The Generalization of the Decomposition of Functions by Energy Operators (Part II) and Some Applications

Abstract

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 Lemma 1, 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 Theorem 1 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.