Research On Oscillatory Properties For Some Classes Of Mixed Partial Difference Equations

This dissertation consists of four chapters,its concern focuses on the oscillatory properties of some classes of mixed partial difference equations by means of the envelop theory.The paper is organized as follows:Chapter 1 briefly describes the studying background of partial difference equa-tion.We also present some general concepts and fundamental knowledge of partial difference equation.Chapter 2 uses the envelop theory to investigate the oscillatory behavior of the second-order mixed partial difference equation with two constant coefficients as following Pum+2,n + qum,n+2-um,n + um + σ,n-τ = 0 and um + 2,n + pum,n+2-umm,n + qum + σ,n-τ = 0,where σ,τ are positive integers,m,n are nonnegative integers,p,q are real numbers.This dissertation obtains the necessary and sufficient conditions for the oscillation of solutions of the equation.Chapter 3 uses the envelop theory to investigate the oscillatory behavior of the second-order mixed partial difference equation with three constant coefficients as following pum+2,n + qum,n+2-um,n + rum+σ,n-τ = 0 and um+2,n + pum,n + 2-qum,rum+σ,n-τ = 0,where σ,τ are positive integers,m,n are nonnegative integers,p,q,r are real numbers.This dissertation obtains the necessary and sufficient conditions for the oscillation of solutions of the equation.Chapter 4 uses the envelop theory to investigate the oscillatory behavior of two kinds of special second-order mixed partial difference equation as following Pum+2,n + qum,n+1-um,n + um + σ,n-τ = 0 and pum + 2,n + qum,n+1-n + rum + σ,= 0 where σ,τ are positive integers,m,n are nonnegative integers,p,q,r are real numbers.This dissertation obtains the necessary and sufficient conditions for the oscillation of solutions of the equation.All of the above researches are accompanied with computer simulations.The experimental results not only illustrate the oscillatory behavior of the solutions,but also validate the presented results.