Analytical study of the pantograph equation ...
Document type :
Compte-rendu et recension critique d'ouvrage
Title :
Analytical study of the pantograph equation using Jacobi theta functions
Author(s) :
Journal title :
Journal of Approximation Theory
Publisher :
Elsevier
Publication date :
2023
ISSN :
0021-9045
English keyword(s) :
Pantograph equation
q-difference equation
Connection problem
Jacobi θ-function
q-difference equation
Connection problem
Jacobi θ-function
HAL domain(s) :
Mathématiques [math]/Equations aux dérivées partielles [math.AP]
English abstract : [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. ...
Show more >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.Show less >
Show more >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.Show less >
Language :
Anglais
Popular science :
Non
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