ENG  Vol.5 No.5 A , May 2013
Higher-Order WHEP Solutions of Quadratic Nonlinear Stochastic Oscillatory Equation
Abstract: This paper introduces higher-order solutions of the quadratic nonlinear stochastic oscillatory equation. Solutions with different orders and different number of corrections are obtained with the WHEP technique which uses the WienerHermite expansion and perturbation technique. The equivalent deterministic equations are derived for each order and correction. The solution ensemble average and variance are estimated and compared for different orders, different number of corrections and different strengths of the nonlinearity. The solutions are simulated using symbolic computation software such as Mathematica. The comparisons between different orders and different number of corrections show the importance of higher-order and higher corrected WHEP solutions for the nonlinear stochastic differential equations.
Cite this paper: M. El-Beltagy and A. Al-Johani, "Higher-Order WHEP Solutions of Quadratic Nonlinear Stochastic Oscillatory Equation," Engineering, Vol. 5 No. 5, 2013, pp. 57-69. doi: 10.4236/eng.2013.55A009.

[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.

[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.