American Journal of Computational Mathematics, 2014, 4, 280-288 Published Online September 2014 in SciRes. http://www.scirp.org/journal/ajcm http://dx.doi.org/10.4236/ajcm.2014.44024

How to cite this paper: Mohammed, W.W., Sohaly, M.A., El-Bassiouny, A.H. and Elnagar, K.A. (2014) Mean Square Conver-gent Finite Difference Scheme for Stochastic Parabolic PDEs. American Journal of Computational Mathematics, 4, 280-288. http://dx.doi.org/10.4236/ajcm.2014.44024

Mean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs W. W. Mohammed, M. A. Sohaly, A. H. El-Bassiouny, K. A. Elnagar Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt Email: wael.mohammed@mans.edu.eg, m_stat2000@yahoo.com, el_bassiouny@mans.edu.eg

Received 23 June 2013; revised 28 July 2014; accepted 3 August 2014

Copyright © 2014 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract Stochastic partial differential equations (SPDEs) describe the dynamics of stochastic processes depending on space-time continuum. These equations have been widely used to model many ap-plications in engineering and mathematical sciences. In this paper we use three finite difference schemes in order to approximate the solution of stochastic parabolic partial differential equations. The conditions of the mean square convergence of the numerical solution are studied. Some case studies are discussed.

Keywords Stochastic Partial Differential Equations, Mean Square Sense, Second Order Random Variable, Finite Difference Scheme

1. Introduction Stochastic partial differential equations (SPDEs) frequently arise from applications in areas such as physics, en-gineering and finance. However, in many cases it is difficult to derive an explicit form of their solution. In re-cent years, some of the main numerical methods for solving stochastic partial differential equations (SPDEs), like finite difference and finite element schemes, have been considered [1]-[9], e.g. [10]-[12], based on a finite difference scheme in both space and time. It is well known that explicit time discretization via standard methods (e.g., as the Euler-Maruyama method) leads to a time step restriction due to the stiffness originating from the discretization of the diffusion operator (e.g. the Courant-Friedrichs-Lewy (CFL) ( )2t C x∆ ≤ ∆ , where Δt and Δx are the time and space discretization, respectively). Mohammed [8] discussed stochastic finite difference

schemes by three points under the following condition of stability: ( )2

10 ,2

rxtβ β

∆

∆≤ = ≤ 0β > and five

W. W. Mohammed et al.

281

points under this condition of stability: ( )2

20 ,5

rxtβ β

∆

∆≤ = ≤ 0.β > Our aim of this paper is to use the sto-

chastic finite difference schemes by seven points that are strong convergences to our problem and much better stability properties than three and five points.

This paper is organized as follows. In Section 2, some important preliminaries are discussed. In Section 3, the Finite Difference Scheme with seven points for solving stochastic parabolic partial differential equation is dis-cussed. In Section 4, some case studies are discussed. The general conclusions are presented in the last section.

2. Preliminaries In this section we will state some definition from [9] as follow:

Definition 1. A sequence of r.v’s { }, ,nkX n k o> converges in mean square ( ).m s to a random variable X if: .lim 0 . . m s

nk nknkX X i e X X

→∞− = →

Definition 2. A stochastic difference scheme n n nk k kL u G= approximating SPDE Lv G= is consistent in

mean square at time ( )1t n t= + ∆ , if for any differentiable function ( ),x tΦ = Φ , we have in mean square:

( ) ( )( )( ) ( )

2, 0

As , 0, 0 and , , .

n n nk kkE L G L k x n t G

k n x t k x n t x t

Φ− − Φ ∆ ∆ − →

→∞ →∞∆ → ∆ → ∆ ∆ →

Definition 3. A stochastic difference scheme is stable in mean square if there exist some positive constants ε , δ and constants k, b such that:

( )

2 21 0e

For all 0 1 , 0 and 0 .

n btkE u k E u

t n t x tε δ

+ ≤

≤ = + ∆ ≤ ∆ ≤ ≤ ∆ ≤

Definition 4. A stochastic difference scheme n n nk k kL u G= approximating SPDE Lv G= is convergent in

mean square at time ( )1t n t= + ∆ , if:

( ) ( )

20

As , , 0, 0 and , , .

nkE u u

n k x t k x n t x t

− →

→∞ →∞ ∆ → ∆ → ∆ ∆ →

3. Stochastic Parabolic Partial Differential Equation (SPPDE) In this section the stochastic finite difference method is used for solving the SPPDE. Consider the following stochastic parabolic partial differential equation in the form:

( ) ( ) ( ) ( ), , , dt xxu x t u x t u x t W tβ σ= + (1)

( ) ( ) [ ] [ ]0,0 , 0, , 0,u x u x x X t T= ∈ ∈ (2)

( ) ( )0, , 0u t u X t= = (3)

where ( )W t is a white noise stochastic process and ,β σ are constants.

