Mathematical Sciences: Asymptotic Solutions of Differential Equations

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8821547 Sibuya This project will continue work focusing on the properties of solutions of ordinary differential equations defined in the real and complex domain. There are four main areas under consideration. The first consists of studies on the asymptotic behavior of solutions of a first-order system of linear differential equations as the real independent variable increases indefinitely. This work extends results in the textbook of Coddington and Levinson. A second line of investigation will consider order linear differential equations of higher order in which the coefficients of the derivatives are given by holomorphic functions. The object of this investigation will be to give a new interpretation in terms of the algebraic formalism of D-modules to the question whether the inhomogeneous equation can be solved in a sector of the complex plane. General results on differential equations are to be derived as elementary consequences of the theory of injective modules. Related work will also be done studying asymptotic solutions of mildly nonlinear systsms of equations of Gevrey type. The difference in this effort will be the introduction of singular perturbations. Solutions are expected to reflect the same growth properties as those from the unperturbed equations for small values of the parameter. The final theme will treat the classical Stokes multipliers of Riemann's theory of hypergeometric equations. The multipliers give invariants of equations. In this context, parameters will be introduced into the equations and the properties of the multipliers will be studied as functions of the parameters. Some results of this type can be derived from the principal investigator's 1975 book.