Frequently Oscillatory Solutions For Several Kinds Of Differential Equations

Difference equation is a tool which used to describe differential equationwich is discrete model. After a long inquiry, difference equation has been as astudy the oscillation of solutions of differential equation of one of the mainmethods. At the same time, in the engineering technology and scientific fields,such as in control theory, life science, economy, finance, we can only usedifference equation which is discrete mathematical model to describe thephenomenon, so the research of difference equation has received a lot ofevaluation.Partial difference equation is difference equation which contains two ormoreindepent variable.When we use the finite difference method to get thesolution of partial differential equation and research some problems ofmolecular orbital, mathematical and physics equation,we often face with thiskind of equation.In the past ten years, partial difference equation of theoscillation theory has been got a rapidly development.The theory of time scales can unify continuous and discrete cases, whichpioneers a new mathematical area. This theory not only unifies differential and difference equations, reveals the essence of continuous and discrete cases,avoids repeat study, but also contains many other cases. The prominentpeculiarity of time scales is unification and generalization, so the study oftime scales has important significance in theory and application.The main contents of this paper are as follows:First of all, this paper summarized the research background and thepresent situation of difference equation, functional partial difference equationand dynamic equation on time scales.Secondly, we use frequency measure method to discuss the frequencyoscillation for a class of the second order nonlinear difference equations withpositive and negative coefficients.Some criterions are got and some examplesare given.Again, we use frequency measure method to study the frequentoscillation for a class of nonlinear neutral variable delay partial differenceequation and a kind of special form of nonlinear neutral partial differenceequation, some examples are given.Finally, we use frequency measure method to study the frequencyoscillation for three order dynamic equation on time scales and three ordervariable delay dynamic equation with positive and negative coefficients.Some lemmas and criterions are got.At the last, a example isgiven.