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arXiv · 1509.02603

Multiplicity of Solutions for Linear Partial Differential Equations Using (Generalized) Energy Operators

J. P. Montillet
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Abstract

Families of energy operators and generalized energy operators have recently been introduced in the definition of the solutions of linear Partial Differential Equations (PDEs) with a particular application to the wave equation [Montillet, 2014, doi: 10.1007/s10440-014-9978-9]. To do so, the author has introduced the notion of energy spaces included in the Schwartz space $\mathbf{S}^-(\mathbb{R})$. In this model, the key is to look at which ones of these subspaces are reduced to {0} with the help of energy operators (and generalized energy operators). It leads to define additional solutions for a nominated PDE. Beyond that, this work intends to develop the concept of multiplicity of solutions for a linear PDE through the study of these energy spaces (i.e. emptiness). The main concept is that the PDE is viewed as a generator of solutions rather than the classical way of solving the given equation with a known form of the solutions together with boundary conditions. The theory is applied to the wave equation with the special case of the evanescent waves. The work ends with a discussion on another concept, the duplication of solutions and some applications in a closed cavity.

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J. P. Montillet. 2015-09-09. Multiplicity of Solutions for Linear Partial Differential Equations Using (Generalized) Energy Operators. https://arxiv.org/abs/1509.02603

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