caa a22 4500 606166181 CHVBK 20210128105227.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s10440-014-9978-9 doi (NATIONALLICENCE)springer-10.1007/s10440-014-9978-9 Montillet J.-P Cascadia Hazards Institute, Ellensburg, WA, USA aut The Generalization of the Decomposition of Functions by Energy Operators (Part II) and Some Applications [Elektronische Daten] [J.-P. Montillet] This work introduces the families of generalized energy operators $([[.]^{p}]_{k}^{+})_{k\in\mathbb{Z}}$ and $([[.]^{p}]_{k}^{-} )_{k\in\mathbb{Z}}$ (p in $\mathbb{Z}^{+}$ ). One shows that with Lemma1, the successive derivatives of $([[f]^{p-1}]_{1}^{+} )^{n}$ (n in $\mathbb{Z}$ , n≠0) can be decomposed with the generalized energy operators $([[.]^{p}]_{k}^{+})_{k\in \mathbb{Z}}$ when f is in the subspace $\mathbf{S}_{p}^{-}(\mathbb{R})$ . With Theorem1 and f in $\mathbf{s}_{p}^{-}(\mathbb{R})$ , one can decompose uniquely the successive derivatives of $([[f]^{p-1}]_{1}^{+} )^{n}$ (n in $\mathbb{Z}$ , n≠0) with the generalized energy operators $([[.]^{p}]_{k}^{+})_{k\in\mathbb{Z}}$ and $([[.]^{p}]_{k}^{-})_{k\in\mathbb{Z}}$ . $\mathbf{S}_{p}^{-}(\mathbb{R})$ and $\mathbf{s}_{p}^{-}(\mathbb{R})$ (p in $\mathbb{Z}^{+}$ ) are subspaces of the Schwartz space $\mathbf{S}^{-}(\mathbb{R})$ . These results generalize the work of Montillet (Acta Appl. Math., doi: 10.1007/s10440-013-9829-0 , 2013). The second fold of this work is the application of the generalized energy operator families onto the solutions of linear partial differential equations. The solutions are functions of two variables and defined in subspaces of $\mathbf{S}^{-}(\mathbb{R}^{2})$ . The theory is then applied to the Helmholtz equation. In this specific case, the use of generalized energy operators in the general solution of this PDE extends the results of Montillet (Int. Math. Forum 5(48):2387-2400, 2010). This work ends with some numerical examples. We also underline that this theory could possibly open some applications in astrophysics and aeronautics. Springer Science+Business Media Dordrecht, 2014 Energy operator nationallicence Generalized energy operator nationallicence Schwartz space nationallicence Decomposition of finite energy function nationallicence Linear PDE nationallicence Application to Helmholtz equation nationallicence Wave nationallicence Acta Applicandae Mathematicae Springer Netherlands 140/1(2015-12-01), 43-70 0167-8019 140:1<43 2015 140 10440 https://doi.org/10.1007/s10440-014-9978-9 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10440-014-9978-9 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Montillet J.-P Cascadia Hazards Institute, Ellensburg, WA, USA aut NATIONALLICENCE 773 0- Acta Applicandae Mathematicae Springer Netherlands 140/1(2015-12-01), 43-70 0167-8019 140:1<43 2015 140 10440 SWISSBIB 56113703X