NUMERICAL SOLUTION OF STOCHASTIC TIME FRACTIONAL HEAT TRANSFER EQUATION WITH ADDITIVE NOISE

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In the present paper, we developed an explicit finite difference scheme for stochastic time fractional heat transfer equation. Furthermore, we proved the conditional stability of the solution of the scheme and analyze the effect of the multiplier of the random noise in mean square stability. Also, we discuss the convergence of the explicit scheme in the mean square sense. The approximate solution of the practical problem is obtain by developed scheme and it is represented graphically by Mathematica software.


INTRODUCTION
Stochastic partial differential equations are very useful in describing random effects occuruing in the fields of science, engineering and mathematical finance. Several numerical methods have been developed to solve stochasstic partial differential equations (SPDEs), Allen [2] has constructed finite element and difference approximation of some linear SPDEs. Kamrani and Hosseini [9] discussed explicite and implicit finite difference methods and their stability convergence for general SPDEs. Furthermore, Soheili et al. [29] presented higher order finite difference scheme and Saul'yev scheme for solving linear parabolic SPDEs. Now a days, fractional calculus is a developing branch of mathematics which deals with the derivative and integrals of non-integer order and attracted lots of attention in several fields such as physics, chemistry, engineering, hydrology and finance. Recently, fractional diffusion equations are becoming more popular in many areas of applications [11,14,23,27,37] and stochastic fractional partial differential equations [1,36] etc. Stochastic partial differential equations with fractional time derivative can be used to describe random effects on transport of particles in medium with thermal memory or particles subjected to sticking and trapping [4].
The theoretical analysis of fractional stochastic partial differential equations have been intensively investigated in the literature [4,15,16], also, Sweilam et al. [31] presented compact finite difference method to solve stochastic fractional advection diffusion equation. Amaneh et al [3] introduced stochastic generalized fractional diffusion equation and used finite difference method for finding numerical solution. Chen et al [4] introduced a class of SPDEs with time fractional derivatives and proved existence and uniqueness of their solutions.
The classical heat equation deals with heat propagation in homogeneous medium and the time fractional diffusion equation has been widely used to model the anomalous diffusion exhibiting sub diffusive behavior due to particle sticking and trapping phenomena [13]. When noise is introduced in partial differential equations, turns it to SPDEs. According to the nature of the noise term there are two main classes of SPDEs like SPDEs with additive noise and SPDEs with multiplicative noise. In additive SPDEs, noise term does not depends on the state of the system, while the noise term depends on the state of the system in multiplicative SPDEs. Additive noise occurs due to temporal fluctuations of internal degrees of freedom, while multiplicative noise arrives due to random variations of some external control parameters [8]. Fractional stochastic partial differential equation considers both memory and environmental noise effect. Now a days, Liu et al [36] studied some properties of a class of fractional stochastic heat equations.
Babaei et al [1] applied a spectral collection method to solve a class of time fractional heat equation driven by Brownian motion. Guang-an Zou [6] presented a Galerkin finite element method for time fractional stochastic heat equation driven by multiplicative noise. Stochastic fractional heat equations driven by additive noise are yet to be investigated. At the same time, it is very difficult to find exact solution of stochastic fractional heat equations by analytical methods but finite difference methods that gives large algebraic system to solve in place of differential equation. In this connection, the theme of this paper is to develop a fractional order explicit finite difference scheme for stochastic time fractional heat transfer equation (STFHTE) driven by additive noise with initial and boundary conditions, given as follows, where, D > 0 is the diffusivity constant, σ > 0 is the constant noise intensity, i.e.additive noise. Now, W (t) is one dimensional standard Winner process, whereẆ (t) = ∂W (t) ∂t , such that white noise ∆W (t) is a random variable generated from Gaussian distribution with zero mean and standard deviation ∆t and S(x,t) is temperature at the position x and at time t. Here, ∂ α S(x,t) ∂t α is Caputo time fractional derivative of order α. We consider the following definitions for further developments.

Definition 1.2. Euler-Maruyama Scheme:
For the d-dimensional Stochastic differential equation of Itó type [25] where f , g : R d → R d , the Euler-Maruyama Scheme for computing approximations S k i takes the form and ∆W k ∼ N (0, ∆t).
Definition 1.6. The symmetric second order difference quotient in space at time level t = t k is given as follows [28] We organize the paper as follows: In section 2 , we develop the explicit fractional order finite

FINITE DIFFERENCE SCHEME
We consider the following Stochastic time fractional heat transfer equation driven by additive noise (STFHTE) with initial and boundary conditions ∂t , such that white noise ∆W (t) is a random variable generated from Gaussian distribution with zero mean and standard deviation ∆t. Note that for α = 1, we recover in the limit the well known Stochastic heat transfer equation of Markovian process For the numerical approximation of the explicit scheme, we define h = L N and τ = T N the space and time steps respectively, such that t k = kτ; k = 0, 1, · · · , N be the integration time 0 ≤ t k ≤ T and x i = ih for i = 0, 1, · · · , N. Define S k i = S(x i ,t k ) and let S k i denote the numerical approximation at the mesh point (x i ,t k ) to the exact solution S(x i ,t k ).

STABILITY
In the present section, we discuss the stability of the solution of fractional order explicit finite difference scheme (9)

Then equation (9) leads to
We assume that Then we prove From equation (10), we obtain N (0, 1) and V n is Normally distributed with mean 0 and variance 1 i.e. N (0, 1) random variable.
We know that E[V n ] = 0 and E[V 2 n ] = 1, therefore, Hence, by induction, we prove Therefore, we prove that the solution of the scheme is conditionally stable in mean square sense.
where T k is vector of truncation error at time level t k . Suppose, |e k i | = |S k i − S k i | is the error vector. Let us assume that For k=0, from equation (9), we obtain For n = k, we assume that For n = k + 1, we prove that Now, from (10), we have Hence, we prove that the scheme is conditionally convergent in mean square sense. This completes the proof.

NUMERICAL SOLUTIONS
We consider the following stochastic time fractional heat transfer equation with initial and boundary conditions initial condition: S(x, 0) = sin πx, 0 ≤ x ≤ 1 (17) boundary conditions: S(0,t) = S(1,t) = 0,t ≥ 0. (18) where 0 < α ≤ 1, σ > 0 is the constant noise intensity. Now, W (t) is one dimensional standard Winner process, whereẆ (t) = ∂W (t) ∂t , such that white noise ∆W (t) is a random variable generated from Gaussian distribution with zero mean and standard deviation ∆t. In Figure 1     In Figure 2, we obtain the numerical solution of the STFHTE (5.1)-(5.3) for α = 1, 0.9, 0.8 with additive noise σ = 1, 2 respectively and observed that these solutions are good approximation to the exact solutions. Therefore the developed fractional order explicit finite difference scheme provides a effective approximate solution. We observed that as α increased the amplitude of the solution behavior is increased. From Figure 3, we observed that the approximate solutions of STFHTE (5.1)-(5.3) for α = 0.8 and α = 0.9 with σ = 1, 2 are approximately same.
Therefore we conclude that if the additive noise intensity changes the numerical solutions are approximately same.

CONCLUSIONS
(1) In present paper, we successfully developed the fractional order explicit finite difference scheme for Stochastic time fractional heat transfer equation.