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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Stochastic Heat and Advection Cauchy

Sohalya MA*, Yassena MT and Elbaza IM

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

*Corresponding Author:
Sohalya MA
Faculty of Science, Department of Mathematics
Mansoura University, Egypt
Tel: +20 50 2383781
E-mail: [email protected]; [email protected]; [email protected]

Received date: January 28, 2017; Accepted date: March 14, 2017; Published date: March 21, 2017

Citation: Sohalya MA, Yassena MT, Elbaza IM (2017) Stochastic Heat and Advection Cauchy. J Appl Computat Math 6:345. doi: 10.4172/2168-9679.1000345

Copyright: © 2017 Sohalya MA, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this paper, the solutions of Cauchy problems for the stochastic advection and stochastic diffusion equations are obtained using the finite difference method. In the case when the flow velocity is a function of stochastic flow velocity and also, the diffusion coefficient in the stochastic heat equation is a function of stochastic diffusion coefficient, the consistency and stability of the finite difference scheme we are used need to be performed under mean square calculus.

Keywords

Stochastic differential equation; Mean square; Cauchy problem

Introduction

Diffusion and advection problems involve randomness not only from the error measurements but may be also due to material defects or from impurities. The basic motivation of this paper focuses on this fundamental difficulty: the numerical consistency and stability of finite difference discretization of a Cauchy problems for stochastic advection and stochastic diffusion equations in the forms:

equation   (1)

equation   (2)

The independent variables are t (time) and x (space). The dependent variable is U which is a function of t and x. p(x,ß) is a discrete stochastic process (s.p.) defined on a common probability space (Ω,F,P) and depending of the stochastic variable ß,(U0(x)),f(x) are an initial deterministic data function.

In order to study the consistency and stability for the problems eqns (1) and (2) with the according difference schemes, it is necessary to use mean square calculus. Many papers have been studied the analytical solutions for stochastic differential models [1-7]. Also, in recent years numerical methods have also been applied to Cauchy problems such as heat equation with stochastic variable input [8]. Our paper is organized as follows, Section 2, deals with preliminaries of some points used in our discussing. Section 3, deals with constructing the stochastic finite difference scheme of eqns (1) and (2). Section 4, deals with the study of stability and consistency of (SFDS) of eqns (1) and (2).

Preliminaries

Definition 1: A real random variable X on a probability space (Ω,F,P) and satisfying the property that equation is called 2nd order random variable (2.r.v) where, E[ ] denotes the expectation value operator [9].

Remark 1:

If X∈L2(Ω), Then the L2 norm is defined as

equation [9]

The approximations of the solution s.p. to eqns (1) and (2), [8], will be constructed in the sense of fixed station for the time, we will work in the following Banach space equation defined by

equation

equation   (3)

Analysis of Stochastic Finite Difference Scheme (SFDS)

The SFDS for stochastic cauchy diffusion eqn (1)

The discretization of the problem eqn (1) will lead to stochastic difference scheme where at each node the unknown represents a r.v. Let us subdivide the domain (–∞,∞) × [0,∞] and define the grid cells for the space to be Δx=(xk–xk–1) for k≥1 and also, define the time steps to be t=(tn–tn–1) for n≥1. Consider equation approximates the exact solution for the problem eqn (1) as, U(x,t) at the point (kΔx,nΔt). By replacing the derivatives in eqn (1) with the difference formulas, we can have the difference scheme as follows:

Using the first-order forward difference formula for approximating Ut

equation

Then,

equation   (4)

Using the first-order backward difference formula for approximating p'(x,ß), Ux and the second-order centred difference formula for approximating Uxx

equation   (5)

Substituting eqns (4) and (5) into eqn (1), we can motivate the scheme

equation

Then,

equation   (6)

The SFDS for stochastic cauchy advection eqn (2)

Now, we will construct the stochastic finite difference scheme by the same approach we use in the diffusion equation. Using eqn (4) for approximating Ut and first-order backward difference formula for approximating p'(x,ß) and Ux.

equation   (7)

Substituting eqns (4) and (7) into eqn (2), we get

equation

Then,

equation   (8)

Mean Square Consistency and Stability of SFDS

Consistency and stability of SFDS eqn (6)

The consistency of numerical scheme is determined by studying the truncation error of the scheme. A finite difference scheme is consistent with the stochastic partial differential equation if the difference between the finite difference equation and the stochastic partial differential equation vanishes as the size of the grid spacing goes to zero independently. If this condition is met, then the numerical scheme is said to be consistent. Consistency of finite difference schemes is determined by using the technique of modified stochastic partial differential equation (MSPDE). This MSPDE is determined by expressing each term in the finite difference equation in a Taylor series at some base point. If the grid points approaches zero and the MSPDE changes back to original SPDE, then the scheme is consistent.

