Numerical solution of second-order stochastic differential equations with Gaussian random parameters

Subject Areas : Ordinary differential equations

R. Farnoosh ¹ , H. Rezazadeh ² , A. Sobhani ³ , D. Ebrahimibagha ⁴

1 - School of Mathematics, Iran University of Science and Technology, 16844, Tehran, Iran
2 - School of Mathematics, Iran University of Science and Technology, 16844, Tehran, Iran
3 - School of Mathematics, Iran University of Science and Technology, 16844, Tehran, Iran
4 - Department of Mathematics, Center Branch, Islamic Azad university, Tehran, Iran

Received: 2013-09-16 Accepted : 2013-12-31 Published : 2014-01-01

Keywords:

References:

[1] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Oxford, UK, (1997).

[2] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, (1974).

[3] K. Burrage, I. Lenane, and G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM J. SCI. Compute., Vol. 29, No. 1, pp. 245264, (2007).

[4] R. Cairoli, J. Walsh, Stochastic integrals in the plane, in Acta Math., 134, pp. 111183., (1975).

[5] Dongbin Xiu, D Daniel M. Tartakovsky, Numerical solution for differential equation in random domain, SIAM J. Sci. Compute. Vol. 28, No. 3, pp. 1167-1185 (2006).

[6] Lawrence C. Evans.:An Introduction to Stochastic Dierential Equations Version 1.2 (2004).

[7] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, 3rd ed., Springer-Verlag, Berlin, (2004).

[8] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review 43, 525-546, (2001).

[9] E. Hairer, S. P. Norsett, and G. Wanner, Solving Ordinary Dierential Equations I: Nonstiff Problems, 2nd ed., Springer-Verlag, Berlin, (1993).

[10] N. V. Krylov, Introduction to the Theory of Diusion Processes, American Math Society, (1995).

[11] J. Lamperti, A simple construction of certain diusion processes, J. Math. Kyoto, 161-170, (1964).

[12] G. N. Milstein and M. V. Tretyakov, Quasi-symplectic methods for Langevin-type equations, IMA J. Numer. Anal., 23, pp. 593626, (2003).

[13] H. McKean, Stochastic Integrals, Academic Press, (1969).

[14] C. A. Marsh and J. M. Yeomans, Dissipative particle dynamics: The equilibrium for nite time steps, Euro-phys. Lett., 37, pp. 511516, (1997).

[15] B. K. Oksendal, Stochastic Dierential Equations: An Introduction with Applications, 4th ed., Springer, (1995).

[16] H. R. Rezazadeha, M. Magasedib, B. Shojaeec.Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation, Journal of Linear and Topological Algebra Vol. 01, No. 02, 79- 89, (2012).

[17] Wuan Luo. Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations. California Institute of Technology Pasadena, California,(2006).