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Approximate Solution of Stochastic Partial Differential Equation with Random Neumann Boundary Condition

A. H. EL_Bassiouny*, W. W. Mohammed, F. Eskander

Department of Mathematics, Faculty of Science, Mansoura University, Egypt

Abstract: In this paper we approximate the solution of a parabolic nonlinear stochastic partial differential equation (SPDE) with cubic nonlinearity and with random Neumann boundary condition via a stochastic ordinary differential equation (SODE) which is a stochastic amplitude equation near a change of stability.

Keywords: amplitude equations, SPDEs, random boundary conditions, multiscale analysis, Ginzburg-Landau equation.

Received February 04, 2015; Revised March 05, 2015; Accepted March 15, 2015

Citation: A. H. EL_Bassiouny, W. W. Mohammed, F. Eskander, Approximate Solution of Stochastic Partial Differential Equation with Random Neumann Boundary Condition. International Journal of Partial Differential Equations and Applications, 3 (1). 20-24. January 2015. doi:10.12691/ijpdea-3-1-4

1. Introduction

Stochastic partial differential equations (SPDEs) appear naturally as models for dynamical systems abided by random influences. The SPDEs have a wide range of applications outside mathematics. For instance, biology, chemistry, epidemiology, economics, microelectronics, mechanics, and finance.

For some applications the noise not affects only inside the medium, but on its physical boundary, too. This happens for heat transfer in a solid in contact with afield [6], the air-sea interactions on the ocean surface [8] and chemical reactor theory [7]. Thus, this topic has a rapidly developing as a fascinating research field with many interesting unanswered questions.

To approximate the SPDEs near a change of stability, we use a rigorous technique, so it is important to make the reduction of the dynamics of SPDEs to obtain simpler equations that are the amplitude or the modulation equations.

In this paper we deal with a parabolic equation (typically, the heat equation) perturbed by a Neumann boundary noise involve additive degenerate noise. More specifically, consider for t ≥ 0;

[picture] | (1)

where A is a non-positive self-adjoint operator with finite dimensional kernel, is a small deterministic perturbation, the constant , F is a cubic nonlinearity, W is a Wiener process, B is a real valued Brownian motion and is the positive noise intensity parameter.

In the case of no homogenous boundary conditions (i.e., ). Sowers [9] investigated general reaction diffusion equation with Neumann boundary conditions. Da Prato and Zabczyk [4,5] explained the difference between the problems with Dirichlet and Neumann boundary noises. Recently, Cerrai and Freidlin [2] have considered a nonlinear stochastic parabolic equation with Neumann boundary noise. The Ginzburg-Landau equation with random Neumann boundary conditions is solved numerically by Xu and Duan [10].

The paper is organized as follows. In the next section we state some definitions, notation and assumptions that we need for our result. In Section 3 we give a formal derivation for the amplitude equation, also we state and prove the main result of this paper. Finally, we give applications to the nonlinear heat equation.

2. Preliminaries

Let H = L²(D) be a Hilbert space with L²-norm denoted by ||.|| and inner product by <.,.>, where D is a bounded domain with smooth boundary .

The linear operator generates an analytic semigroup on H. Moreover, denote by , which forms a complete orthonormal basis in H; a family of eigenfunctions of A and for the eigenvalues with . If we take in the form

then . Define N: = ker A = {1}, and the orthogonal component of N in H. Also, define the projection and . Let the projections Pc and Ps are commute with A.

Definition 1. For ; we define the space as

where is an orthonormal basis of H and are real numbers.

Lemma 2. For all t > 0 and , there are constants M > 0 and such that for all

[picture] | (2)

Definition 3. (Stopping time) For the N × S- valued stochastic process (a,ψ) defined in the next section. We define, for some T₀ > 0 and , the stopping time as

[picture] | (3)

Also we have the following hypotheses.

H₁: Assume that the nonlinearity with is trilinear, symmetric and satisfies the following conditions, for some C > 0,

and

We use F(u) = F(u,u,u) and Fc = PcF for short.

H₂: Let W be a cylindrical Wiener process on H. Suppose for t ≥ 0;

where the are independent, standard Brownian motions in and the are real numbers for all k. Also, we assume that

3. Amplitude Equation and Main Result

In this section we state and prove the main theorem after we derive the amplitude equation of the Equation (1). First, let us derive the Amplitude equation with error term. According to [3] the mild solution of Equation (1) is

[picture] | (4)

where is the Neumann map and it is defined for any by the solution of

Fortunately, we have an explicit expression for the Neumann map as follows:

Define Z(t) as

In the following, we write Z as explicit formula in terms of Fourier series

[picture] | (5)

Hence,

[picture] | (6)

By substituting from (6) into (5) we have

[picture] | (7)

Now, we can rewrite the mild solution (4) in the following form

[picture] | (8)

where and Z(t) is defined in (7). Thus

[picture] | (9)

In order to rescale (9) to the slow time-scale, we consider the following ansatz

[picture] | (10)

to obtain

[picture] | (11)

where

with . To get the amplitude equation with error term, let

[picture] | (12)

where and Substituting from (12) into (11) to have

[picture] | (13)

Taking projection onto Pc for (13) we obtain

[picture] | (14)

Taking projection onto Ps for (13) we obtain

[picture] | (15)

In the next lemma, we can easy to show that the non-dominant modes are not too large as asserted in Definition 3 for .

