Stochastic Oscillators with Quadratic Nonlinearity Using WHEP and HPM Methods

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References

[1] A. Nayfeh, “Problems in Perturbation,” John Wiley, New York, 1993.

[2] S. Crow and G. Canavan, “Relationship between a Wiener-Hermite Expansion and an Energy Cascade,” Journal of Fluid Mechanics, Vol. 41, No. 2, 1970, pp. 387-403.
doi:10.1017/S0022112070000654

[3] P. Saffman, “Application of Wiener-Hermite Expansion to the Diffusion of a Passive Scalar in a Homogeneous Turbulent Flow,” Physics of Fluids, Vol. 12, No. 9, 1969, pp. 1786-1798. doi:10.1063/1.1692743

[4] W. Kahan and A. Siegel, “Cameron-Martin-Wiener Method in Turbulence and in Burger’s Model: General Formulae and Application to Late Decay,” Journal of Fluid Mechanics, Vol. 41, No. 3, 1970, pp. 593-618.

[5] J. Wang and S. Shu, “Wiener-Hermite Expansion and the Inertial Subrange of a Homogeneous Isotropic Turbulence,” Physics of Fluids, Vol. 17, No. 6, 1974, pp. 1130-1134.

[6] H. Hogge and W. Meecham, “Wiener-Hermite Expansion Applied to Decaying Isotropic Turbulence Using a Renormalized Time-Dependent Base,” Journal of Fluid of Mechanics, Vol. 85, No. 2, 1978, pp. 325-347.
doi:10.1017/S002211207800066X

[7] M. Doi and T. Imamura, “An Exact Gaussian Solution for Two-Dimensional Incompressible Inviscid Turbulent Flow,” Journal of the Physical Society of Japan, Vol. 46, No. 4, 1979, pp. 1358-1359. doi:10.1143/JPSJ.46.1358

[8] R. Kambe, M. Doi and T. Imamura, “Turbulent Flows Near Flat Plates,” Journal of the Physical Society of Japan, Vol. 49, No. 2, 1980, pp. 763-778.
doi:10.1143/JPSJ.49.763

[9] A. J. Chorin, “Gaussian Fields and Random Flow,” Journal of Fluid of Mechanics, Vol. 63, No. 1, 1974, pp. 21-32. doi:10.1017/S0022112074000991

[10] Y. Kayanuma, “Stochastic Theory for Non-Adiabatic Crossing with Fluctuating Off-Diagonal Coupling,” Journal of the Physical Society of Japan, Vol. 54, No. 5, 1985, pp. 2037-2046. doi:10.1143/JPSJ.54.2037

[11] M. Joelson and A. Ramamonjiarisoa, “Random Fields of Water Surface Waves Using Wiener-Hermite Functional Series Expansions,” Journal of Fluid of Mechanics, Vol. 496, 2003, pp. 313-334.
doi:10.1017/S002211200300644X

[12] C. Eftimiu, “First-Order Wiener-Hermite Expansion in the Electromagnetic Scattering by Conducting Rough Surfaces,” Radio Science, Vol. 23, No. 5, 1988, pp. 769-779.
doi:10.1029/RS023i005p00769

[13] N. J. Gaol, “Scattering of a TM Plane Wave from Periodic Random Surfaces,” Waves Random Media, Vol. 9, No. 11, 1999, pp. 53-67.

[14] Y. Tamura and J. Nakayama, “Enhanced Scattering from a Thin Film with One-Dimensional Disorder,” Waves in Random and Complex Media, Vol. 15, No. 2, 2005, pp. 269-295.

[15] Y. Tamura and J. Nakayama, “TE Plane Wave Reflection and Transmission from One-Dimensional Random Slab,” IEICE Transactions on Electronics, Vol. E88-C, No. 4, 2005, pp. 713-720.

[16] N. Skaropoulos and D. Chrissoulidis, “Rigorous Application of the Stochastic Functional Method to Plane Wave Scattering from a Random Cylindrical Surface,” Journal of Mathematical Physics, Vol. 40, No. 1, 1999, pp. 156-168. doi:10.1063/1.532766

[17] A. Jahedi and G. Ahmadi, “Application of Wiener-Hermite Expansion to Non-Stationary Random Vibration of a Duffing Oscillator,” Journal of Applied Mechanics, Transactions ASME, Vol. 50, No. 2, 1983, pp. 436-442.

[18] I. I. Orabi and G. Ahmadi, “Functional Series Expansion Method for Response Analysis of Nonlinear Systems Subjected to Ransom Excitations,” International Journal of Nonlinear Mechanics, Vol. 22, No. 6, 1987, pp. 451-465. doi:10.1016/0020-7462(87)90036-9

[19] I. I. Orabi, “Response of the Duffing Oscillator to a Non-Gaussian Random Excitation,” Journal of Applied Mechanics, Transaction of ASME, Vol. 55, No. 3, 1988, pp. 740-743.

[20] E. Abdel Gawad, M. El-Tawil and M. A. Nassar, “Nonlinear Oscillatory Systems with Random Excitation,” Modeling, Simulation and Control B, Vol. 23, No. 1, 1989, pp. 55-63.

