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

For q>1, a new Green’s function provides solutions of inhomogeneous multiplicatively advanced ordinary differential equations (iMADEs) of form yNt−Ayqt=ft for t∈[0,∞). Such solutions are extended to global solutions on ℠. 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 ft 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 ft 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.