Approximate Solution And Oscillation Criteria Of Second Order Nonlinear Damped Q-symmetric Difference Equation

With the broad application of q-calculus in cosmic strings, black holes, conformal quantum mechanics and other physics domain, more and more scientist have turned to the study of q-calculus since 1990s. So far, q-calculus has played an important role in quantum physics, Tsallis nonextensive statistics, nonlinear physics system and so on. This paper aims at finding the approximate solution of the second-order non-linear damped q-symmetric functional difference equation and justifying whether the equation oscillates.Firstly, the paper introduces the background and significance of q-symmetric dif-ference equation, states the obtained results given by the experts and scholars at home and abroad in q-calculus domain, and points out the main problems the paper studied.Then, the paper studies the approximate solution of the following second-order nonlinear damped q-symmetric functional difference equation: dq2x+(2γ+∈γ1x)dqx+Ω2x+x2= 0. (0.4) where γ and γ1 are linear damping parameters, ε is nonlinear parameter, Ω is un-derdamping parameter. With the basic definitions and properties of q-symmetric dif-ference operator and q-symmetric integral operator, this paper transforms the above equation into algebra equation by differential transformation method:where k=0,1,2,3,…. Then we can solve the approximate solution by iteration. Furtherly, the paper also represents the numerical solution of (1) with the help of Taylor formula on time scale, i.e., if n ∈N, the real-valued function f is continuous and dqn exists, a,t ∈ qN. Then at the point x= a the function f can be represented as: Thus we make a comparison between approximate solution and numerical solution.Lastly,the paper applies kernel functionΨ(t,s,l)and 0perator Ap[;l,t]to consider the oscillation criteria of the second-order nonlinear damped q-symmetric difference equation dq2x+(2γ+∈γ1x)dqx(q-1t)+Ω2x+x2=0. (0.6) basing on the idea that the solution should haVe infinity roots,the paper gives the following oscillation criteria of the above equation:Suppose that εγ1dqx(q-1t)+Ω2x(t)+x2(t)=f(t,x(t),dqx(q-1t)), and ∈γ1xdqx(q-1t)+Ω2x+x2≥μq(t)|x(t)|,If for every l≥to,there exists the first order q.symmetric difference derivative of continuous and monotonic increase functionp(s):[Tq,∞)→(0,∞),Ψ∈Z,and positive number M, which satisfies: lim sup Ap[μq(t)/Koq-[4γ2(Koq)2+(K1Ψp(qs)/p(s)+K2dqp(s)/p(s))M];l,t]>0,where Ap,Ψ is the kernel function,then the equation(6)oscillates.