The Study Of Monomial Summability Of The Formal Solution To A Singular Partial Differential Equation

Since the20th century, with the study on divergent power series solution and the exploringof the meaning of them, the classical asymptotic expansion theory and summability theory havebeen developed rapidly. And now they become Gevrey asymptotic expansion theory andmultisummability theory, which have more extansive used and wider vistas of research.Part one: We prove the existence and uniqueness of the formal power series solution to asingular partial differential equation by comparing coeffients of an identity, given the specificform of the series solution, we construct holomorphic solution of the problem with the help ofthe formal Borel transform, and prove that the formal power series solution is1-Gevrey.Part two: We use the fixed point theorem and part of the results in Gevrey asymptotictheory and summability theory to prove that the above formal solution is1-summable in amonomial.