The Solutions Of Singular Partial Differential Equations Are Studied Based On Asymptotic Expansion

The asymptotic expansion theory plays a critical role in the research of the properties of solutions of singular PDE,it has been researched extensively.While,the classical power series comparison coefficient method and undetermined coefficient method are used to research the solutions of the NLPDE.However,the methods sometimes have errors for the complex equations.To make accurate research,mathematicians construct Banach space and introduce compression operator method.However,there are few studies on the problem of constructing asymptotic solutions.In this paper,we construct asymptotic solutions for the NLPDE and discuss their initial value problems,and then prove the existence and uniqueness of the solutions.It is an application of asymptotic expansion theory and enriches the research results of PDE.The following is the main research work of this paper.Firstly,the concept of asymptotic expansion of monomials and how to construct formal power series solutions according to Gevrey asymptotic expansion are introduced,and the summability of solutions and related theorems and propositions are introduced.Secondly,introduces the integral transformation of singular PDE by Borel-Laplace method,and then constructs the formal asymptotic solution according to the transformed form,and proves the existence of the solution of PDE by giving a special case.Finally,The Riesz-Thorim method is used to prove the solution of the equation.Then,the existence and summability of the solution of the singular partial differential equation are proved through the operation and the transformed form.