Mathematical Sciences: Asymptotic Solutions of Differential Equations

8821547 Sibuya该项目将继续开展工作，重点是研究在实数和复数域中定义的常微分方程解的性质。目前正在考虑的主要领域有四个。第一部分研究了一阶线性微分方程解的渐近行为。这一工作推广了Coddington和Levinson教科书中的结果。第二类研究将考虑高阶线性微分方程组，其中导数的系数由全纯函数给出。本研究的目的是用D-模的代数形式对非齐次方程能否在复平面的一个扇区内求解的问题给予新的解释。关于微分方程式的一般结果将作为内射模理论的基本结果来推导。相关工作也将被用来研究弱非线性Gevrey型方程组的渐近解。这一努力的不同之处将是引入了奇异微扰。对于较小的参数值，解应该反映与未扰动方程相同的增长特性。最后一个主题将讨论黎曼超几何方程理论的经典斯托克斯乘子。乘子给出了方程的不变量。在这种情况下，将参数引入方程，并将乘子的性质作为参数的函数来研究。这种类型的一些结果可以从首席研究员1975年的书中得出。

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.