On The Eigenvalues Of A Class Of Differential Operators

Differential operator is one of the mostly basic and widely used operator of the linear operators. A large amount of problems can be summarized as the problem of differential operator in mathematical-physics, engineering and technology courses, the research field of which includes the theory of deficiency index, self-conjugate extention, spectral analysis, numerical method, inverse problem and other important branches. This article deeply discuss the numerical method of the differential operator, which is important in terms of the theoretical research, application and offers unified resolution models and theoretical framework to the problems of differential equation.Firstly, we use the method of transform theory to solve the eigenvalue of the second—differential operator, obtaining the eigenfunction and spectral analysis. Then we use the numerical method to solve the same equation with second-differential operator, getting the similar spectral analysis compared with the traditional theory, which explains that using the numerical method to solve the eigenvalue of the second-differential operator is available. Comparing with the transform theory, the advantage of numerical method is that the procedure is quite simple and efficient, without using the sophisticated tools from hypergeometric equations.Secondly, the problem of the second-differential operator has been applied into the fourth-differential operator in analyzing the eigenvalue of the fourth-differential operator and the KP equation with the fourth-differential operator. Through the above contents, we conclude that the eigenvalue of the fourth-differential operator related with the KP equation is not unlimited, which is the main purpose of this article.This paper contains five parts:First part is introduction; Second part is the transform theory on solving the second-differential operator; Third part is the numerical method on solving the second-differential operator; Fourth part is the numerical method on solving the fourth-differential operator; Fifth part is conclusion and prospect.