A linear Fuchsian equation with variable indices
type: Item
instance of: scholarly article
title: A linear Fuchsian equation with variable indices
zbMATH Open document ID: 1030.35004
DOI: 10.1016/S0022-0396(03)00029-9
author: Ali Bentrad
author: Satyanad Kichenassamy
published in: Journal of Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: The authors are interested in formal solutions to Fuchsian partial differential equation of the form \[ Lu:= (t\partial_t- c)(t\partial_t- x- d) u- t^2 P_2(x,\partial_x)u= f, \] where \(P_2\) is a second-order operator with analytic coefficients. The following two problems are under discussion: \(\bullet\) Let \(f\equiv 0\). What is the influence of \(c\) and \(d\) on the existence of power series representations with respect to \(t\) for solutons of \(Lu\equiv 0\), that is, \(u= \sum_{j\geq 0} u_j(x, t)t^{a+ j}\)? \(\bullet\) How does a special structure of \(f\) (e.g., \(f(t,x)= t^k x^m\), \(f(t,x)= t^k(x^m- p(t)))\), of \(p\) especially, influence the existence of power series representations with respect to \(t\) for solutions of \(Lu\equiv f\)? The authors give several answers to these questions. They \(\bullet\) characterize the function space to which \(u_j\) belong to, these are spaces of analytic functions in new basic variables, containing \(x^{-k}\), \(t^x\) and \(\ln t\); \(\bullet\) explain the degree of freedom, that is, the number of functions one can choose arbitrarily; \(\bullet\) show how a special behaviour of \(p= p(t)\) is reflected to a solution of \(Lu\equiv f\) (polynomial behaviour, logarithmic behaviour).
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35C20
zbMATH DE Number: 1915653
zbMATH Keywords: Fuchsian reduction
zbMATH Keywords: hypergeometric functions
zbMATH Keywords: Euler-Poisson-Darboux equation
zbMATH Keywords: formal solutions
zbMATH Keywords: power series representations
reviewed by: Michael Reissig
MaRDI profile type: Publication
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full work available at URL: s0022-0396(03)00029-9
OpenAlex ID: W2082702560
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description: scientific article; zbMATH DE number 1915653