A New Type Of Sturm-liouville Problem

Sturm-Liouville problems originated in the early 19th century,it is a mathematical model which is set up in order to solve the problem of Fouri-er heat conduction.Its theory is widely used in the mathematics,physics,the earth of meteorology and other modern science.The Sturm-Liouville prob-lems gradually become a very important mathematics and quantum mechanics theory research branch,and attract the attention of many experts and schol-ars.As an important milestone of the boundary value problem of ordinary dif-ferential equations in the 19th century,the Sturm-Liouville problems is worth studying and discussing in a depth and comprehensive way.Its theoretical and practical value is mainly manifested in the following aspects:Firstly,people found that many research of practical problems can boil down to the study of eigenvalue and eigenfunction about the Sturm-Liouville problems,and many practical problems can be solved by solving the nature of eigenvalue and eigenfunction about the Sturm-Liouville problems.As a result,it put forward the concept of eigenvalue and eigenfunction for the first time,and it is the first theoretical system of the eigenvalue and eigenfunction.Secondly,it is a very good mathematical model for solving a class of typical mathemat-ical physics problems.Both the heat conduction problem,string vibration and wave equation,can be reflected in its theoretical framework.Third-ly,it lays the foundation for the theory of orthogonal functions.Fourthly,to some extent,it indicates the theory of spectrum,the self-adjointness,the Hilbert space and so on,it is a pioneer of the operator spectrum theory and also is a important part of functional analysis.It had a long history of more than one hundred years to study the Sturm-Liouville problem.In the early nineteenth century,Liouville and Sturm pro-posed the theory of spectrum firstly when they studied the solution of string vibration equation.Then Brikhoff,Hilbert,Neumann,Steklov and many mathematicians made a depth research and promoted the spectral theory[1-4].More than one hundred years,Sturm-Liouville problems,especially the regular S-L problems’s research had been very complete on both the theory and methods[5-8].But we gradually found that the regular conditions often cannot be satisfied in practice.Therefore,in recent years,the research of studying the discontinuity problems about differential equation’s solution or the derivative of solution within the range attracted the interests of more and more experts and scholars[9-11].So many problems is derived from the physical problems,such as the thin laminated plate of the heat conduction problem[16].Due to the difficulty of the singular problem and it’s importance on the physical theory,there are still lots of problems to be solved.Compared with the regular S-L problem research,the theory of singular problem is not per-fect and system enough.Especially in this new type Sturm-Liouville problem which is from the earth fluid mechanics that we are going to discuss,it’s re-search methods and tools are still explored and summarized.This kind of problem is different from the regular problems,it is also different from the pulse problems,it is not included in the existing spectrum theory.My tutor Jiangang Qi discussed the related properties of initial value problems in the literature[12],in this article we will consolidate and expand the nature of the initial value problems in[12],then we will obtain the nature of the bound-ary value problem and study the basic theory of eigenvalue and eigenfunction.These conclusions will be added to the existing spectrum theory and make the theory of discontinuous Sturm-Liouville problem more perfected.In this paper,we study a new type Sturm-Liouville problem which is founded in the earth fluid mechanics.We will get a series of conclusions of the initial value problem firstly,including the existence and uniqueness,continuity and analyticity of the solution,as well as to the differentiability and continuous dependency of parameters and so on.And then we will pro-mote these conclusions to the boundary value problem,so we can lay the foundation for studying the eigenvalue theory.Moreover we we will get the nature of the eigenvalue and eigenfunction of the new type Sturm-Liouville problem,including the Green function and the completeness of eigenfunc-tion,the eigenvalue is real、single、countable,the number of zero in the eigenfunctions,and the asymptotic estimation formula of eigenvalue,etc.