THERMAL SCIENCE

International Scientific Journal

Suleyman Sengul

,

Zafer Bekiryazici

,

Mehmet Merdan

WONG-ZAKAI METHOD FOR STOCHASTIC DIFFERENTIAL EQUATIONS IN ENGINEERING

ABSTRACT
In this paper, Wong-Zakai approximation methods are presented for some stochastic differential equations in engineering sciences. Wong-Zakai approximate solutions of the equations are analyzed and the numerical results are compared with results from popular approximation schemes for stochastic differential equations such as Euler-Maruyama and Milstein methods. Several differential equations from engineering problems containing stochastic noise are investigated as numerical examples. Results show that Wong-Zakai method is a reliable tool for studying stochastic differential equations and can be used as an alternative for the known approximation techniques for stochastic models.
KEYWORDS
Wong-Zakai Approximation, Stochastic Differential Equation, Euler-Maruyama Method, Milstein method, solar irradiance, Brownian motion
PAPER SUBMITTED: 2020-05-28
PAPER REVISED: 2020-10-14
PAPER ACCEPTED: 2020-10-22
PUBLISHED ONLINE: 2021-01-24
DOI REFERENCE: https://doi.org/10.2298/TSCI200528014S
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2021, VOLUME 25, ISSUE Special issue 1, PAGES [131 - 142]
REFERENCES
  1. Mehmet, M., et al., Comparison of stochastic and random models for bacterial resistance, Advances in Difference Equations, 1 (2017), 2017, pp. 133
  2. Salgia, R., et. al., Modeling small cell lung cancer (SCLC) biology through deterministic and stochastic mathematical models, Oncotarget, 9 (2018), 40, pp. 26226
  3. Møller, J. K., et. al., Parameter estimation in a simple stochastic differential equation for phytoplankton modelling, Ecological modelling, 222 (2011), 11, pp. 1793-1799
  4. Khodabin, M., Rostami, M., Mean square numerical solution of stochastic differential equations by fourth order Runge-Kutta method and its application in the electric circuits with noise, Advances in Difference Equations, 2015 (2015), 1, pp. 62
  5. Iversen, E. B., et. al., Probabilistic forecasts of solar irradiance using stochastic differential equations, Environmetrics, 25 (2014), 3, pp. 152-164
  6. Kloeden, P. E., Platen, E., Numerical solution of stochastic differential equations, Springer Science & Business Media, New York, USA, 2013
  7. Gard, T. C., Introduction to Stochastic Differential Equations. Monographs and Text-books in pure and applied mathematics, Marcel Dekker Inc., New York, USA, 1988
  8. El-Tawil, M., et. al., A proposed technique of SFEM on solving ordinary random differential equation, Applied Mathematics and Computation, 161 (2005), 1, pp. 35-47
  9. Casabán, M. C., et. al., Determining the first probability density function of linear random initial value problems by the random variable transformation (RVT) technique: a comprehensive study, Abstract and Applied Analysis, 2014 (2014)
  10. Wong, E., Zakai, M., On the convergence of ordinary integrals to stochastic integrals, The Annals of Mathematical Statistics, 36 (1965), 5, pp. 1560-1564
  11. Wong, E., Zakai, M., On the relation between ordinary and stochastic differential equations, International Journal of Engineering Science, 3 (1965), 2, pp. 213-229
  12. Xiaohu, W., et. al., Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, Journal of Differential Equations, 264 (2018), 1, pp. 378-424
  13. Flandoli, F., et. al., Well-posedness of the transport equation by stochastic perturbation, Inventiones mathematicae, 180 (2010), 1, pp.1-53
  14. Londo, J. A., et. al., Numerical performance of some Wong-Zakai type approximations for stochastic differential equations, International Journal of Pure and Applied Mathematics, 107 (2016), 2, pp. 301-315
  15. Kelly, D., et. al., Smooth approximation of stochastic differential equations, Institute of Mathematical Statistics, 44 (2016), 1, pp. 479-520
  16. Zhang, T., Strong convergence ofWong-Zakai approximations of reflected SDEs in a multidimensional general domain, Potential Analysis, 41 (2014), 3, pp. 783-815
  17. Nakao, S., On weak convergence of sequences of continuous local martingales, Annales de l'IHP Probabilités et statistiques, 22 (1986), 3, pp. 371-380
  18. Hairer, M., Pardoux, É., A Wong-Zakai theorem for stochastic PDEs, Journal of the Mathematical Society of Japan, 4 (2015), 67, pp. 1551-1604
  19. Quarteroni, A., Sacco, R., Saleri, F., Numerical mathematics, Springer Science & Business Media, New York, USA, 2010
  20. Cyganowski, S., Kloeden, P., Ombach, J., From elementary probability to stochastic differential equations with MAPLE®, Springer Science & Business Media, 2001
  21. Voyant, C., et. al., Numerical weather prediction (NWP) and hybrid ARMA/ANN model to predict global radiation, Energy, 39 (2012), 1, pp. 341-355
  22. Solar Irradiance Data for 41.02845 N, 50.5157E 15 August 2018 - 21 August 2018, www.sodapro.com/
  23. Wang, J., Ahmed, H. M., Null controllability of nonlocal Hilfer fractional stochastic differential equations, Miskolc Mathematical Notes, 18 (2017), 2, pp. 1073-1083
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Wong-Zakai method for stochastic differential equations in engineering

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence