Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations

• Published in: Axioms Vol. 9; no. 3; p. 83
• Main Authors: Pravica, David W., Randriampiry, Njinasoa, Spurr, Michael J.
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.09.2020
• Subjects: Convergence; eigenfunction; Eigenvalues; Eigenvectors; Fourier transform; Fourier transforms; MADE; Parameters; Partial differential equations; Trigonometric functions
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms9030083

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.