Solutions of an Inhomogeneous MADE using an Analysis of Wavelet Coefficients
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Date
2023-12-08
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Authors
Faircloth, Dallas Ali
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Publisher
East Carolina University
Abstract
In this paper solutions to an Inhomogeneous Multiplicatively Advanced Differential Equation (MADE) of the form $y^\prime(t) - Ay(qt)=f(t)$, for $q>1$ and certain functions $f(t)\in \L^2(\R)$, are provided. Such solutions are obtained with the help of wavelet frames generated by two special functions called $_qCos(t)$ and $_qSin(t)$. The introduction provides basic definitions and useful tools. Then a specific solution of our main MADE above is obtained, where $f(t)$ is equal to either $_qCos(\alpha t+\beta)$ or $_qSin(\alpha t+\beta)$. After obtaining the solutions for these specific MADE's, we will prove that the solutions are actually Schwartz wavelets whose series expansion converge uniformly and absolutely with all moments vanishing. The general MADE is then solved by obtaining a series expansion for $f(t)$ in terms of $_qCos(t)$ or $_qSin(t)$. A series solution, $y(t)$, of the general MADE then follows by linearity. Assumptions on the wavelet coefficients will be obtained sufficient for the series solution $y(t)$ to converge uniformly and absolutely. Finally, pictures of these solutions are provided along with questions about confluence between the solutions of these MADE's and their non-advanced differential equation counterparts.