[21] I. I. Orabi and G. Ahmadi, “New Approach for Response Analysis of Nonlinear Systems under Random Excitation,” American Society of Mechanical Engineers, Design Engineering Division (Publication) DE, Vol. 37, 1991, pp. 147-151.

[22] E. Gawad and M. El-Tawil, “General Stochastic Oscillatory Systems,” Applied Mathematical Modelling, Vol. 17, No. 6, 1993, pp. 329-335.
doi:10.1016/0307-904X(93)90058-O

[23] M. El-Tawil and G. Mahmoud, “The Solvability of Parametrically Forced Oscillators Using WHEP Technique,” Mechanics and Mechanical Engineering, Vol. 3, No. 2, 1999, pp. 181-188.

[24] Y. Tamura and J. Nakayama, “A Formula on the Hermite Expansion and Its Aoolication to a Random Boundary Value Problem,” IEICE Transactions on Electronics, Vol. E86-C, No. 8, 2003, pp. 1743-1748.

[25] M. El-Tawil, “The Application of WHEP Technique on Stochastic Partial Differential Equations,” International Journal of Differential Equations and Applications, Vol. 7, No. 3, 2003, pp. 325-337.

[26] Y. Kayanuma and K. Noba, “Wiener-Hermite Expansion Formalism for the Stochastic Model of a Driven Quantum System,” Chemical Physics, Vol. 268, No. 1-3, 2001, pp. 177-188. doi:10.1016/S0301-0104(01)00305-6

[27] O. Kenny and D. Nelson, “Time-Frequency Methods for Enhancing Speech,” Proceedings of SPIE—The International Society for Optical Engineering, Vol. 3162, 1997, pp. 48-57.

[28] E. Isobe and S. Sato, “Wiener-Hermite Expansion of a Process Generated by an Ito Stochastic Differential Equations,” Journal of Applied Probability, Vol. 20, No. 4, 1983, pp. 754-765. doi:10.2307/3213587

[29] R. Rubinstein and M. Choudhari, “Uncertainty Quantification for Systems with Random Initial Conditions Using Wiener-Hermite Expansions,” Studies in Applied Mathematics, Vol. 114, No. 2, 2005, pp. 167-188.
doi:10.1111/j.0022-2526.2005.01543.x

[30] T. Imamura, W. Meecham and A. Siegel, “Symbolic Calculus of the Wiener Process and Wiener-Hermite Functionals,” Journal of Mathematical Physics, Vol. 6, No. 5, 1983, pp. 695-706. doi:10.1063/1.1704327

[31] J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, 1999, pp. 257-292.
doi:10.1016/S0045-7825(99)00018-3

[32] J. H. He, “A Coupling Method of a Homotopy Technique and a Perturbation Technique for Nonlinear Problems,” International Journal of Nonlinear Mechanics, Vol. 35, 2000, pp. 37-43. doi:10.1016/S0020-7462(98)00085-7

[33] J. H. He, “Homotopy Perturbation Method: A New Nonlinear Analytical Technique,” Applied Mathematics and Computation, Vol. 135, 2003, pp. 73-79.

[34] J. H. He, “The Homotopy Perturbation Method for Nonlinear Oscillators with Discontinuities,” Applied Mathematics and Computation, Vol. 151, 2004, pp. 287-292.

[35] J. H. He, “Some Asymptotic Methods for Strongly Nonlinear Equations,” International Journal of Modern Physics B, Vol. 20, No. 10, 2006, pp. 1141-1199.

[36] S. J. Liao, “On the Proposed Homotopy Analysis Techniques for Nonlinear Problems and Its Applications,” Ph.D. Dissertation, Shanghai Jiao Tong University, 1992.

[37] S. J. Liao, “Beyond Perturbation: Introduction to the Homotopy Analysis Method,” Chapman & Hall\CRC Press, Boca Raton, 2003.

[38] S. J. Liao, “An Approximate Solution Technique Which Does Not Depend upon Small Parameters: A Special Example,” International Journal of Nonlinear Mechanics, Vol. 30, 1995, pp. 371-380.
doi:10.1016/0020-7462(94)00054-E

[39] T. Hayat, M. Khan and S. Asghar, “Homotopy Analysis of MHD Flows of an Oldroyd 8-Constant Fluid,” Acta Mechanica, Vol. 168, 2004, pp. 213-232.
doi:10.1007/s00707-004-0085-2

[40] S. Asghar, M. Mudassar and M. Ayub, “Rotating Flow of a Third Fluid by Homotopy Analysis Method,” Applied Mathematics and Computation, Vol. 165, 2005, pp. 213-221. doi:10.1016/j.amc.2004.04.047

[41] S. P. Zhu, “A Closed Form Analytical Solution for the Valuation of Convertible Bonds with Constant Dividend Yield,” ANZIAM Journal, Vol. 47, No. Part 4, 2006, pp. 477-494.

[42] M. A. El-Tawil and A. S. Al-Johani, “Approximate Solution of a Mixed Nonlinear Stochastic Oscillator,” Computers & Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2236-2259.
doi:10.1016/j.camwa.2009.03.057