3.1. Stochastic Difference Scheme (with Seven Points) For the Equations (1)-(3) the difference scheme is:

( ) ( )1 13 2 1 1 2 32 27 270 490 270 27 2 ,

180n n n n n n n n n n n nk k k k k k k k k k k k

ru u u u u u u u u tu W Wβ σ+ +− − − + + += + − + − + − + + ∆ − (4)

( )00 ,k ku u x= (5)

0 0n nxu u= = , (6)

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282

where ( )2r

xt∆

=∆

and it can be written in the form:

( ) ( )1 13 2 1 1 2 3

491 2 27 270 270 27 2 .18 180

n n n n n n n n n n nk k k k k k k k k k k

ru r u u u u u u u tu W Wββ σ+ +− − − + + +

= − + − + + − + + ∆ −

(7)

3.1.1. Consistency In this subsection we study the consistency in mean square sense of the Equation (7).

Theorem 3.1. The stochastic difference scheme (7) is consistent in mean square sense. Proof. Assume that ( ),x tΦ be a smooth function then:

( ) ( )( ) ( ) ( )( ) ( ) ( )( )1 1, 1 , , d , d ,

n t n tnxxk n t n t

L k x n t k x n t k x s s k x s W sβ σ+ ∆ + ∆

∆ ∆Φ = Φ ∆ + ∆ −Φ ∆ ∆ − Φ ∆ − Φ ∆∫ ∫

( )( ) ( ) ( )( ) ( )( )(( )( ) ( ) ( )( ) ( )( )

( )( )) ( ) ( )( ) ( )( )

, 1 , 2 3 , 27 2 ,180

270 1 , 490 , 270 1 , 27 2 ,

2 3 , , 1 ,

nk

rL k x n t k x n t k x n t k x n t

k x n t k x n t k x n t k x n t

k x n t t k x s W n t W n t

β

σ

Φ = Φ ∆ + ∆ −Φ ∆ ∆ − Φ − ∆ ∆ − Φ − ∆ ∆

+ Φ + ∆ ∆ − Φ ∆ ∆ + Φ + ∆ ∆ − Φ + ∆ ∆

+ Φ + ∆ ∆ − ∆ Φ ∆ + ∆ − ∆

then we have:

( ) ( )( ) ( ) ( )( )

( )( ) ( )( ) ( )( )(( ) ( )( ) ( )( ) ( )( ))( ) ( )( ) ( )( )

2 1 1

2

, d , d

2 3 , 27 2 , 270 3 ,180490 , 270 1 , 27 2 , 2 3 ,

, 1

n t n tn nk xxk n t n t

E L L E k x s s k x s W s

r k x n t k x n t k x n t

k x n t k x n t k x n t k x n t

t k x s W n t W n t

E

β σ

β

σ

β

+ ∆ + ∆

∆ ∆Φ − Φ = − Φ ∆ − Φ ∆

+ Φ − ∆ ∆ − Φ − ∆ ∆ + Φ + ∆ ∆

− Φ ∆ ∆ + Φ + ∆ ∆ − Φ + ∆ ∆ + Φ + ∆ ∆

+ ∆ Φ ∆ + ∆ − ∆

−=

∫ ∫

( ) ( )( ) ( )( )(( )

( )( ) ( ) ( )( )

( )( ) ( )( )

( ) ( ) ( ) ( )( ) ( )( )( )

1

21

, d 2 3 , 27 2 ,180

270 1 , 490 , 270 1 ,

27 2 , 2 3 ,

, d , 1

n txxn t

n t

n t

rk x s s k x n t k x n t

k x n t k x n t k x n t

k x n t k x n t

k x s W s t k x s W n t W n tσ

+ ∆

∆

+ ∆

∆

Φ ∆ − Φ − ∆ ∆ − Φ − ∆ ∆

+ Φ + ∆ ∆ − Φ ∆ ∆ + Φ + ∆ ∆

− Φ + ∆ ∆ + Φ + ∆ ∆

Φ ∆ −∆ Φ ∆ + ∆ − ∆ −

∫

∫

since:

