Chebyshev wavelet is proposed. Firstly, the Chebyshev wavelet operational matrices for fractional integration operator is derived. Then, the FOLS is converted to an algebraic equation by using the the Chebyshev wavelet operational matrices. Finally, the parameters and differentiation orders are estimated by minimizing the error between the output of real system and that of identified systems. Experimental results show the effectiveness of the proposed method.

scholarly journals Pathway fractional integral operators of generalized k-wright function and k4-function

2017 ◽  
Vol 35 (2) ◽  
pp. 235 ◽  
Author(s):  
Dinesh Kumar ◽  
Ram Kishore Saxena ◽  
Jitendra Daiya
Keyword(s):  
Fractional Integral ◽  
Fractional Integration ◽  
Integral Operators ◽  
Wright Function ◽  
Finite Product ◽  
Fractional Integral Operators ◽  
Integral Formulas ◽  
Fractional Integration Operator ◽  
Composition Formula ◽  
Special Cases

In the present work we introduce a composition formula of the pathway fractional integration operator with finite product of generalized K-Wright function and K4-function. The obtained results are in terms of generalized Wright function.Certain special cases of the main results given here are also considered to correspond with some known and new (presumably) pathway fractional integral formulas.

scholarly journals Measures of Growth of Entire Solutions of Generalized Axially Symmetric Helmholtz Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Devendra Kumar ◽  
Rajbir Singh
Keyword(s):  
Helmholtz Equation ◽  
Lower Order ◽  
Entire Solution ◽  
Axially Symmetric ◽  
Entire Solutions ◽  
Lower Type ◽  
Order And Type

For an entire solution of the generalized axially symmetric Helmholtz equation , measures of growth such as lower order and lower type are obtained in terms of the Bessel-Gegenbauer coefficients. Alternative characterizations for order and type are also obtained in terms of the ratios of these successive coefficients.

scholarly journals A note on Riesz fractional integrals in the limiting case α(x)p(x) ≡ n

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Stefan Samko
Keyword(s):  
Fractional Integration ◽  
Fractional Integrals ◽  
Variable Order ◽  
Integration Operator ◽  
Hölder Continuous ◽  
Open Set ◽  
Fractional Integration Operator ◽  
Limiting Case

AbstractWe show that the Riesz fractional integration operator I α(·) of variable order on a bounded open set in Ω ⊂ ℝn in the limiting Sobolev case is bounded from L p(·)(Ω) into BMO(Ω), if p(x) satisfies the standard logcondition and α(x) is Hölder continuous of an arbitrarily small order.

scholarly journals Relation between Schrödinger’s Equation and Einstein’s Equation

2021 ◽  
Author(s):  
Aman Yadav
Keyword(s):  
Wave Equation ◽  
Field Equation ◽  
Helmholtz Equation ◽  
Space Time ◽  
Wave Functions ◽  
Quantum Level ◽  
Schrödinger’S Equation ◽  
Einstein's Equation ◽  
The Relationship ◽  
Schrödinger's Equation

The relationship between Einstein's Field Equation and Schrodinger's Equation is examined in thiswork. I adjusted Schrodinger's Equation to offer the solution, and utilizing the wave equation, Icame up with two cases: In case 1, I discovered the structure and dimension of the equations in amanner similar to Einstein's Field Equation, and in case 2, the Helmholtz equation replaces themodified Schrodinger's equation. Finally, the findings suggested that wave functions may haverelevance beyond determining the position of a particle, and that they may be used to determinethe structure of space-time at the quantum level.

scholarly journals Regularities of Time-Fractional Derivatives of Semigroups Related to Schrodinger Operators with Application to Hardy-Sobolev Spaces on Heisenberg Groups

2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Zhiyong Wang ◽  
Chuanhong Sun ◽  
Pengtao Li
Keyword(s):  
Sobolev Spaces ◽  
Equivalence Relation ◽  
Fractional Derivatives ◽  
Sobolev Type ◽  
Square Function ◽  
Heisenberg Groups ◽  
Square Functions ◽  
Regularity Properties ◽  
Reverse Hölder ◽  
Derivatives Of

In this paper, assume that L=−Δℍn+V is a Schrödinger operator on the Heisenberg group ℍn, where the nonnegative potential V belongs to the reverse Hölder class BQ/2. By the aid of the subordinate formula, we investigate the regularity properties of the time-fractional derivatives of semigroups e−tLt>0 and e−tLt>0, respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type space HL1,αℍn associated with L. Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with L.

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