Definition 2: The stochastic finite difference scheme (SFDS)

equation

is said to be mean square equation –consistent with the stochastic partial differential equation (RPDE) Lu=F, if the solution stochastic process (s.p.) of (RPDE), U, satisfies

equation   (9)

equation

Theorem 1: Let us consider the stochastic cauchy problem (1), and assume that

(1) p(x) and p'(x) are functions of the 2-r.v. ß such that 0<m<p(x,ω)<M; m,M are positive constants.

(2) Uxxx(x,t)(ω), Utt(x,t)(ω) are uniformly bounded for (x,t)∈[xk δ,xk+δ]×[tn–δ,tn+δ]. Then, the SFDS (6) is mean square equation – consistent.

Proof: Assuming that for each ω∈Ω, the solution U(x,t)(ω) of (1) admits taylor expansion of the form

equation   (10)

equation   (11)

equation   (12)

According to expression (9) with G=0, and using the expansions (10)-(12), we have

equation

Since p(x,ω) is mean square differentiable, then we have

equation

equation    (13)

Assuming, Uxxx(x,t)(ω), Utt(x,t)(ω) are uniformly bounded for (x,t)∈[xk–δ,xk+δ]×[tn–δ,tn+δ] Then, from eqn (13) we have

equation

Then, from eqn (9) we get

equation

Then,

equation

fß(ß) is the probability density function of the stochastic variable. Hence, the SFDS (6) is mean square consistent.

A finite difference scheme is said to be stable if it produces a bounded solution for a stable stochastic partial differential equation and it is unstable if it produces an unbounded solution of the stochastic partial differential equations. There are several methods used to analyse stability of numerical schemes, for example the discrete perturbation method, the Von-Neumann method and the matrix method. In this work, we use the mean square calculus for stability analysis of numerical scheme.

Definition 3: A stochastic difference scheme equation is mean square stable, if there exist some positive constants ε,δ, non-negative constants l,m and u0 is an initial data such that:

equation   (14)

for all, equation

Theorem 2: The SFDS (6) that according to the problem (1) is mean square stable under the condition:equation , 0<m<p(x,ω)<M where m,M are positive constants.

Proof: Since, our scheme is

equation

equation

Under the hypotheses equation , 0<Δt≤δ and 0<m<p(x,ω)<M, we have equation Then,

equation   (15)

equation

Hence,

equation

Then,

equation

Finally, we have:

equation

Therefore,

equation

Hence, the SFDS (6) is mean square stable with l=1, m=0 and under the conditions eqn (15).

Consistency and stability of SFDS (8)

Taking into account definition eq (2), we can state that theorem:

Theorem 3: The SFDS (8) is mean square equation-consistent Let us consider the random cauchy problem (2), and assume that

(1) p(x) and p'(x) are functions of the 2-r.v. ß such that 0<m<p(x,ω)<M; m,M are positive constants.

(2) Uxxx(x,t)(ω), Utt(x,t)(ω) are uniformly bounded for (x,t)∈[xk– δ,xk+δ]×[tn–δ,tn+δ]

Proof: Assuming that for each ω∈Ω, the solution U(x,t)(ω) of (2) admits taylor expansion of the form

equation   (16)

equation   (17)

According to expression (10) with G=0, and using the expansions (16) and (17), we have

equation

Since p(x,ω) is mean square differentiable, then we have

equation    (18)

Assuming, Uxxx(x,t)(ω), Utt(x,t)(ω) are uniformly bounded for (x,t)∈[xk–δ,xk+δ]×[tn–δ,tn+δ] Then, from eqn (18) we have

Then, from eqn (9) we get

equation

Then,

equation

fß(ß) is the probability density function of the stochastic variable. Hence, the SFDS (8) is mean square consistent.

Now, taking into account definition (3), we can state that theorem

Theorem 4: The SFDS (8) that according to the problem (2) is mean square stable under the condition: equation , 0<m<p(x,ω)<M where m,M are positive constants.

Proof: Since, our scheme is

equation

equation

equation

equation

equation

equation

equation

equation

Under the hypotheses equation , 0<Δt≤δand 0<m<p(x,ω)<M, we have for 0<rM≤1 Then,

equation   (19)

equation

Hence,

equation

Then,

equation

Finally, we have:

equation

Therefore,

equation

Hence, the SFDS (8) is mean square stable with l=1, m=0 and under the conditions eqn (19).

References

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