Research On Solutions To Boundary Value Problems Of Several Types Of Differential Equations

With the development of mathematics,functional analysis plays an extraordinary role in analytical science.Functional analysis is a relatively new subject,which has been promoted by quantum mechanics and mathematical physics equations with the continuous development.Partial differential equations are generally used to solve some important problems in life,which to some extent express the commonness of most kinds of physical phenomena.However,the definite solution condition constrains some problems and reflects the personality of the problem,which puts forward different requirements for different cases.The so-called definite solution problem usually refers to the process of transforming the equation and the conditions of the definite solution into a whole.In order to obtain the definite solution problem,we must first find the general solution of the definite solution problem,and then determine the function.This is the basic step to solve the partial differential equation.In general,however,solving the general solution is not so simple,so the above terms of solution are used to determine the function.However,we can also use the separation coefficient method to solve the partial differential equation.The solution of the solution includes the bounded space and the unbounded space.The solutions are the Fourier series method,the separation coefficient method and the separation variable method.However,the two cases are different because of the different problems discussed.Some methods are also different.For example,some methods can not obtain the definite solution of the unbounded space,but can solve the bounded space.Of course,there are more than these methods to solve the fixed solution problem.When we meet the solution of the equation in one dimension space,we can also use the Laplasse transform method to solve it flexibly.The finite difference method and variational method are commonly used in our solution.The finite difference method is not to let us use the computer to calculate first,but to transform the definite solution problem into the variational problem and then to calculate.The variational method is to transform the bounded problem into a variational problem and then to obtain the first or second order approximate solution of the variational problem.With the development of physics science,partial differential equations gradually become the center of mathematics.This paper is divided into three chapters.The first chapter is the introduction.It introduces the research background of partial differential equation and a global summary of the knowledge of variational method,second order elliptic equation,functional analysis and so on.The second chapter studies the solvability of boundary value problems for a class of semilinear elliptic equations.In this chapter,the existence and uniqueness theorems of solutions are studied by studying the boundary value problems of semilinear elliptic equations.The main part of our first part is devoted to examining the problem(1).The boundary value problem of this class of semilinear elliptic equations is studied.By proving two theorems,it is finally proved that the problem(1)has only one solution from itself to verify the uniqueness theorem of the solution.In chapter 3,we study the solvability of boundary value problems of differential equations by using Reese's method and fixed point theorem.This chapter is divided into two parts.The first part introduces the principle and steps of the Rhys method in finding the approximate solution of the boundary value problem of differential equations.The second part studies the form of This kind of boundary value problem of ordinary differential equation is transformed into the extremum function of variational problem to obtain the approximate solution of the equation.