Analysis Of A Perturbed Solution For The One-Dimensional Nonlinear Beam Equation

The nonlinear vibratory beam equation is a very important kind of equation in the study of engineering practical problems. Nonlinear partial differential equations usually have not the exact solutions. So the approximate method is usually used instead. There are two kinds of methods in common use. One is numerical method, and the another one is analytic approximate which is represented by perturbation method.In this paper, we investigate the initial-boundary value problem for the following one-dimensional nonlinear beam equation with fixed ends ((?)4u)/((?)x4)-ε(?)[((?)u)/((?)x)]2dx[((?)2u)/((?)x2)+((?)2u)/((?)t2)=0 the given initial-boundary value condition u(x,0)=φ(x),((?)u)/((?)t)(x,0)=ψ(x) u(0,t)=u(1,t)=[((?)2u)/((?)x2)](0,t)=[((?)2u)/((?)x2)](1,t)=0Under certain conditions, Dickey etc. have proved the existence and the uniqueness of the solution about the above problem. When the initial displacement and velocity are represented by the Fourier sine series, we obtain the first term of the approximate solution by using the multiple scales method, and estimate the growth of the time about the second term of the approximate solution. Furthermore, in order to get the error estimation about the first term of the approximate solution, we give some prior estimation of the approximate solution by using the energy method at first and prove the uniform boundedness of the approximate solution, for all 0<ε≤ε0.Then, we estimate the error of the approximate solution by respectively using integral equations and energy method, combining the nonlinear Gronwall inequality, when the initial value respectively is a finite Fourier sine series and an infinite Fourier sine series. And we come to the following results:If the initial-value is a finite Fourier sine series, then for any given T>0, there is a positive constantε0, such that the error between the exact solution and the first-term approximation solution is bounded byεmultiplying a constant depending onε0,T and N, for all 0≤x≤1,0≤εt≤T and 0<ε≤ε0.If the initial-value is an infinite Fourier sine series, andφ(x)∈C6, ψ(x)∈C4, there are positive constants T0 andε1 such that error between the exact solution and the first-term approximation solution is bounded byεmultiplying a constant depending onε1 and T0, for all 0≤x≤1,0<ε≤ε1 and 0≤εt≤T0.