Solutions of a class of multiplicatively advanced differential equations

Pravica, David W.; Randriampiry, Njinasoa; Spurr, Michael J.

doi:10.1016/j.crma.2018.05.011

Harmonic analysis/Ordinary differential equations

Solutions of a class of multiplicatively advanced differential equations
[Solutions d'une classe d'équations différentielles multiplicativement avancées]

Présenté par : the Editorial Board

Pravica, David W.^1,² ; Randriampiry, Njinasoa ¹ ; Spurr, Michael J.^1,²

¹ Department of Mathematics, East Carolina University, Greenville, NC, USA
² School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa

Comptes Rendus. Mathématique, Tome 356 (2018) no. 7, pp. 776-817.

Résumé
Abstract

Des équations différentielles multiplicativement avancées (MADE) de la forme $f^{(n)} (t) = α f (β t)$ avec $α \neq 0$ , $β > 1$ sont étudiées dans le cadre des solutions de type $f_{μ, λ} (t)$ définies sur $[0, \infty)$ . Pour $λ \in Q^{+}, μ \in R$ , les solutions $f_{μ, λ} (t)$ sont prolongées sur $(- \infty, \infty)$ d'une manière non unique pour obtenir des solutions ondelettes dans l'espace de Schwartz $F_{μ, λ} (t)$ de l'originale MADE, avec tous les moments de $F_{μ, λ} (t)$ nuls. Des exemples sont étudiés en détail. La transformée de Fourier de chaque $F_{μ, λ} (t)$ est calculée et, dans un certain nombre d'exemples, est liée à la fonction thêta de Jacobi. Des conditions supplémentaires suffisantes pour l'unicité de la solution de certaines MADE avec condition initiale sont données. Les conditions de décroissance et de non-décroissance à −∞ sont obtenues. Les taux de décroissance à ±∞ en termes de fonctions familières sont établis.

The multiplicatively advanced differential equations (MADEs) of form $f^{(n)} (t) = α f (β t)$ with $α \neq 0$ , $β > 1$ are studied along with a class of their solutions of type $f_{μ, λ} (t)$ defined on $[0, \infty)$ . For $λ \in Q^{+}, μ \in R$ , the solutions $f_{μ, λ} (t)$ are extended to $(- \infty, \infty)$ in a non-unique manner to obtain Schwartz wavelet solutions $F_{μ, λ} (t)$ of the original MADE, with all moments of $F_{μ, λ} (t)$ vanishing. Examples are studied in detail. The Fourier transform of each $F_{μ, λ} (t)$ is computed and, in a number of examples, is related to the Jacobi theta function. Additional conditions sufficient for the uniqueness of certain MADE initial value problems are given. Conditions for decay and non-decay at −∞ are obtained. Decay rates at ±∞ in terms of familiar functions are established.

Reçu le : 2018-05-06
Accepté le : 2018-05-18
Publié le : 2018-06-05

DOI : 10.1016/j.crma.2018.05.011

Affiliations des auteurs :

Pravica, David W. ^{1, 2} ; Randriampiry, Njinasoa ¹ ; Spurr, Michael J. ^{1, 2}

¹ Department of Mathematics, East Carolina University, Greenville, NC, USA
² School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa

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     title = {Solutions of a class of multiplicatively advanced differential equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {776--817},
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Pravica, David W.; Randriampiry, Njinasoa; Spurr, Michael J. Solutions of a class of multiplicatively advanced differential equations. Comptes Rendus. Mathématique, Tome 356 (2018) no. 7, pp. 776-817. doi : 10.1016/j.crma.2018.05.011. http://www.numdam.org/articles/10.1016/j.crma.2018.05.011/

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