Engineering Transactions, 50, 1-2, pp. 43–54, 2002

Nonlinear Mathieu Equation and its Approximation Without a Small Parameter

Ji-Huan He
Shanghai Donghua University
China

In this paper, the linear and nonlinear Mathieu equations without a small parameter are considered, which cannot be solved by the perturbation techniques. However, using the variational iteration method, their periodic solutions can be readily obtained with high accuracy. In addition, some special cases have been discussed, where the perturbation solutions are meaningless even when there exists a small parameter.
Full Text: PDF

References

J.H. HE, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mech. and Engineering, 167, 57–68, 1998.

J.H. HE, Approximate solution for nonlinear differential equations with convolution product nonlinearities, Computer Methods in Applied Mech. and Engineering, 167, 69–73, 1998.

J.H. HE, Variational iteration method: a kind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics, 34, 4, 699–708, 1999.

J.H. HE, Variational iteration method for autonomous ordinary differential system, Applied Math. and Computer, 114, 2/3, 115–123, 2000.

J.H. HE, A review on some new recently developed nonlinear analytical technique, International Journal of Nonlinear Sciences and Numerical Simulation, l, l, 51–70, 2000.

A.H. NAYFEH, Introduction to perturbation techniques, Wiley & Son, 1981.

X.H. SHAO and X. ZN. WANG, Free vibration of a class of Hill's equation having a small parameter, Applied Math. and Mech., (English edition), 11, 4, 355–361, 1990.

M. INOKUTI, H. SEKINE and T. MURA, General use of the Lagrange multiplier in nonlinear mathematical physics, [in:] Variational Method in the Mechanics of Solids, S. NEMAT-NASSER [Ed.], Pergamon Press, 156–162, 1978.

B. A. FINLAYSON, The method of weighted residual and variational principles, Acad. Press, 1972.




Copyright © 2014 by Institute of Fundamental Technological Research
Polish Academy of Sciences, Warsaw, Poland