Analytical study of the pantograph equation ...
Type de document :
Compte-rendu et recension critique d'ouvrage
Titre :
Analytical study of the pantograph equation using Jacobi theta functions
Auteur(s) :
Titre de la revue :
Journal of Approximation Theory
Éditeur :
Elsevier
Date de publication :
2023
ISSN :
0021-9045
Mot(s)-clé(s) en anglais :
Pantograph equation
q-difference equation
Connection problem
Jacobi θ-function
q-difference equation
Connection problem
Jacobi θ-function
Discipline(s) HAL :
Mathématiques [math]/Equations aux dérivées partielles [math.AP]
Résumé en anglais : [en]
The aim of this paper is to use the analytic theory of linear q-difference equations for the study of the functional-differential equation y ′ (x) = ay(q x) + by(x), where a and b are two non-zero real or complex numbers. ...
Lire la suite >The aim of this paper is to use the analytic theory of linear q-difference equations for the study of the functional-differential equation y ′ (x) = ay(q x) + by(x), where a and b are two non-zero real or complex numbers. When 0 < q < 1 and y(0) = 1, the associated Cauchy problem admits a unique power series solution, ∑ n≥0 (-a/b; q) n/n! (bx) n , that converges in the whole complex x-plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a byproduct, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the asymptotic behavior of the solutions over the real axis.Lire moins >
Lire la suite >The aim of this paper is to use the analytic theory of linear q-difference equations for the study of the functional-differential equation y ′ (x) = ay(q x) + by(x), where a and b are two non-zero real or complex numbers. When 0 < q < 1 and y(0) = 1, the associated Cauchy problem admits a unique power series solution, ∑ n≥0 (-a/b; q) n/n! (bx) n , that converges in the whole complex x-plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a byproduct, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the asymptotic behavior of the solutions over the real axis.Lire moins >
Langue :
Anglais
Vulgarisation :
Non
Collections :
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