Harmonic analysis/Ordinary differential equations
Solutions of a class of multiplicatively advanced differential equations
Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 776-817.

The multiplicatively advanced differential equations (MADEs) of form f(n)(t)=αf(βt) with α0, β>1 are studied along with a class of their solutions of type fμ,λ(t) defined on [0,). For λQ+,μR, the solutions fμ,λ(t) are extended to (,) in a non-unique manner to obtain Schwartz wavelet solutions Fμ,λ(t) of the original MADE, with all moments of Fμ,λ(t) vanishing. Examples are studied in detail. The Fourier transform of each Fμ,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.

Des équations différentielles multiplicativement avancées (MADE) de la forme f(n)(t)=αf(βt) avec α0, β>1 sont étudiées dans le cadre des solutions de type fμ,λ(t) définies sur [0,). Pour λQ+,μR, les solutions fμ,λ(t) sont prolongées sur (,) d'une manière non unique pour obtenir des solutions ondelettes dans l'espace de Schwartz Fμ,λ(t) de l'originale MADE, avec tous les moments de Fμ,λ(t) nuls. Des exemples sont étudiés en détail. La transformée de Fourier de chaque Fμ,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.

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DOI: 10.1016/j.crma.2018.05.011
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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Pravica, David W.; Randriampiry, Njinasoa; Spurr, Michael J. Solutions of a class of multiplicatively advanced differential equations. Comptes Rendus. Mathématique, Volume 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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