Solutions of Inhomogeneous Multiplicatively Advanced ODEs and PDEs with a q ‐Fredholm Theory and Applications to a q ‐Advanced Schrödinger Equation

• Published in: Abstract and applied analysis Vol. 2024; no. 1
• Main Authors: Pravica, David W., Randriampiry, Njinasoa, Spurr, Michael J.
• Format: Journal Article
• Language: English
• Published: New York John Wiley & Sons, Inc 01.01.2024; Hindawi Limited
• Subjects: Decay rate; Differential equations; Green's functions; Integral equations; Mathematical research; Operators (mathematics); Partial differential equations; Quantum mechanics; Schrodinger equation; Series (mathematics); United States
• ISSN: 1085-3375; 1687-0409
• DOI: 10.1155/2024/8130561

For q > 1, a new Green’s function provides solutions of inhomogeneous multiplicatively advanced ordinary differential equations (iMADEs) of form y ( N ) ( t ) − A y ( q t ) = f ( t ) for t ∈ [0, ∞). Such solutions are extended to global solutions on ℝ . Applications to inhomogeneous separable multiplicatively advanced partial differential equations are presented. Solutions to a linear free forced q ‐advanced Schrödinger equation are obtained, opening an avenue to applications in quantum mechanics. New q ‐Mittag‐Leffler functions q E α , β and Υ N , p govern the allowable decay rate of the inhomogeneities f ( t ) in the above iMADE. This provides a refinement to standard distribution theory, as we show is necessary for this study of iMADEs. A q ‐Fredholm theory is developed and related to the above approach. For f ( t ) whose antiderivatives provide eigenfuntions of the noncompact integral operator K below, we exhibit solutions of the iMADE. Examples are provided, including a certain class of Dirichlet series.