TITLE:
Random Crank-Nicolson Scheme for Random Heat Equation in Mean Square Sense
AUTHORS:
M. T. Yassen, M. A. Sohaly, Islam Elbaz
KEYWORDS:
Random Partial Differential Equations (RPDEs), Mean Square Sense (m.s), Second Order Random Variable (2r.v.'s), Random Crank-Nicolson Scheme, Convergence, Consistency, Stability
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.6 No.2,
June
3,
2016
ABSTRACT: 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.