Reaserch On The Approximate Solution To The Perutrbed Duffing Equation In Difference Form

The Solution for nonlinear equations has been paid extensive attention in many fields, such as mathematics, physics and engineering science. Because generally there is no analytical expression for exact solutions, the approximate method was usually used instead. There are two kinds of method in common use. One is numerical method, and the another one is analytic approximation which is represented by perturbation method. Since both of them are approximate solutions, attention to the estimation of error has to be paid.In this paper, we study the initial value problem of Duffing equation in the difference form with multiple-scale method, which is one of the perturbation method. By the multiple scale method, the analytical expression of the first term of approximate solution to the difference equation is obtained. And furthermore the second term of solution for difference equation is discussed. Actually, the first term of the approximate solution for this difference equation had been obtained in some literature, but to obtain the second term is more difficult and require to solve a non-homogeneous linear differential system of the first order with periodic coefficients.For simplicity, we focus on the estimation of error for the first term of the approximate solution in the case that y₀ = 0, y₁ = 1. Referring to the errorestimate of the leading term of the approximate solution for Duffing equation in the differential form, we find the difference equation satisfied by the error of first term and show that the error of first term satisfies a nonlinear Gronwal's inequality in discrete form, and then to prove that there existsε₀ > 0,L > 0, such that the error for the first term of approximate solution is less than M(εn)εif 0≤ε≤ε₀,εn≤L, where M(Ï„)≥0 is a continuous function on [0, L].