The
goal of computational science is to develop models that predict phenomena
observed in nature. However, these models are often based on parameters that
are uncertain. In recent decades, main numerical methods for solving SPDEs have
been used such as, finite difference and finite element schemes [1]-[5]. Also,
some practical techniques like the method of lines for boundary value problems
have been applied to the linear stochastic partial differential equations, and
the outcomes of these approaches have been experimented numerically [7]. In
[8]-[10], the author discussed mean square convergent finite difference method
for solving some random partial differential equations. Random numerical
techniques for both ordinary and partial random differential equations are
treated in [4] [10]. As regards applications using explicit analytic solutions
or numerical methods, a few results may be found in [5] [6] [11]. This article
focuses on solving random heat equation by using Crank-Nicol- son technique
under mean square sense and it is organized as follows. In Section 2, the mean
square calculus preliminaries that will be required throughout the paper are
presented. In Section 3, the Crank-Nicolson scheme for solving the random heat
equation is presented. In Section 4, some case studies are showed. Short
conclusions are cleared in the end section.

Allen, E.J., Novosel, S.J. and Zhang, Z.C. (1998) Finite Element and Difference Approximation of Some Linear Stochastic Partial Differential Equations. Stochastics and Stochastic Reports, 64, 117-142. http://dx.doi.org/10.1080/17442509808834159

Davie, A.J. and Gaines, J.G. (2001) Convergence of Numerical Schemes for the Solution of Parabolic Stochastic Partial Differential Equations. Mathematics of Computations, 70, 121-134. http://dx.doi.org/10.1090/S0025-5718-00-01224-2

Cortes, J.C., Lucas Jódar, L. and Villafuerte, R.J. (2007) Villanueva: Computing Mean Square Approximations of Random Diffusion Models with Source Term. Mathematics and Computers in Simulation, 76, 44-48. http://dx.doi.org/10.1016/j.matcom.2007.01.020

El-Tawil, M.A. and Sohaly, M.A. (2009) Mean Square Numerical Methods for Initial Value Random Differential Equations. Open Journal of Discrete Mathematics (OJDM), 1, 66-84,

El-Tawil, M.A. and Sohaly, M.A. (2012) Mean Square Convergent Three Points Finite Difference Scheme for Random Partial Differential Equations. Journal of the Egyptian Mathematical Society, 20, 188-204. http://dx.doi.org/10.1016/j.joems.2012.08.017

Sohaly, M.A. (2014) Mean Square Heun’s Method Convergent for Solving Random Differential Initial Value Problems of First Order. American Journal of Computational Mathematics, 4, 280-288.

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.

Mohammed, W., Sohaly, M.A., El-Bassiouny, A. and Elnagar, K. (2014) Mean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs. American Journal of Computational Mathematics, 4, 280-288. http://dx.doi.org/10.4236/ajcm.2014.44024