Random Crank-Nicolson Scheme for Random Heat Equation in Mean Square Sense

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.


Introduction
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.

Preliminaries
Definition 2.1.Let us take in to consideration that, the properties of a class of real random variables are finite.In this case they are called second order random variables ( )

Random Crank-Nicolson Scheme (RCNS)
If we have the linear random heat problem of the form: Where α is a second order random variable.
( ) ( ) Then, we can find the random Crank-Nicolson scheme for this problem as follows: Take a uniform mesh with step size x ∆ and t ∆ on x-axis and t-axis respectively.Additionally, Let ∆ .On this mesh we have: Hence for (1): ∆ Hence, the RCNS for our problem is:

Consistency of RCNS
We can rewrite the above scheme as: The above scheme is a random Crank-Nicolson version of (1 -3).For a RPDE, say Lv = G where L is a differentiable operator and On the other hand, we represent finite difference scheme at the point ( ) , if for any continuously differentiable function The random Crank-Nicolson difference scheme ( 4)-( 6) with second order random variable is to be consistent in mean square sense as: and at time ( ) Hence, the random Crank-Nicolson scheme (4)-( 6) is consistent in mean square sense.∎( )

Exponential Stability Analysis of RCNS
The random Crank-Nicolson scheme ( 4)-( 6) with second order random variable is unconditionally stable in mean square sense as with k = 1 and b = 0.
Proof: Since, , , , , , Finally, we have: ( ) Hence, the random Crank-Nicolson difference scheme with second order random variable is unconditionally stable with The random Crank-Nicolson difference scheme ( 4)-( 6) with second order random variables is convergent in mean square sense. Proof.
Since, the RCNS is consistent and unconditionally exponential stable, thus, the scheme ( 4)-( 6) is convergent in mean square sense.∎

Case Studies
Consider the linear random parabolic partial differential equation: and the boundary conditions And  is a second order random variable.

The Numerical Solution
The Random Crank-Nicolson Difference Scheme for this problem is ( ) ( ) where Substituting by 1, 2, 3 k = in (9) we have: Putting n = 0 in the above system then we have: )

Conclusion
The random heat equation can be solved numerically by using mean square convergent Crank-Nicolson scheme.The random variable in the Crank-Nicolson scheme is must second order random variable and the random Crank-Nicolson scheme is unconditionally stable in the area of mean square sense.Many complicated equations in linear and nonlinear parabolic partial differential problems can be discussed using finite difference schemes in mean square sense.

Definition 3 . 2 . 1 .
A random Crank-Nicolson difference scheme in mean square if there exist some positive constants a, c and constants k, b.Such that: