Analysis Of The Perturbed Solution On Initial-boundary Value Problem For The Kirchhoff Equation

The method of multiple scales has been used for a long time to derive asymptotic expansions for solutions to singular perturbation problems. In this paper, we discuss the initial-value problem for the Kirchhoff equation with initial value u(x,0)=φ(x),ut(x,0)=ψ(x) and homogeneous Dirichlet condition. Firstly, we pursue the priori estimate for the solution by the energy method and obtain the following result:there is a positive numberε0 such that u(x,t) is uniformly bounded for all 0<ε≤ε0, and there is a constant T0>0 such that dx is uniformly bounded for all 0<ε≤ε0 and 0≤εt≤To.In the case when the initial-value are respectively a finite Fourier sine series and an infinite Fourier sine series, we apply the method of multiple scales to obtain the leading term approximation uo(x,t;ε) to the solutions. Finally, the errors of the approximate solutions are estimated respectively by integral equation and the energy method. 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 difference 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)∈C7,ψ(x)∈C6, there are positive constants T1 andε1 such that the difference between the exact solution and the first-term approximation solution is bounded byεmultiplying a constant depending onε1 and T1, for all 0≤x≤1,0<ε≤ε1 and 0≤εt≤T1.