Approximate Solutions of the Space-Fractional Di⁄usion Equations with additive noise

In this article we take into account a class of stochastic space dif-fusion equations with polynomials forced by additive noise. We derive rigorously limiting equations which de(cid:133)ne the critical dynamics. Also, we approximate solutions of stochastic fractional space di⁄usion equations with polynomial term by limiting equations, which are ordinary di⁄erential equations. Moreover, we address the e⁄ect of the noise on the solution(cid:146)s stabilization. Finally, we apply our results to Fisher(cid:146)s equation and Ginzburg(cid:150)Landau models.


Introduction
In the last few decades, fractional derivatives have drawn tremendous interest mainly because of their possible implementations in di¤erent areas for example in physics [1,2,3,4], biology [5], …nance [6,7,8], biochemistry and chemistry [9], hydrology [10,11]. These fractional-order equations are better suited than equations with integer-order, because derivatives of fractional order are allowed the memory and hereditary properties of di¤erent substances to be represented [12].
In normal di¤usion with time, the mean square displacement of an equation particle linearly increases, i.e. x 2 (t) nt: On the other side, anomalous di¤usion is a di¤usion process not following this linear relation. In some cases, they have a power-law scaling relation, x 2 (t) n t r ; that is present in various types of equations. r is de…ned as the anomalous exponent of di¤usion, in the case r = 1 of normal di¤usion; whereas r = 2; r 2 (0; 1) and r 2 (1; 2) correspond to a ballistic di¤usion, a sub-di¤usion and a Levy super-di¤usion, respectively [2]. By the transformation of Fourier, the anomalously di¤usive operator ( ) r 2 is de…ned [6,13] as: Lf( ) r 2 'g( ) = j j r Lf'g( ); where Lf'g is the Fourier transform of ': In this paper, we are concerned with fractional space-di¤usion equation perturbed by additive noise on a bounded domain G R: (1.1) where " 1 is a small parameter, D is the di¤usion coe¢ cient, A r 2 is the fractional operator with r 2 (1; 2] (One standard example is fractional Laplacian ( ) r 2 ), P is a polynomial with degree m and it is representing reaction kinetics, W is a …nite dimensional Wiener process. It is interesting to note that if we put P(') = '(1 ')(' a); then Equation (1.1) becomes the stochastic fractional space Fitzhugh-Nagumo equation, which is used in the …led of biology and population genetics in circuit theory [14], also is used to model nerve impulse transmission [15,16]. While, if we set P(') = ' ' 3 ; we get to stochastic space-fractional heat equation, which is used in physics and describes the heat distribution within a given time interval in a given region. Furthermore, If P(') = '(1 ' N ); then (1.1) is giving rise to stochastic space-fractional Fisher equation which is used as the spatial and temporal propagation model in an in…nite medium of a virile gene. Also, it is used in chemical kinetics [17], auto catalytic chemical reaction [18], ‡ame propagation [19], neurophysiology [20], nuclear reactor theory [21].
Recently, Equation (1.1) with r = 2 was studied analytically by [22,23] in the deterministic case, i.e without noise. While in the stochastic case this equation with r = 2 was addressed by [24,25,26,27]. Moreover, several numerical and analytical methods have recently been suggested to solve the space fractional partial di¤erential Equation (1.1) without noise see for instance [28,29,30,31]. Here, by perturbation method, we analytically approximate the solution of Equation (1.1), where this equation has not been solved with this method before.
We are not looking at the speci…cs of the existence and uniqueness of solutions in this paper, which is a well known issue. Always, we are assuming at least one local solution exists. We refer to monographs in [32,33,34,35,36,37,38] and the reference therein for the existence and uniqueness of solutions.
Our aim here is to show that the approximate solution of (1.1) is given by The polynomial G( ); is de…ned later in (5.2), has degree m 2. The stochastic process (t; x) in equation (1.2), will be de…ned later in (2.6), is called a fast Ornstein-Uhlenbeck process. We note that the ordinary di¤erential Equation (1.3) contains the same polynomial P with a further polynomial G that exists due to interact between nonlinear term and additive noise. The following real-valued Ginzburg-Landau equation, with Neumann boundary conditions on [0; ], is considered for clari-…cation of our results We demonstrate in approximation Theorem 17 the solution of Ginzburg-Landau Eq. (1.4) shall be of kind (1.2) where is the solution of when r = 2, then we have the old result that obtained by [25].
In this paper, one great innovation of our approach is the explicit estimation of error in terms of arbitrarily high moments of error, while usually only weak convergence is handled against approximation. Moreover, this paper is the …rst paper, to our best knowledge, to …nd analytically the approximate solution for stochastic space-fractional partial di¤erential equations.
The remainder of this article is set out as follows. In next section, we present some notations, assumptions and preliminaries, that we need in this paper, while we estimate an equation represent the high modes and give bounds on it in Section 3. We will state a general case of the averaging over OU-process In Section 4. After that we deduce the limiting equation and prove the main result 17 in Section 5. While in Section 6; there are two examples to clarify our results, such as the Ginzburg-Landau and the Fisher's equations. Finally, we give the conclusion of this paper.

Preliminaries
Let H = L 2 ([0; ]) be a separable Hilbert space with inner product h ; i and k k norm.
De…ne A = , since the operator A is self-adjoint, there exists an complete orthonormal system fe j g where I is the identity operator on H: For r > 0; let the fractional space H r be the domain of A r 2 ;which can be de…ned by with the induced norms 2 ) for t 0 be analytic semigroup generated by the fractional Laplacian ( ) r 2 and satisfy For the nonlinear P in Equation (1.1); we assume: where m is the degree of P.
For short, we are using P c (') = P c P(') and P s (') = P s P(').
We note that from the assumption 2 if we put v = w = 0; then we have For the noise in Equation (1.1): Assumption 3 Suppose the Wiener process W (t); for t 0; is …nite dimensional and acts only on H s . Corresponding to [39] one can write it as where j 2 R for all j 2 f1; 2; ::::; N g and ( j ) j2f1; 2;::::; N g are mutually independent real-valued Brownian motions.

