The Studies Of Positive Solutions About Nonlinear Neumann Boundary Value Problems And Numerical Solutions Of Integral Equations

The boundary value problem(BVP)of ordinary differential equations are an important part of the researches of differential equations,which can well explain a variety of natural phenomena,such as atmospheric convection,the stability of aircraft flight,etc.It also has great theoretical significance to study them.In this thesis,the Green’s functions are used to transform the BVPs of ordinary differential equations into the corresponding integral equations.Then the existence and multiplicity of the positive solutions of the integral equations are studied.At the same time,the numerical solutions of the integral equations are discussed.The main contents of this thesis are as follows:In the first chapter,we briefly summarize the development history and current situation of the BVPs of ordinary differential equations as well as the integral equations;we also give the relevant lemmas and concepts commonly used in the thesis.In the second chapter,we mainly study finding the Green’s function methods for the BVP of a class of third-order ordinary differential equations.The calculating methods of their Green’s functions are summarized,and the practical examples are given as well.In the third chapter,we mainly research the existence of multiple positive solutions and the non-existence of positive solutions for the Neumann BVP of second-order nonlinear ordinary equations.Using the fixed point index theory in nonlinear functional analysis,we obtain the existence of at least two positive solutions for the BVP.The conditions for the non-existence of positive solutions are given.In the fourth chapter,the numerical solutions of the second kind of Fredholm integral equations are mainly studied.Firstly,the current situation and classification of the integral equations are briefly summarized.Secondly,the existence and uniqueness of the solutions for the second kind of Fredholm integral equations are proved,and the specific examples are given.Finally,a numerical integration solving method for the equations is given,for which the analytical solutions are difficult to be found.The fifth chapter is the summary and prospect of this thesis.