Higher-Order WHEP Solutions of Quadratic Nonlinear Stochastic Oscillatory Equation

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References

[1] A. Jahedi and G. Ahmadi, “Application of Wiener-Hermite Expnasion to Nonstationary Random Vibration of a Duffing Oscillator,” Transactions of the ASME, Vol. 50, 1983, pp. 436-442.

[2] J. C. Cortes, J. V. Romero, M. D. Rosello and R. J. Villanueva, “Applying the Wiener-Hermite Random Technique to Study the Evolution of Excess Weight Population in the Region of Valencia (Spain),” American Journal of Computational Mathematics, Vol. 2, No. 4, 2012, pp. 274-281. doi:10.4236/ajcm.2012.24037

[3] W. Lue, “Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations,” PhD Thesis, California Institute of Technology, Pasadena, 2006.

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

[5] M. A. El-Tawil and A. S. Al-Johani, “Approximate Solution of a Mixed Nonlinear Stochastic Oscillator,” Computers & Mathematics with Applications, Vol. 58, No. 1112, 2009, pp. 2236-2259.
doi:10.1016/j.camwa.2009.03.057

[6] M. A. El-Tawil and A. S. El-Johani, “On Solutions of Stochastic Oscillatory Quadratic Nonlinear Equations Using Different Techniques, A Comparison Study,” Journal of Physics: Conference Series, Vol. 96, No. 1, 2008.
http://iopscience.iop.org/1742-6596/96/1/012009

[7] M. A. El-Tawil and A. Fareed, “Solution of Stochastic Cubic and Quintic Nonlinear Diffusion Equation Using WHEP, Pickard and HPM Methods,” Open Journal of Discrete Mathematics, Vol. 1, No. 1, 2011, pp. 6-21.
doi:10.4236/ojdm.2011.11002

[8] M. A. El-Tawil and A. A. El-Shekhipy, “Approximations for Some Statistical Moments of the Solution Process of Stochastic Navier-Stokes Equation Using WHEP Technique,” Applied Mathematics & Information Sciences, Vol. 6, No. 3S, 2012, pp. 1095-1100.

[9] M. A. El-Tawil and A. A. El-Shekhipy, “Statistical Analysis of the Stochastic Solution Processes of 1-D Stochastic Navier-Stokes Equation Using WHEP Technique,” Applied Mathematical Modelling, Vol. 37, No. 8, 2013, pp. 5756-5773. doi:10.1016/j.apm.2012.08.015

[10] A. S. El-Johani, “Comparisons between WHEP and Homotopy Perturbation Techniques in Solving Stochastic Cubic Oscillatory Problems,” AIP Conference Proceedings, Vol. 1148, 2010, pp. 743-752.
doi:10.1063/1.3225426

[11] N. Wiener, “Nonlinear Problems in Random Theory,” MIT Press, John Wiley, Cambridge, 1958.

[12] R. H. Cameron and W. T. Martin, “The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals,” Annals of Mathematics, Vol. 48, 1947, pp. 385-392.

[13] 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, 1965, pp. 695-706.

[14] W. C. Meecham and D. T. Jeng, “Use of the WienerHermite Expansion for Nearly Normal Turbulence,” Journal of Fluid Mechanics, Vol. 32, 1968, pp. 225-235.

[15] X. Yong, X. Wei and G. Mahmoud, “On a Complex Duffing System with Random Excitation,” Chaos, Solitons and Fractals, Vol. 35, No. 1, 2008, pp. 126-132.
doi:10.1016/j.chaos.2006.07.016

[16] P. Spanos “Stochastic Linearization in Structural Dynamics,” Applied Mechanics Reviews, Vol. 34, 1980, pp. 1-8.

[17] W. Q. Zhu, “Recent Developments and Applications of the Stochastic Averaging Method in Random Vibration,” Applied Mechanics Reviews, Vol. 49, No. 10, 1996, pp. 72-80. doi:10.1115/1.3101980

[18] E. F. Abdel-Gawad and M. A. El-Tawil, “General Stochastic Oscillatory Systems,” Applied Mathematical Modelling, Vol. 17, No. 6, 1993, pp. 329-335.

[19] J. Atkinson, “Eigenfunction Expansions for Randomly Excited Nonlinear Systems,” Journal of Sound and Vibration, Vol. 30, No. 2, 1973, pp. 153-172.
doi:10.1016/S0022-460X(73)80110-5

[20] A. Bezen and F. Klebaner, “Stationary Solutions and Stability of Second Order Random Differential Equations,” Physica A, Vol. 233, No. 3-4, 1996, pp. 809-823.
doi:10.1016/S0378-4371(96)00205-1

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

[22] M. A. El-Beltagy and M. A. El-Tawil, “Toward a Solution of a Class of Non-Linear Stochastic Perturbed PDEs Using Automated WHEP Algorithm,” Applied Mathematical Modelling, in Press.
doi:10.1016/j.apm.2013.01.038

[23] MathML Website. http://www.w3.org/Math/