Mathematical Sciences: Asymptotic Solutions of Differential Equations

该项目继续在数学研究 常微分方程组 目的是研究 代数微分方程的形式幂级数解 方程 具体而言，工作将集中力量 系数不大于Gevrey型级数 而不是仅仅依赖于 系数和单个参数。 问题的来源 这里进行的是1903年的一篇论文，它给出了一个条件， 方程，确保解决方案可以表达这样一个 幂级数 这个结果的每一个后续证明都需要 特殊复杂性的递归公式，其中最好的 参数的值几乎是不可能估计的。 一个新 解决这个问题的方法最近被发现， 在线性设置中提供良好的结果。 工作要做 现在将集中在非线性方程和奇异摄动 其中系数估计是 预计将是尖锐的。

This project continues mathematical investigations in the field of ordinary differential equations. The object is to study the formal power series solutions of algebraic differential equations. Specifically, the work will concentrate on power series of Gevrey type in which the coefficients are not greater than certain bounds depending only on the index of the coefficient and a single parameter. The source of the questions undertaken here is a paper of 1903 which gives conditions on a equation which ensure that a solution may be expressed by such a power series. Every subsequent proof of this result requires recurrence formulas of exceptional complexity in which the best value of the parameter is all but impossible to estimate. A new approach to the problem has recently been uncovered which provides good results in the linear setting. Work to be done will now focus on nonlinear equations and singular perturbations of regular equations in which the coefficient estimates are expected to be sharp.