Lemma 4. Assume the hypothesis H₁ and H₂ hold. Then for all p 1 there is a constant C > 0 such that

for .

Proof. See the proof of the Corollary 4.3 in [1].

Lemma 5. Under the hypothesis H₁ and and for from the definition of , then

[picture] | (16)

with

Proof. We have from the previous lemma that

[picture] | (17)

Substituting into (14) and integrating the resulted equation from 0 to T, we obtain

where

We can find that the bound of R is when we use equation (17).

Lemma 6. Let the hypotheses H1 and H2 hold. De.ne the stochastic process b(T) in N with as the solution of

[picture] | (18)

Then for T0 > 0 there exists a constant C > 0 such that

Proof. We define X as

[picture] | (19)

Substituting into (18), we obtain

[picture] | (20)

Taking the scalar product h:;XiR on both sides of (20)

Using Cauchy-Schwartz and Young inequalities and the hypothesis H₁ we have

By integrating the above equation from 0 to .we obtain

Taking -th power and using Gronwall's lemma, then the supremum and expectation, we obtain

[picture] | (21)

Using (21) and (19), we have

Definition 3: Define the set such that all these estimates

[picture] | (22)

[picture] | (23)

[picture] | (24)

hold on

Theorem 1: Assume that the hypotheses H₁ and H₂ hold. Let be the solution of (16) and be the solution of (18). If the initial condition satisfies , then

and

for on

Proof: Define as

From (16) we obtain

[picture] | (25)

Subtracting (25) from (18) and defining , we obtain

Thus,

[picture] | (26)

Taking the scalar product on both sides of (26), we have

Using Cauchy-Schwartz and Young inequalities, we obtain the following linear ordinary differential inequality

[picture] | (27)

holds on By substituting from (23) and (24) into (27). As long as ,

Integrating from 0 toand using Gronwall's lemma, we obtain

Hence,

Then,

for For the second part of the theorem, using the triangle inequality, we have

Theorem 2. (Approximation theorem): Under hypotheses H₁ and H₂, let be the solution of (1) defined in (10) and (12) with the initial conditions where and , b is the solution of (18) with Then for and for k , there exists such that

Proof: First we note that by using triangle inequality, we obtain

and

For the probability of we have,

Hence,

[picture] | (28)

we used Chebychev's inequality. Thus

where

4. Application

We apply our results to heat equation. The heat equation is a partial differential equation that describe the distribution of heat in a given area in a given time interval. Generally, given a certain area in space, because of heat movement from warmer are ask to colder ones, the warm spots will cool down and the colder spots will begin to warm up. Solutions for which there is no heat moving are called "equilibrium solutions".

Also, we can set boundary conditions for this PDE. For instance, if we have a rod with one end on a block of ice and the other end attached to a heater. Here we find that the interior point on the rod will not excede the temperature of the heater and will not drop below the temperature of the ice. Therefore we can apply our work on this kind (heat equation) with Neumman boundary condition which has the form

[picture] | (29)

Now, we can satisfy the conditions of stability:

For and

and

The main theorem states that the solution of the heat Equation (29) is approximated by

and

where is the solution of the amplitude equation that takes the form

References

[1]   | D. Blomker and W. W. Mohammed. Amplitude equations for SPDEs with cubic nonlinearities. Stochastics. An International Journal of Probability and Stochastic Processes. 85 (2): 181-215, (2013).
[2]   | S. Cerrai and M. Freidlin. Fast transport asymptotics for stochastic RDE.s with boundary noise, Ann. Probab. 39: 369.405, (2011).
[3]   | G. Da Prato, S. Kwapien, and J. Zabczyk. Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics, 23:1-23, (1987).
[4]   | G. Da Prato and J. Zabczyk. Evolution equations with white-noise boundary conditions. Stochastics Rep. 42: 167-182, (1993).
[5]   | G. Da Prato and J. Zabczyk. Ergodicity for In.nite Dimensional Systems, volume 229 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, (1996).
[6]   | R. E. Langer. A problem in diffusion or in the. ow of heat for a solid in contact with afuid. Tohoku Math. J.35: 260-275, (1932).
[7]   | L. Lapidus and N. Amundson. Chemical reactor theory, Prentice-Hall, (1977).
[8]   | J. P. Peixoto and A. H. Oort. Physics of climate. Springer, New York, (1992).
[9]   | R. B. Sowers. Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Probab. 22: 2071.2121, (1994).
[10]   | S. Xu, J. Duan. A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions. Applied Mathematics and Computation. 217, 9532-9542, (2011).