Mean Square Heun’s Method Convergent for Solving Random Differential Initial Value Problems of First Order

Author(s)
M. A. Sohaly

ABSTRACT

This paper deals with the construction of Heun’s method of random initial value problems. Sufficient conditions for their mean square convergence are established. Main statistical properties of the approximations processes are computed in several illustrative examples.

This paper deals with the construction of Heun’s method of random initial value problems. Sufficient conditions for their mean square convergence are established. Main statistical properties of the approximations processes are computed in several illustrative examples.

KEYWORDS

Stochastic Partial Differential Equations, Mean Square Sense, Second Order Random Variable,

Stochastic Partial Differential Equations, Mean Square Sense, Second Order Random Variable,

Cite this paper

Sohaly, M. (2014) Mean Square Heun’s Method Convergent for Solving Random Differential Initial Value Problems of First Order.*American Journal of Computational Mathematics*, **4**, 474-481. doi: 10.4236/ajcm.2014.45040.

Sohaly, M. (2014) Mean Square Heun’s Method Convergent for Solving Random Differential Initial Value Problems of First Order.

References

[1] Burrage, K. and Burrage, P.M. (1996) High Strong Order Explicit Runge-Kutta Methods for Stochastic Ordinary Differential Equations. Applied Numerical Mathematics, 22, 81-101.

http://dx.doi.org/10.1016/S0168-9274(96)00027-X

[2] Burrage, K. and Burrage, P.M. (1998) General Order Conditions for Stochastic Runge-Kutta Methods for Both Commuting and Non-Commuting Stochastic Ordinary Equations. Applied Numerical Mathematics, 28, 161-177.

http://dx.doi.org/10.1016/S0168-9274(98)00042-7

[3] Cortes, J.C., Jodar, L. and Villafuerte, L. (2007) Numerical Solution of Random Differential Equations, a Mean Square Approach. Mathematical and Computer Modeling, 45, 757-765.

http://dx.doi.org/10.1016/j.mcm.2006.07.017

[4] Cortes, J.C., Jodar, L. and Villafuerte, L. (2006) A Random Euler Method for Solving Differential Equations with Uncertainties. Progress in Industrial Mathematics at ECMI.

[5] Lamba, H., Mattingly, J.C. and Stuart, A. (2007) An Adaptive Euler-Maruyama Scheme for SDEs, Convergence and Stability. IMA Journal of Numerical Analysis, 27, 479-506.

http://dx.doi.org/10.1093/imanum/drl032

[6] Platen, E. (1999) An Introduction to Numerical Methods for Stochastic Differential Equations. Acta Numerica, 8, 197-246.

http://dx.doi.org/10.1017/S0962492900002920

[7] Higham, D.J. (2001) An Algorithmic Introduction to Numerical Simulation of SDE. SIAM Review, 43, 525-546.

http://dx.doi.org/10.1137/S0036144500378302

[8] Talay, D. and Tubaro, L. (1990) Expansion of the Global Error for Numerical Schemes Solving Stochastic Differential Equation. Stochastic Analysis and Applications, 8, 483-509.

http://dx.doi.org/10.1080/07362999008809220

[9] Burrage, P.M. (1999) Numerical Methods for SDE. Ph.D. Thesis, University of Queensland, Brisbane.

[10] Kloeden, P.E., Platen, E. and Schurz, H. (1997) Numerical Solution of SDE through Computer Experiments. 2nd Edition, Springer, Berlin.

[11] El-Tawil, M.A. (2005) The Approximate Solutions of Some Stochastic Differential Equations Using Transformation. Applied Mathematics and Computation, 164, 167-178.

http://dx.doi.org/10.1016/j.amc.2004.04.062

[12] El-Tawil, M.A. and Sohaly, M.A. (2011) Mean Square Numerical Methods for Initial Value Random Differential Equations. Open Journal of Discrete Mathematics, 1, 66-84.

http://dx.doi.org/10.4236/ojdm.2011.12009

[13] Kloeden, P.E. and Platen, E. (1999) Numerical Solution of Stochastic Differential Equations. Springer, Berlin.

[1] Burrage, K. and Burrage, P.M. (1996) High Strong Order Explicit Runge-Kutta Methods for Stochastic Ordinary Differential Equations. Applied Numerical Mathematics, 22, 81-101.

http://dx.doi.org/10.1016/S0168-9274(96)00027-X

[2] Burrage, K. and Burrage, P.M. (1998) General Order Conditions for Stochastic Runge-Kutta Methods for Both Commuting and Non-Commuting Stochastic Ordinary Equations. Applied Numerical Mathematics, 28, 161-177.

http://dx.doi.org/10.1016/S0168-9274(98)00042-7

[3] Cortes, J.C., Jodar, L. and Villafuerte, L. (2007) Numerical Solution of Random Differential Equations, a Mean Square Approach. Mathematical and Computer Modeling, 45, 757-765.

http://dx.doi.org/10.1016/j.mcm.2006.07.017

[4] Cortes, J.C., Jodar, L. and Villafuerte, L. (2006) A Random Euler Method for Solving Differential Equations with Uncertainties. Progress in Industrial Mathematics at ECMI.

[5] Lamba, H., Mattingly, J.C. and Stuart, A. (2007) An Adaptive Euler-Maruyama Scheme for SDEs, Convergence and Stability. IMA Journal of Numerical Analysis, 27, 479-506.

http://dx.doi.org/10.1093/imanum/drl032

[6] Platen, E. (1999) An Introduction to Numerical Methods for Stochastic Differential Equations. Acta Numerica, 8, 197-246.

http://dx.doi.org/10.1017/S0962492900002920

[7] Higham, D.J. (2001) An Algorithmic Introduction to Numerical Simulation of SDE. SIAM Review, 43, 525-546.

http://dx.doi.org/10.1137/S0036144500378302

[8] Talay, D. and Tubaro, L. (1990) Expansion of the Global Error for Numerical Schemes Solving Stochastic Differential Equation. Stochastic Analysis and Applications, 8, 483-509.

http://dx.doi.org/10.1080/07362999008809220

[9] Burrage, P.M. (1999) Numerical Methods for SDE. Ph.D. Thesis, University of Queensland, Brisbane.

[10] Kloeden, P.E., Platen, E. and Schurz, H. (1997) Numerical Solution of SDE through Computer Experiments. 2nd Edition, Springer, Berlin.

[11] El-Tawil, M.A. (2005) The Approximate Solutions of Some Stochastic Differential Equations Using Transformation. Applied Mathematics and Computation, 164, 167-178.

http://dx.doi.org/10.1016/j.amc.2004.04.062

[12] El-Tawil, M.A. and Sohaly, M.A. (2011) Mean Square Numerical Methods for Initial Value Random Differential Equations. Open Journal of Discrete Mathematics, 1, 66-84.

http://dx.doi.org/10.4236/ojdm.2011.12009

[13] Kloeden, P.E. and Platen, E. (1999) Numerical Solution of Stochastic Differential Equations. Springer, Berlin.