Mean Square Numerical Methods for Initial Value Random Differential Equations
Magdy A. El-Tawil, Mohammed A. Sohaly
DOI: 10.4236/ojdm.2011.12009   PDF    HTML     6,573 Downloads   12,562 Views   Citations


In this paper, the random Euler and random Runge-Kutta of the second order methods are used in solving random differential initial value problems of first order. The conditions of the mean square convergence of the numerical solutions are studied. The statistical properties of the numerical solutions are computed through numerical case studies.

Share and Cite:

El-Tawil, M. and Sohaly, M. (2011) Mean Square Numerical Methods for Initial Value Random Differential Equations. Open Journal of Discrete Mathematics, 1, 66-84. doi: 10.4236/ojdm.2011.12009.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] K. Burrage, and P.M. Burrage, “High Strong Order Explicit Runge-Kutta Methods for Stochastic Ordinary Differential Equations,” Applied Numerical Mathematics, Vol. 22, No. 1-3, 1996, pp. 81-101. doi:10.1016/S0168-9274(96)00027-X
[2] K. Burrage, and P. M. Burrage, “General Order Conditions for Stochastic Runge-Kutta Methods for Both Commuting and Non-Commuting Stochastic Ordinary Equations,” Applied Numerical Mathematics, Vol. 28, No. 2-4, 1998, pp. 161-177. doi:10.1016/S0168-9274(98)00042-7
[3] J. C. Cortes, L. Jodar and L. Villafuerte, “Numerical Soluion of Random Differential Equations, a Mean Square Approach,” Mathematical and Computer Modelling, Vol. 45, No.7, 2007, pp. 757-765. doi:10.1016/j.mcm.2006.07.017
[4] J. C. Cortes, L. Jodar, and L.Villafuerte, “A Random Euler Method for Solving Differential Equations with Uncertainties,” Progress in Industrial Mathematics at ECMI, Madrid, 2006.
[5] H. Lamba, J. C. Mattingly and A. Stuart, “An adaptive Euler-Maruyama Scheme for SDEs, Convergence and Stability,” IMA Journal of Numerical Analysis, Vol. 27, No. 3, 2007, pp. 479-506. doi:10.1093/imanum/drl032
[6] E. Platen, “An Introduction to Numerical Methods for Stochastic Differential Equations,” Acta Numerica, Vol. 8, 1999, pp. 197-246. doi:10.1017/S0962492900002920
[7] D. J. Higham, “An Algorithmic Introduction to Numerical Simulation of SDE,” SIAM Review, Vol. 43, No. 3, 2001, pp. 525-546. doi:10.1137/S0036144500378302
[8] D. Talay, and L. Tubaro, “Expansion of The Global Error for Numerical Schemes Solving Stochastic Differential Equation,” Stochastic Analysis and Applications, Vol. 38, No. 4, 1990, pp. 483-509. doi:10.1080/07362999008809220
[9] P. M. Burrage, “Numerical Methods for SDE,” Ph.D. Thesis, The University of Queensland, 1999.
[10] P. E. Kloeden, E. Platen and H. Schurz, “Numerical Solution of SDE Through Computer Experiments,” Second Edition, Springer, 1997.
[11] M. A. El-Tawil, “The Approximate Solutions of Some Stochastic Differential Equations Using Transformation,” Journal of Applied Mathematics and Computing, Vol. 164, No. 1, 2005, pp. 167-178. doi:10.1016/j.amc.2004.04.062
[12] P. E. Kloeden, and E. Platen, “Numeical Solution of Stochastic Differential Equations,” Springer, Berlin, 1999.
[13] T. T. Soong, “Random Differential Equations in Science and Engineering,” Academic Press, New York, 1973.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.