An Open Problem On The Optimality Of An Asymptotic Solution To Duffing’s Nonlinear Oscillation Problem

Based on the renormalization group (or RG) method, Kirkinis [E. Kirkinis, SIAM Review 54 (2012) 374-388] obtained an asymptotic solution to Duffing’s nonlinear oscillation problem. Kirkinis then proposed some open problems, among that asked if the asymptotic solution is optimal. In this paper, an affirmative answer to the open problem is given.The main tool used is the homotopy analysis method.In this paper, the first chapter mainly introduces renormalization group method and homo-topy analysis method of some basic concepts, methods and procedure. In the second chapter, to solve the open problem, the HAM is applied to seek a series solution to Duffing’s nonlin-ear oscillation problem; then by choosing an appropriate value of the parameter c0, Kirkinis’s asymptotic solution is recovered; finally by examining the averaged residual error of the approx-imation, the optimality of Kirkinis’s asymptotic solution is verified. In the third chapter, a HAM approximation to a more general oscillation problem is obtained which recovers and is superior to the existing asymptotic solution in the literature. In the forth chapter, to further demonstrate its effectiveness, the HAM is applied to solve a fourth order boundary value problem containing a parameter y [19] which cannot be solved by other analytic methods for large values of y. In the end, some concluding remarks are given. HAM is a very useful and effective analytical method.