On the solution of stochastic oscillatory quadratic nonlinear equations using different techniques, a comparison study

In this paper, nonlinear oscillators under quadratic nonlinearity with stochastic inputs are considered. Different methods are used to obtain first order approximations, namely; the WHEP technique, the perturbation method, the Pickard approximations, the Adomian decompositions and the homotopy perturbation method (HPM). Some statistical moments are computed for the different methods using mathematica 5. Comparisons are illustrated through figures for different case-studies.

[1]  D. Chrissoulidis,et al.  Rigorous application of the stochastic functional method to plane-wave scattering from a random cylindrical surface , 1999 .

[2]  Ji-Huan He,et al.  Homotopy perturbation method: a new nonlinear analytical technique , 2003, Appl. Math. Comput..

[3]  A. Siegel,et al.  The Cameron—Martin—Wiener method in turbulence and in Burgers’ model: general formulae, and application to late decay , 1970, Journal of Fluid Mechanics.

[4]  Ji-Huan He.  SOME ASYMPTOTIC METHODS FOR STRONGLY NONLINEAR EQUATIONS , 2006 .

[5]  Douglas J. Nelson,et al.  Time-frequency methods for enhancing speech , 1997, Optics & Photonics.

[6]  P. Saffman.  Application of the Wiener-Hermite Expansion to the Diffusion of a Passive Scalar in a Homogeneous Turbulent Flow , 1969 .

[7]  Yasuhiko Tamura,et al.  A Formula on the Hermite Expansion and Its Application to a Random Boundary Value Problem , 2003 .

[8]  A. Chorin.  Gaussian fields and random flow , 1974, Journal of Fluid Mechanics.

[9]  Robert Rubinstein,et al.  Uncertainty Quantification for Systems with Random Initial Conditions Using Wiener–Hermite Expansions , 2005 .

[10]  Ali H. Nayfeh,et al.  Problems in Perturbation , 1985 .

[11]  Yasuhiko Tamura,et al.  TE Plane Wave Reflection and Transmission from a One-Dimensional Random Slab , 2005, IEICE Trans. Electron..

[12]  William C. Meecham,et al.  Symbolic Calculus of the Wiener Process and Wiener‐Hermite Functionals , 1965 .

[13]  J. C. T. Wang,et al.  Wiener‐Hermite expansion and the inertial subrange of a homogeneous isotropic turbulence , 1974 .

[14]  J. Nakayama,et al.  Scattering of a TM plane wave from periodic random surfaces , 1999 .

[15]  G. H. Canavan,et al.  Relationship between a Wiener–Hermite expansion and an energy cascade , 1970, Journal of Fluid Mechanics.

[16]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[17]  M. Joelson,et al.  Random fields of water surface waves using Wiener–Hermite functional series expansions , 2003, Journal of Fluid Mechanics.

[18]  Gamal M. Mahmoud,et al.  The solvability of parametrically forced oscillators using WHEP technique , 1999 .

[19]  Goodarz Ahmadi,et al.  Application of Wiener-Hermite Expansion to Nonstationary Random Vibration of a Duffing Oscillator , 1983 .

[20]  E. F. Abdel Gawad,et al.  General stochastic oscillatory systems , 1993 .

[21]  W. C. Meecham,et al.  The Wiener-Hermite expansion applied to decaying isotropic turbulence using a renormalized time-dependent base , 1978, Journal of Fluid Mechanics.

[22]  Tsutomu Imamura,et al.  Turbulent Flows near Flat Plates , 1980 .

[23]  Y. Kayanuma,et al.  Wiener–Hermite expansion formalism for the stochastic model of a driven quantum system , 2001 .

[24]  Tsutomu Imamura,et al.  An Exact Gaussian Solution for Two-Dimensional Incompressible Inviscid Turbulent Flow , 1979 .

[25]  C. Eftimiu.  First‐order Wiener‐Hermite expansion in the electromagnetic scattering by conducting rough surfaces , 1988 .

[26]  G. Ahmadi,et al.  Response of the Duffing Oscillator to a Non-Gaussian Random Excitation , 1988 .