Mean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs

W. W. Mohammed, M. A. Sohaly, A. H. El-Bassiouny, K. A. Elnagar

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt.

DOI: 10.4236/ajcm.2014.44024                

Abstract

Stochastic partial differential equations (SPDEs) describe the dynamics of stochastic processes depending on space-time continuum. These equations have been widely used to model many applications in engineering and mathematical sciences. In this paper we use three finite difference schemes in order to approximate the solution of stochastic parabolic partial differential equations. The conditions of the mean square convergence of the numerical solution are studied. Some case studies are discussed.

Keywords

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Barth, A. (2009) Stochastic Partial Differential Equations: Approximations and Applications. Ph.D. Thesis, University of Oslo, Oslo.
[2]	Barth, A. (2010) A Finite Element Method for Martingale-Stochastic Partial Differential Equations. Communications on Stochastic Analysis, 4, 355-375.
[3]	Cortes, J.C. (2007) Computing Mean Square Approximations of Random Differential Models. Mathematics and Computers in Simulation, 76, 44-48. http://dx.doi.org/10.1016/j.matcom.2007.01.020
[4]	Cortés, J.C., Jódar, L., Villanueva, R.-J. and Villafuerte, L. (2010) Mean Square Convergent Numerical Methods for Nonlinear Random Differential Equations. Transactions on Computational Science, 7, 1-21.
[5]	Cortés, J.C., Jódar, L., Villafuerte, L. and Rafael, R.J. (2007) Computing Mean Square Approximations of Random Diffusion Models with Source Term. Mathematics and Computers in Simulation, 76, 44-48.
[6]	Gardon, A. (2004) The Order of Approximation for Solutions of Ito-Type Stochastic Differential Equations with Jumps. Stochastic Analysis and Applications, 22, 679-699.
[7]	Khan, I.R., Ohba, R. and Hozumi, N. (2003) Mathematical Proof of Closed Form Expressions for Finite Difference Approximations Based on Taylor Series. Journal of Computational and Applied Mathematics, 150, 303-309. http://dx.doi.org/10.1016/S0377-0427(02)00667-2
[8]	Mohammed, A.S. (2014) Mean Square Convergent Three and Five Points Finite Difference Scheme for Stochastic Parabolic Partial Differential Equations. Electronic Journal of Mathematical Analysis and Applications, 2, 164-171.
[9]	Soong, T.T. (1973) Random Differential Equations in Science and Engineering. Academic Press, New York.
[10]	Alabert, A. and Gyongy, I. (2006) On Numerical Approximation of Stochastic Burgers’ Equation. In: Kabanov, Y., Liptser, R. and Stoyanov, J., Eds., From Stochastic Calculus to Mathematical Finance, Springer, Berlin, 1-15. http://dx.doi.org/10.1007/978-3-540-30788-4_1
[11]	Davie, A.M. and Gaines, J.G. (2001) Convergence of Numerical Schemes for the Solution of Parabolic Stochastic Partial Differential Equations. Mathematics of Computation, 70, 121-134. http://dx.doi.org/10.1090/S0025-5718-00-01224-2
[12]	Printems, J. (2001) On the Discretization in Time of Parabolic Stochastic Partial Differential Equations. ESAIM: Mathematical Modelling and Numerical Analysis, 35, 1055-1078. http://dx.doi.org/10.1051/m2an:2001148