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.

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 Preconditioned Krylov Subspace Methods for Sixth Order Compact Approximations of the Helmholtz Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Yury Gryazin
Keyword(s):  
Helmholtz Equation ◽  
Lower Order ◽  
Krylov Subspace ◽  
High Efficiency ◽  
Order Approximation ◽  
Separation Of Variables ◽  
Sixth Order ◽  
Test Problems ◽  
Iterative Approach ◽  
Preconditioned Matrix

We consider an efficient iterative approach to the solution of the discrete Helmholtz equation with Dirichlet, Neumann, and Sommerfeld-like boundary conditions based on a compact sixth order approximation scheme and lower order preconditioned Krylov subspace methodology. The resulting systems of finite-difference equations are solved by different preconditioned Krylov subspace-based methods. In the analysis of the lower order preconditioning developed here, we introduce the term “kth order preconditioned matrix” in addition to the commonly used “an optimal preconditioner.” The necessity of the new criterion is justified by the fact that the condition number of the preconditioned matrix in some of our test problems improves with the decrease of the grid step size. In a simple 1D case, we are able to prove this analytically. This new parameter could serve as a guide in the construction of new preconditioners. The lower order direct preconditioner used in our algorithms is based on a combination of the separation of variables technique and fast Fourier transform (FFT) type methods. The resulting numerical methods allow efficient implementation on parallel computers. Numerical results confirm the high efficiency of the proposed iterative approach.

ENTIRE SOLUTIONS OF A CURVATURE FLOW IN AN UNDULATING CYLINDER

2018 ◽  
Vol 99 (1) ◽  
pp. 137-147
Author(s):  
LIXIA YUAN ◽  
BENDONG LOU
Keyword(s):  
Curvature Flow ◽  
Entire Solution ◽  
Travelling Wave ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Entire Solutions ◽  
Smooth Functions ◽  
Periodic Traveling Waves ◽  
Periodic Travelling Wave ◽  
Positive Functions

We consider a curvature flow $V=\unicode[STIX]{x1D705}+A$ in a two-dimensional undulating cylinder $\unicode[STIX]{x1D6FA}$ described by $\unicode[STIX]{x1D6FA}:=\{(x,y)\in \mathbb{R}^{2}\mid -g_{1}(y)<x<g_{2}(y),y\in \mathbb{R}\}$, where $V$ is the normal velocity of a moving curve contacting the boundaries of $\unicode[STIX]{x1D6FA}$ perpendicularly, $\unicode[STIX]{x1D705}$ is its curvature, $A>0$ is a constant and $g_{1}(y),g_{2}(y)$ are positive smooth functions. If $g_{1}$ and $g_{2}$ are periodic functions and there are no stationary curves, Matano et al. [‘Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit’, Netw. Heterog. Media1 (2006), 537–568] proved the existence of a periodic travelling wave. We consider the case where $g_{1},g_{2}$ are general nonperiodic positive functions and the problem has some stationary curves. For each stationary curve $\unicode[STIX]{x1D6E4}$ unstable from above/below, we construct an entire solution growing out of it, that is, a solution curve $\unicode[STIX]{x1D6E4}_{t}$ which increases/decreases monotonically, converging to $\unicode[STIX]{x1D6E4}$ as $t\rightarrow -\infty$ and converging to another stationary curve or to $+\infty /-\infty$ as $t\rightarrow \infty$.

scholarly journals Entire solutions of the Fisher–KPP equation on the half line

2019 ◽  
Vol 31 (3) ◽  
pp. 407-422 ◽  
Author(s):  
BENDONG LOU ◽  
JUNFAN LU ◽  
YOSHIHISA MORITA
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Traveling Wave ◽  
Wave Solution ◽  
Dirichlet Boundary ◽  
Entire Solution ◽  
Traveling Wave Solution ◽  
Half Line ◽  
Entire Solutions ◽  
Kpp Equation

In this paper, we study the entire solutions of the Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation ut = uxx + f(u) on the half line [0, ∞) with Dirichlet boundary condition at x = 0. (1) For any $c \ge 2\sqrt {f'(0)} $, we show the existence of an entire solution ${{\cal U}^c}(x,t)$ which connects the traveling wave solution φc(x + ct) at t = −∞ and the unique positive stationary solution V(x) at t = +∞; (2) We also construct an entire solution ${{\cal U}}(x,t)$ which connects the solution of ηt = f(η) at t = −∞ and V(x) at t = +∞.

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

scholarly journals A Linear Homogeneous Partial Differential Equation with Entire Solutions Represented by Laguerre Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Xin-Li Wang ◽  
Feng-Li Zhang ◽  
Pei-Chu Hu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Laguerre Polynomials ◽  
Convergent Series ◽  
Entire Solutions ◽  
Partial Differential ◽  
Order And Type

We study a homogeneous partial differential equation and get its entire solutions represented in convergent series of Laguerre polynomials. Moreover, the formulae of the order and type of the solutions are established.

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