Special functions arising in the study of semi-linear equations in circular domains
type: Item
instance of: scholarly article
title: Special functions arising in the study of semi-linear equations in circular domains
zbMATH Open document ID: 1125.33001
published in: Journal of Computational and Applied Mathematics
publication date: 2007-04-13T00:00:00Z
review text: Rayleigh functions are defined by the formula \(\sigma_{l}(\nu)=\sum_{n=1}^{\infty}\frac{1}{\lambda_{\nu,n}^{2l}}\), where \(l=1,2,\ldots\) and \(\lambda_{\nu,n}\neq 0\) are zeros of the Bessel function \(J_{\nu}(x)\) and \(n=1,2,\ldots\)is the number of the zero. These functions appear in the classical problems of vibrating circular membranes, heat conduction in cylinders and diffraction through circular apertures. In this paper the author studies a new family of special functions, namely, convolutions of Rayleigh functions with respect to the Bessel index, \[ R_{l}(m)=\sum_{p,k=-\infty;\;p+k=m}^{\infty}\,\sum_{q,s=1}^{\infty}\frac{1}{\lambda_{p,q}^{2l}}\,\frac{1}{\lambda_{k,s}^{2l}} \] for \(l=1,2,\ldots ;m=0,\pm1,\pm2,\ldots\) appearing in constructing solutions of semi-linear evolution equations in circular domains. The author considers as an example a forced Cahn-Hilliard equation in a unit disc with homogeneous boundary and initial conditions. Construction of its global in time solutions involves the use of \(R_{1}(m)\) and \(R_{2}(m)\). A general representation of \(R_{l}(m)\) is deduced and on the basis of that a particular result of \(R_{2}(m)\) is obtained convenient for computing its asymptotics as \(| m| \rightarrow \infty\). The latter issue is important for establishing a function space to which the solution of the corresponding problem belongs.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 33C90
zbMATH DE Number: 5142283
zbMATH Keywords: Convolutions of Rayleigh functions
zbMATH Keywords: representation
zbMATH Keywords: asymptotics
zbMATH Keywords: Cahn-Hilliard equation
author: V. V. Varlamov
MaRDI profile type: Publication
full work available at URL: j.cam.2005.10.040
OpenAlex ID: W2032727088
cites work: The zeta function constructed from the zeros of the Bessel function
cites work: Free Energy of a Nonuniform System. I. Interfacial Free Energy
cites work: Q3536312
cites work: Sum rules for zeros of Bessel functions and an application to spherical Aharonov-Bohm quantum bags
cites work: Riccati equations and convolution formulae for functions of Rayleigh type
cites work: The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types
cites work: The Rayleigh Function
cites work: Q5634306
cites work: Q3231359
cites work: Convolution formulae for functions of Rayleigh type
cites work: Q4065978
cites work: Q3028909
cites work: Dynamics of evolutionary equations
cites work: Nonlinear Heat Equation with a Fractional Laplacian in a Disk
cites work: On the spatially two-dimensional Boussinesq equation in a circular domain
cites work: Convolution of Rayleigh functions with respect to the Bessel index
cites work: Forced nonlinear oscillations of elastic membranes
cites work: Q5562752
cites work: Coarsening dynamics of the convective Cahn-Hilliard equation
cites work: Q5531981
DOI: 10.1016/J.CAM.2005.10.040
description: scientific article; zbMATH DE number 5142283