Direct and inverse interpolation for Jacobian elliptic functions, zeta function of Jacobi and complete elliptic integrals of the second kind
type: Item
instance of: scholarly article
title: Direct and inverse interpolation for Jacobian elliptic functions, zeta function of Jacobi and complete elliptic integrals of the second kind
zbMATH Open document ID: 0910.33012
DOI: 10.1016/0898-1221(96)00166-6
author: S. D. Panteliou
author: Andrew D. Dimarogonas
author: I. N. Katz
publication date: 1997-02-19T00:00:00Z
review text: The problems mentioned in this paper for the ellipsoidal shape of the internal boundary of the cavity or inclusion suggests the use of the ellipsoidal coordinates. The corresponding coordinate transformation, which relates the Cartesian coordinates \((x,y,z)\) to the ellipsoidal coordinates \(a_i\), \(i= 1,2,3\) is most conveniently introduced by using Jacobian elliptic functions. Computing the value of the Jacobian elliptic functions, given the argument \(u\) and the parameter \(m\), is a problem, whose solution can be found either tabulated in tables of elliptic functions or by use of existing software, such as Mathematica, etc. The inverse problem, finding the argument, given the Jacobian elliptic function and the parameter \(m\), is a problem whose solution is found only in tables of elliptic functions. Standard polynomial inverse procedures fail, due to ill conditioning of the system of linear equations of the unknown. This paper contains a numerical procedure for inverse interpolation which gives good results in the computation of the argument of the Jacobian elliptic function given the Jacobian elliptic function and the parameter. Also, a direct interpolation is described which gives the zeta function of Jacobi and the complete elliptic integral of the second kind given the argument and the parameter. These new interpolation procedures are important in problems involving cavities or inclusions of ellipsoidal shape encountered in the mechanical design of bearing, filters and composite materials. They are also important in the modeling of porosity of bones. This porosity may lead to osteoporosis, a disease which affects bone material density in human with bad consequences.
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 33C90
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 92C05
zbMATH DE Number: 980340
zbMATH Keywords: Jacobian elliptic functions
reviewed by: R. S. Dahiya
describes a project that uses: Mathematica
MaRDI profile type: Publication
full work available at URL: 0898-1221(96)00166-6
OpenAlex ID: W2130494126
cites work: Q5802496
cites work: Q3136490
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 980340