De…nition 4
The fast OU-process (OU-process, for short) is de…ned as In the following de…nition we assume that the solution of Equation (1.1) is not too large.

High modes and Its Bounds
In this section we deduce an equation represent high modes and bound it. We start by split the solution ' of (1.1) into By projecting to H s we obtain This equations can be written in the integral form as where is de…ned in De…nition 4. In the following lemma we will show ' s (t) equals (t) plus a small term.
Lemma 6 Assume that Assumption 1 satis…es. Then there exists for p 1 and > 0 from the de…nition of .
Proof. Using the triangle inequality for (3.4), yields : where we used (2.1), Assumption 1 and the de…nition of ; respectively. Now let us state without proof the uniform bounds on (t). For the proof, see Lemma 4.2 in [25].
for every p 1 and 0 > 0: The following corollary declare that ' s (t) is much smaller than " as stated in de…nition of stopping time . Using Lemmas 7 and 6 to …nish the proof: T r (" 2 Ds)' s (0) n ds C" 2 for n 1: 4 Averaging over OU-Process Here we state a general case of Lemma 5.1 from [25] over the fast OU-process j . This lemma declare that odd powers of j are small of order O(" 1 0 ); while even power of j average to a constant.

Lemma 10
Assume that is a real-valued stochastic process with (0) = O(" ) for some if all n i are even. (4.1) In the following lemma, we apply the earlier Lemma 10 iteratively and check the outcome that we need later in our example.
Lemma 11 Assume that is as in Lemma 10. Then there is a constant C 2k for k 2 N such that where is de…ned in De…nition 4.
Proof. We address three cases as follows.
First case when k = 1 : Second case when k = 2 : Again, from Lemma 10 we get

Limiting Equation and Main Theorem
Here, the limiting equation is derived for Equation (1.1). Also, the main theorem of this paper is stated and proved.

Lemma 12
Assume that the Assumptions 1, 2 and 3 are satisfying. If ' Now, applying Taylor's expansion to P c to get Applying Taylor's expansion again to polynomial P c where m is the degree of P: Using (4.2) we get where To bound the error R; …rst we take E sup t2[0; ] k k p r on both sides of (5.8) to get E sup Integrating from 0 to t taking expectation Using Gronwall's lemma to obtain for t 2 [0; T 0 ] j (t)j j (0)j e CT0 : (5.12) Taking expectation on both sides after supremum of equation (5.12) to obtain (5.11). In fact we can not control of the error terms, that are de…ned in terms of ' s or ' c . Therefore we are limited to a su¢ ciently large subset of , where all our estimates of errors are true. We see that the set has probability close to one as follows.
Proposition 15 Assume that Assumptions 1 and 2 are satisfying, then has probability P( ) 1 C" p : Proof. We notice that First using Chebychev inequality and after then using Lemmas 6, 12, 13, Corollary 8 we get P( ) 1 C[" q + " q + " q ] 1 C" q 1 C" p : k' c (t) (t)k r C" 1 2m ; (5.18) and sup Taking the scalar product h ; i on both sides and using Assumption 2 on : Using Gronwall's lemma, we obtain sup [0; ] j j C" 1 2m on : We …nish the …rst part by using For the second part. Consider for all p > 0: Proof. We notice that for : k' s k r < " g : Now, we obtain by using Equation (3.1) and the triangle inequality where we used (5.14) and (5.18). Hence, By using (5.17), yields (5.20).

Application
Throughout chemistry, physics, biology and other …elds of reaction-di¤usion equations with nonlinearities of polynomials, there are many models where the main theory of approximation is applied. For example, Fisher's and Fitzhugh-Nagumo equations in biology and real-valued Ginzburg-Landau equation in physics. We are looking at two models, one from physics and the other from biology, as follows:

Physical Example
The …rst example is Ginzburg-Landau equation [41]. The Ginzburg-Landau equations is used for modeling a wide variety of physical systems. Also, it was …rst formulated in the sense of pattern formation as a long-wave amplitude equation in the case of convection in binary mixtures close to the onset of instability. The fractional space Ginzburg-Landau equations with additive noise is where the variable '(t; x) is a real-valued function of t and x: To check Assumption 1. We note that P(') = ' ' 3 , then for r > 1 2 kP(')k r = ' ' 3 r k'k r + ' 3 where we used Young inequality. Moreover, we use (5.2) with k = 1 to obtain hence, the limiting equation is Now, the solution of (6.1) by our main theorem approximates by where is a solution of (6.2) and is de…ned in (4). If we suppose that the noise acts only in one mode, i.e W (t) = j j cos(jx): Then Equation ( If we choose j such that 2 j < 2j r 3 for r 2 (1; 2]; then the term (1 3 2 j 2j r ) is negative. We may say in this case that the dynamics of the dominant modes was stabilized by the degenerated additive noise.

Biological Example
The second example is the Fisher's equation [42].

Conclusions
In this paper we approximated the solutions of stochastic fractional space diffusion equations via the solutions of ordinary di¤erential equations which is called limiting equations. We illustrated our results by applying to Fisher's equation and Ginzburg-Landau models. We discussed the in ‡uence of degenerate additive noise on the stabilization of the solutions. These solutions are of considerable importance in understanding many important complex physical phenomena as fractional di¤usion equations arise in the modeling of turbulent ‡ow, contaminant transport in groundwater ‡ow and chaotic dynamics of classical conservative systems.