The Homotopy Wiener-Hermite Expansion and Perturbation Technique (WHEP)

The Wiener-Hermite expansion linked with perturbation technique (WHEP) was used to solve perturbed non-linear stochastic differential equations. In this article, the homotopy perturbation method is used instead of the conventional perturbation methods which generalizes the WHEP technique such that it can be applied on non-linear stochastic differential equations without the necessity of the presence of the small parameter. The technique is called homotopy WHEP and is demonstrated through many non-linear problems.

[1]  Tasawar Hayat,et al.  Rotating flow of a third grade fluid by homotopy analysis method , 2005, Appl. Math. Comput..

[2]  T. Hayat,et al.  Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid , 2004 .

[3]  S. Liao,et al.  Beyond Perturbation: Introduction to the Homotopy Analysis Method , 2003 .

[4]  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.

[5]  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.

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

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

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

[9]  Shunsuke Sato,et al.  Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation , 1983 .

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

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

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

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

[14]  Ji-Huan He.  A coupling method of a homotopy technique and a perturbation technique for non-linear problems , 2000 .

[15]  Shijun Liao,et al.  AN APPROXIMATE SOLUTION TECHNIQUE WHICH DOES NOT DEPEND UPON SMALL PARAMETERS: A SPECIAL EXAMPLE , 1995 .

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

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

[18]  Y. Kayanuma.  Stochastic Theory for Nonadiabatic Level Crossing with Fluctuating Off-Diagonal Coupling , 1985 .

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

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

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

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

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

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

[25]  Y. Tamura,et al.  Enhanced scattering from a thin film with one-dimensional disorder , 2005 .

[26]  Clifford Goodman,et al.  American Society of Mechanical Engineers , 1988 .

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

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

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

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

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

[32]  Ji-Huan He,et al.  The homotopy perturbation method for nonlinear oscillators with discontinuities , 2004, Appl. Math. Comput..

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

[34]  Song‐Ping Zhu.  A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield , 2006, The ANZIAM Journal.

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

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

[37]  S. Liao.  An approximate solution technique not depending on small parameters: A special example , 1995 .

[38]  Goodarz Ahmadi,et al.  A functional series expansion method for response analysis of non-linear systems subjected to random excitations , 1987 .