Research On Oscillation For Partial Difference Equations

Partial difference equations have extensive applications in many areas such as finite difference schemes, population dynamics with spatial migrations, random walk, material mechanics, mathematical physics problems as well as image process-ing. In recent years, the oscillation theory of partial difference equations has received much attention and developed rapidly. A lot of research results concerning the oscil-latory behavior of partial difference equations have been obtained. This dissertation focuses on the oscillation of partial difference equations by means of the envelope theory. The main research contents are as follows:1. Oscillation behavior for delay partial difference equations with two constant coefficientsThe oscillatory behavior of delay partial difference equations with two constant coefficients was discussed by means of envelope curve method for the first time. With no conditions attached on constant coefficients, we obtain a series of oscillation criteria. Some simulation examples are given to verify the theoretical results.2. Oscillation behavior for delay partial difference equations with three constant coefficientsBased on the envelope surface method, a series of sufficient and necessary con-ditions for the oscillation of delay partial difference equations with three constant coefficients were established without additional conditions on constant coefficients for the first time. Numerical examples are given to demonstrate the results obtained.3. Oscillation behavior for advanced partial difference equations with three constant coefficientsThe oscillatory behavior of advanced partial difference equations with three constant coefficients was investigated from the viewpoint of the envelope theory of the family of planes for the first time. Some oscillation criteria are derived with no conditions attached on constant coefficients. Numerical examples are provided to demonstrate the results obtained.4. Oscillation behavior for mixed partial difference equations with three con-stant coefficientsThe oscillatory behavior of mixed partial difference equations with three con-stant coefficients was considered from the perspective of the envelope theory of the family of planes for the first time. Some effective criteria to determine the oscilla-tion behavior of mixed partial difference equations are presented with no conditions attached on constant coefficients. Numerical examples are provided to illustrate the results obtained.5. Oscillation behavior for a special partial difference equationThe oscillatory behavior of a special partial difference equation with three con-stant coefficients was investigated from the perspective of the envelope theory of the family of planes for the first time. A sufficient and necessary condition for the oscillation of the special partial difference equation is presented with no conditions attached on constant coefficients. At the same time, a sufficient and necessary con-dition for the oscillation of the corresponding ordinary difference equation is also given. Numerical examples are provided to illustrate the results obtained.