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

Abstract

For q > 1, a new Green’s function provides solutions of inhomogeneous multiplicatively advanced ordinary differential equations (iMADEs) of form y(N)(t) − Ay(qt) = 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 qEα,β and ΥN,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.