Research On Several Problems In Nonlinear Analysis

Recently, many authors have studied the nonlinear “good” Boussinesq evolutionpartial differential equation, which is a typical mathematical model to characterize thepropagation of the nonlinear shallow water wave along the two directions. This papersuggests and uses the multisymplectic scheme to investigate this equation. TheMATLAB has become one of most popular engineering calculation software tools inthe world, and it has been used widely in the fields such as scientific operation,automatic control, signal processing, etc. In the thesis we study a discrete controlsystem via the MATLAB. In addition, the boundary value problem of nonlinearordinary differential equation has very rich sources in the classical mechanics andelectricity, and also it is an important part involved in the discipline of ordinarydifferential equations, and meanwhile the questions about the existence of positivesolutions of a nonlinear ordinary differential equation boundary value problem haverich practical applied background. In the paper we as well discuss the existence ofpositive solutions of a nonlinear ordinary differential equation boundary valueproblem. This thesis is divided into four chapters.Chapter1presents a completely explicit multisymplectic scheme to investigatethe nonlinear "good" Boussinesq partial differential equation based on its Bridge’smultisymplectic form, and the efficiency of the present scheme is illustrated by usingthe motions of single solitary wave and collision between two solitons, as well as thenonlinear stability condition of the scheme is estimated to get by some numericalexperiments.Chapter2applies the MATLAB to analyse a discrete control system. First weestablish its mathematical model, we then find out the response of the discretizationsystem for the continuous system, and finally we give the stability analysis of thediscrete system.Chapter3studies the existence of positive solutions of three-point boundaryvalue problem of a nonlinear ordinary differential equation in super-linear andsub-linear cases, and generalizes some reasearch results about three-point boundaryvalue problems by Ma Ruyun et al.,1999.Chapter4discusses the two-point boundary value problem of a nonlinearordinary differential equation under the Nagumo’s condition, and obtains the existence of positive solutions of the two-point boundary value problem with condition ofy (0)0and y (1)1.