Abstract
A family of Schwartz functions W ( t ) are interpreted as eigensolutions of MADEs in the sense that W ( δ ) ( t ) = E W ( q γ t ) where the eigenvalue E ∈ R is independent of the advancing parameter q > 1 . The parameters δ , γ ∈ N are characteristics of the MADE. Some issues, which are related to corresponding q-advanced PDEs, are also explored. In the limit that q → 1 + we show convergence of MADE eigenfunctions to solutions of ODEs, which involve only simple exponentials and trigonometric functions. The limit eigenfunctions ( q = 1 + ) are not Schwartz, thus convergence is only uniform in t ∈ R on compact sets. An asymptotic analysis is provided for MADEs which indicates how to extend solutions in a neighborhood of the origin t = 0 . Finally, an expanded table of Fourier transforms is provided that includes Schwartz solutions to MADEs.
Highlights
The introduction of a relaxing parameter q > 1 in differential equations was found to provide stability properties for their corresponding solutions. This is a phenomenon well-known in numerical analysis where if the Ordinary Differential Equation (ODE)
That such a principle holds for ODEs as ∆t → 0+ was established through the study of Multiplicatively Advanced Differential Equations (MADEs) as q → 1+, and will be discussed further in this article
The study of multiply advanced differential equations falls within the area of functional differential equations, as is studied for instance in Fox, et al [2], Kato, et al [3] and Dung [5]
Summary
The introduction of a relaxing parameter q > 1 in differential equations was found to provide stability properties for their corresponding solutions. This is a phenomenon well-known in numerical analysis where if the Ordinary Differential Equation (ODE). If one can obtain yn+1 explicitly in terms of yn the iteration scheme often converges much faster, and for longer time intervals, than that provided by the forward Euler method [1], p. Part of our analysis of stability will require obtaining uniform apriori bounds This will be achieved in a somewhat general setting, and the consequences will be presented in the form of examples of advanced differential equations
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