On The Formal Solution Of A Type Of Singular Partial Differential Equations

For the past few years, the new progress made in asymptotic theory and in summability theory of divergent series, has a great promotion effect in the studying of formal solutions to singular differential equations. And it provides effective researching methods. This dissertation mainly studies a singular nonlinear differential equation(PDE). Firstly, the existence and uniqueness of the formal power series solution to the PDE is proven. Secondly, the PDE is transformed to a sequence of ordinary differential equations(ODEs). Based on solutions to ODEs, several Banach spaces and corresponding on them are constructed. By applying the fixed point theorem, the existence and uniqueness of solutions which are holomorphic and bounded on extended corresponding sectors with opening large enough to the PDE is shown. Then it is proved that differences of solutions on common extended sectors are exponentially small. Furthermore, the proof for monomial summability of this formal solution is given according to an important theorem in monomial summability theory.It has, to some extent, a pushing effect in reducing of differential equations.This graduation thesis consists mainly of three parts.In the first part, we give an overview of the origin and the development of the asymptotic theory and summability theory of divergent series.In the second part, we enumerate the definitions and some important theorems of multisummability, and give the definitions and relative theorems of monomial summability.In the last part(mainly including chapter 3 and chapter 4), we get the partial differential equation which we are going to study prom the reducing of an ordinary differential equation. Under some hypotheses, we prove the existence and uniqueness of the formal power series solution to the PDE and show that it is summable in a monomial.