Growth Estimates of Entire Function Solutions of Generalized Bi-Axially Symmetric Helmholtz Equation

2017 ◽  
Vol 8 ◽  
pp. 12-26 ◽  
Author(s):  
Devendra Kumar
Keyword(s):  
Entire Function ◽  
Helmholtz Equation ◽  
Axially Symmetric ◽  
Order And Type

Growth estimates for entire function solutions of the generalized bi-axially symmetric Helmholtz equation ∂2u/∂x2 + ∂2u/∂y2 + (2µ/x)·(∂u/∂x) + (2ν/y)·(∂u/∂y) +k2u = 0, (µ,ν Є R+), in terms of their Jacobi Bessel coefficients and ratio of these coefficients have been studied. Some relations for order and type also have been obtained in terms of Taylor and Neumann coefficients. Our results generalize and extend some results of Gilbert and Howard, McCoy, Kumar and Singh.

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 (p,q)-Growth of an Entire GASHE Function and the Coefficient βn = |an|/Γ(n+μ+1)

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Mohammed Harfaoui ◽  
Abdellah Mourassil ◽  
Loubna Lakhmaili
Keyword(s):  
Entire Function ◽  
Helmholtz Equation ◽  
Axially Symmetric ◽  
Function Solution

The main purpose of this paper is to extend the work concerning the measures of growth of an entire function solution of the generalized axially symmetric Helmholtz equation ∂2u/∂x2+∂2u/∂y2+(2μ/y)(∂u/∂y)=0,  μ>0, by studying the general measures of growth ((p,q)-order, lower (p,q)-order, (p,q)-type, and lower (p,q)-type) in terms of coefficients |an|/Γ(n+μ+1) and the ratios of these successive coefficients.

The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
George Csordas ◽  
Anna Vishnyakova
Keyword(s):  
Entire Function ◽  
Open Problem ◽  
A Priori ◽  
Sufficient Conditions ◽  
Class Definition ◽  
Lindelöf Theorem ◽  
Order And Type ◽  
Geometric Result ◽  
Real Entire Function ◽  
Inequality Theorem

AbstractThe principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.

A FINITE ELEMENT CODE FOR THE NUMERICAL SOLUTION OF THE HELMHOLTZ EQUATION IN AXIALLY SYMMETRIC WAVEGUIDES WITH INTERFACES

1999 ◽  
Vol 07 (02) ◽  
pp. 83-110 ◽  
Author(s):  
NIKOLAOS A. KAMPANIS ◽  
VASSILIOS A. DOUGALIS
Keyword(s):  
Finite Element ◽  
Helmholtz Equation ◽  
Nonlocal Boundary Condition ◽  
Bottom Topography ◽  
Nonlocal Boundary ◽  
Small Scale ◽  
Far Field ◽  
Axially Symmetric ◽  
Fortran Code ◽  
Iterative Linear Solvers

We consider the Helmholtz equation in an axisymmetric cylindrical waveguide consisting of fluid layers overlying a rigid bottom. The medium may have range-dependent speed of sound and interface and bottom topography in the interior nonhomogeneous part of the waveguide, while in the far-field the interfaces and bottom are assumed to be horizontal and the problem separable. A nonlocal boundary condition based on the DtN map of the exterior problem is posed at the far-field artificial boundary. The problem is discretized by a standard Galerkin/finite element method and the resulting numerical scheme is implemented in a Fortran code that is interfaced with general mesh generation programs from the MODULEF finite element library and iterative linear solvers from QMRPACK. The code is tested on several small scale examples of acoustic propagation and scattering in the sea and its results are found to compare well with those of COUPLE.

Fundamental solutions of the bi-axially symmetric Helmholtz equation

2018 ◽  
Vol 2018 (1) ◽  
pp. 55-64 ◽  
Author(s):  
Tukhtasin Ergashev ◽  
◽  
Anvar Hasanov ◽  
Keyword(s):  
Helmholtz Equation ◽  
Fundamental Solutions ◽  
Axially Symmetric
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