Mean Square Numerical Methods for Initial Value Random Differential Equations

ABSTRACT

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.

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.

KEYWORDS

Random Differential Equations, Mean Square Sense, Second Random Variable, Initial Value Problems, Random Euler Method, Random Runge Kutta-2 Method

Random Differential Equations, Mean Square Sense, Second Random Variable, Initial Value Problems, Random Euler Method, Random Runge Kutta-2 Method

Cite this paper

nullM. El-Tawil and M. Sohaly, "Mean Square Numerical Methods for Initial Value Random Differential Equations,"*Open Journal of Discrete Mathematics*, Vol. 1 No. 2, 2011, pp. 66-84. doi: 10.4236/ojdm.2011.12009.

nullM. El-Tawil and M. Sohaly, "Mean Square Numerical Methods for Initial Value Random Differential Equations,"

References

[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.

[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.