Mathematical Sciences: Asymptotic Solutions of Differential Equations

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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.