Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations

• Main Authors: David W. Pravica, Njinasoa Randriampiry, Michael J. Spurr
• Format: Article
• Language: English
• Published: MDPI AG 2020-07-01
• Series: Axioms
• Subjects: MADE; eigenfunction; convergence; Fourier transform
• Online Access: https://www.mdpi.com/2075-1680/9/3/83

A family of Schwartz functions <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">W</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> are interpreted as eigensolutions of MADEs in the sense that <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi mathvariant="script">W</mi> <mrow> <mo>(</mo> <mi>δ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>E</mi> <mspace width="0.166667em"></mspace> <mi mathvariant="script">W</mi> <mrow> <mo>(</mo> <msup> <mi>q</mi> <mi>γ</mi> </msup> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> where the eigenvalue <inline-formula> <math display="inline"> <semantics> <mrow> <mi>E</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula> is independent of the advancing parameter <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. The parameters <inline-formula> <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>,</mo> <mspace width="0.166667em"></mspace> <mi>γ</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </semantics> </math> </inline-formula> are characteristics of the MADE. Some issues, which are related to corresponding <i>q</i>-advanced PDEs, are also explored. In the limit that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>→</mo> <msup> <mn>1</mn> <mo>+</mo> </msup> </mrow> </semantics> </math> </inline-formula> we show convergence of MADE eigenfunctions to solutions of ODEs, which involve only simple exponentials and trigonometric functions. The limit eigenfunctions (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <msup> <mn>1</mn> <mo>+</mo> </msup> </mrow> </semantics> </math> </inline-formula>) are not Schwartz, thus convergence is only uniform in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula> on compact sets. An asymptotic analysis is provided for MADEs which indicates how to extend solutions in a neighborhood of the origin <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. Finally, an expanded table of Fourier transforms is provided that includes Schwartz solutions to MADEs.