About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 685137, 21 pages
http://dx.doi.org/10.1155/2013/685137
Research Article

Numerical Approximation of Higher-Order Solutions of the Quadratic Nonlinear Stochastic Oscillatory Equation Using WHEP Technique

1Engineering Mathematics & Physics Department, Faculty of Engineering, Cairo University, Giza, Egypt
2Mathematics Department, Faculty of Science, Northern Borders University, Arar, Saudi Arabia

Received 21 March 2013; Accepted 4 July 2013

Academic Editor: Turgut Öziş

Copyright © 2013 Mohamed A. El-Beltagy and Amnah S. Al-Johani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Jahedi and G. Ahmadi, “Application of Wiener-Hermite expansion to nonstationary random vibration of a Duffing oscillator,” Transactions of the ASME. Journal of Applied Mechanics, vol. 50, no. 2, pp. 436–442, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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, pp. 274–281, 2012. View at Publisher · View at Google Scholar
  3. W. Lue, Wiener chaos expansion and numerical solutions of stochastic partial differential equations [Ph.D. thesis], California Institute of Technology, Pasadena, Calif, USA, 2006.
  4. M. A. El-Tawil, “The application of WHEP technique on stochastic partial differential equations,” International Journal of Differential Equations and Applications, vol. 7, no. 3, pp. 325–337, 2003. View at Zentralblatt MATH · View at MathSciNet
  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. 11-12, pp. 2236–2259, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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, vol. 96, no. 1, 2008.
  7. M. A. El-Tawil and A. F. 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, pp. 6–21, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  8. M. A. El-Tawil and A. El-Shikhipy, “Approximations for some statistical moments of the solution process of stochastic Navier-Stokes equation using WHEP technique,” Applied Mathematics & Information Sciences, vol. 6, no. 3, pp. 1095–1100, 2012. View at MathSciNet
  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, pp. 5756–5773, 2013. View at Publisher · View at Google Scholar
  10. A. S. El-Johani, “Comparisons between WHEP and homotopy perturbation techniques in solving stochastic cubic oscillatory problems,” in Proceedings of the AIP Conference, vol. 1148, pp. 743–752, 2010. View at Publisher · View at Google Scholar
  11. N. Wiener, Nonlinear Problems in Random Theory, MIT Press, John Wiley, Cambridge, Mass, USA, 1958. View at MathSciNet
  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, pp. 385–392, 1947. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. T. Imamura, W. C. Meecham, and A. Siegel, “Symbolic calculus of the Wiener process and Wiener-Hermite functionals,” Journal of Mathematical Physics, vol. 6, pp. 695–706, 1965. View at Publisher · View at Google Scholar · View at MathSciNet
  14. W. C. Meecham and D. T. Jeng, “Use of the Wiener-Hermite expansion for nearly normal turbulence,” Journal of Fluid Mechanics, vol. 32, pp. 225–235, 1968.
  15. E. F. Abdel-Gawad and M. A. El-Tawil, “General stochastic oscillatory systems,” Applied Mathematical Modelling, vol. 17, no. 6, pp. 329–335, 1993.
  16. 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, vol. 37, no. 12-13, pp. 7174–7192, 2013.
  17. X. Yong, X. Wei, and G. Mahmoud, “On a complex duffing system with random excitation,” Chaos, Solitons and Fractals, vol. 35, no. 1, pp. 126–132, 2008. View at Publisher · View at Google Scholar
  18. P. Spanos, “Stochastic linearization in structural dynamics,” Applied Mechanics Reviews, vol. 34, pp. 1–8, 1980.
  19. W. Q. Zhu, “Recent developments and applications of the stochastic averaging method in random vibration,” Applied Mechanics Reviews, vol. 49, no. 10, pp. 72–80, 1996. View at Publisher · View at Google Scholar
  20. J. Atkinson, “Eigenfunction expansions for randomly excited nonlinear systems,” Journal of Sound and Vibration, vol. 30, no. 2, pp. 153–172, 1973. View at Publisher · View at Google Scholar
  21. A. Bezen and F. Klebaner, “Stationary solutions and stability of second order random differential equations,” Physica A, vol. 233, no. 3-4, pp. 809–823, 1996. View at Publisher · View at Google Scholar
  22. A. H. Nayfeh, Problems in Perturbation, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  23. MathML, http://www.w3.org/Math/.