The Generalization of the Decomposition of Functions by Energy operators (Part II) and some Applications

TL;DR

This paper introduces generalized energy operators to decompose derivatives of functions in specific subspaces, extending previous work, and applies these operators to solve the Helmholtz equation with potential applications in astrophysics and aeronautics.

Contribution

The paper generalizes the decomposition of derivatives using energy operators and applies this framework to linear PDE solutions, notably the Helmholtz equation.

Findings

Successful decomposition of derivatives in subspaces of Schwartz space.

Extension of previous energy operator work to broader function classes.

Application to Helmholtz equation solutions with numerical examples.

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 $\big ([[$f$]^{p-1}]_1^+ \big)^n$ ($n$ in $\mathbb{Z}$, $n\neq 0$) can be decomposed with the generalized energy operators $\big ([[.]^p]_k^+\big)_{k\in\mathbb{Z}}$ 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 $\big ([[$f$]^{p-1}]_1^+ \big)^n$ ($n$ in $\mathbb{Z}$, $n\neq 0$) with the generalized energy operators $\big ([[.]^p]_k^+\big)_{k\in\mathbb{Z}}$ and $\big ([[.]^p]_k^-\big)_{k\in\mathbb{Z}}$. $\mathbf{S}_p^-(\mathbb{R})$ 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 [arxiv:1208.3385]. 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 [montilletIMF45-48-2010]. This work ends with some numerical examples. We also underline that this theory could possibly open some applications in astrophysics and aeronautics.