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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.

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 [

Definition 2.1. Let us take in to consideration that, the properties of a class of real random variables

called second order random variables

Definition 2.2. A sequence of

If we have the linear random heat problem of the form:

Where

Then, we can find the random Crank-Nicolson scheme for this problem as follows:

Take a uniform mesh with step size

Then,

Similarly,

Then,

Hence for (1):

Put

Hence, the RCNS for our problem is:

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

Definition 3.1.1. A random difference scheme

As

Theorem 3.1.1. The random Crank-Nicolson difference scheme (4)-(6) with second order random variable is to be consistent in mean square sense as:

Proof. Assume that

Then,

As:

Hence, the random Crank-Nicolson scheme (4)-(6) is consistent in mean square sense.∎

Definition 3.2.1. A random Crank-Nicolson difference scheme

For:

Theorem 3.2.1. 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,

Then,

Finally, we have:

At:

Hence, the random Crank-Nicolson difference scheme with second order random variable is unconditionally stable with

Definition 3.3.1. A random difference scheme

Theorem 3.3.1. 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.∎

Consider the linear random parabolic partial differential equation:

With initial condition

and the boundary conditions

And is a second order random variable.

The Random Crank-Nicolson Difference Scheme for this problem is

where

Substituting by

Putting n = 0 in the above system then we have:

Then, we have the system:

From this system we have:

(1) Changing step size

・ Choosing:

・ Choosing:

(2) Changing step size

・ Choosing:

・ Choosing:

(3) Changing the expectations

・ Choosing:

・ Choosing:

From these tables we note that the error is acceptable if:

1) The changing happens in

2) The changing happens in

3) The changing happens in

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.

M. T. Yassen,M. A. Sohaly,Islam Elbaz, (2016) Random Crank-Nicolson Scheme for Random Heat Equation in Mean Square Sense. American Journal of Computational Mathematics,06,66-73. doi: 10.4236/ajcm.2016.62008