( ) 1

1 1 for

2 1S s s

s

sE X Y k E X E Y k

s−

≤+ ≤ + = ≥

,

then:

( ) ( )( ) ( )( )(( )( ) ( )( ) ( )

( )( ) ( )( ) ( )( )

( ) ( ) ( )( )

2 122

2

212

2 , d 2 3 ,180

27 2 , 270 1 , 490 ,

270 1 , 27 2 , 2 3 ,

2 , , d .

n tn nk xxk n t

n t

n t

tE L L E k x s s k x n tx

k x n t k x n t k x n t

k x n t k x n t k x n t

E k x s k x s W s

β

σ

+ ∆

∆

+ ∆

∆

∆Φ − Φ ≤ Φ ∆ − Φ − ∆ ∆ ∆

− Φ − ∆ ∆ + Φ + ∆ ∆ − Φ ∆ ∆

+ Φ + ∆ ∆ − Φ + ∆ ∆ + Φ + ∆ ∆

+ Φ ∆ −Φ ∆

∫

∫

and from the inequality

W. W. Mohammed et al.

283

( ) ( ) ( ) ( )0 0

221

0, d 2 1 , d .n

t tn nn

st t

E f s w W t t n n E f s w s− ≤ − − ∫ ∫

Applying in the part

( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )

( ) ( )( )

21 1 2

1 2

, , d 1 1 2 1 , , d

, , d ,

n t n t

n t n t

n t

n t

E k x s k x s W s n n t E k x s k x s s

t E k x s k x s s

+ ∆ + ∆

∆ ∆

+ ∆

∆

Φ ∆ −Φ ∆ ≤ + − ∆ − Φ ∆ −Φ ∆

= ∆ Φ ∆ −Φ ∆

∫ ∫

∫

and ( ),x tΦ is deterministic function we have:

( ) ( )( ) ( )( )(

( )( ) ( )( ) ( )

( )( ) ( )( ) ( )( )

( ) ( )( )

2 122

2

1 22

2 , d 2 3 ,180

27 2 , 270 1 , 490 ,

270 1 , 27 2 , 2 3 ,

2 , , d

n tn nk xxk n t

n t

n t

tE L L E k x s s k x n tx

k x n t k x n t k x n t

k x n t k x n t k x n t

t E k x s k x s s

β

σ

+ ∆

∆

+ ∆

∆

∆Φ − Φ ≤ Φ ∆ − Φ − ∆ ∆ ∆

− Φ − ∆ ∆ + Φ + ∆ ∆ − Φ ∆ ∆

+ Φ + ∆ ∆ − Φ + ∆ ∆ + Φ + ∆ ∆

+ ∆ Φ ∆ −Φ ∆

∫

∫

as time ( )1t n t= + ∆ , 0t∆ → , 0x∆ → , and ( ) ( ), ,k x n t x t∆ ∆ → , then:

( ) ( )2

, 0,n nkkE L L k x n tΦ − Φ ∆ ∆ →

hence the stochastic difference scheme (7) is consistent in mean square sense.

3.1.2. Stability Theorem 3.2. The stochastic difference scheme (7) is stable in mean square sense.

Proof. Since

( ) ( )1 13 2 1 1 2 3

491 2 27 270 270 27 2 ,18 180

n n n n n n n n n n nk k k k k k k k k k k

ru r u u u u u u u tu W Wββ σ+ +− − − + + +

= − + − + + − + + ∆ −

then:

( ) ( )2

21 13 2 1 1 2 3

491 2 27 270 270 27 2 .18 180

n n n n n n n n n n nk k k k k k k k k k k

rE u E r u u u u u u u tu W Wββ σ+ +− − − + + +

= − + − + + − + + ∆ −

Since ( ) ( ){ }., ., W t W s− is normally distributed with mean zero and variance t s− increments of the winner process are independent n

ku and

( ) ( )( ) ( )( )1 , 1 ,n nk kW W W k x n t W k x n t+ − = ∆ + ∆ − ∆ ∆ .

Then we have:

( )

( ) ( ) ( ) ( )

22 21

3 2 1 1 2 3

22 222

3 3 2 2 1 1

49 491 1 2 27 270 270 27 218 90 18

2 27 270180

n n n n n n n n nk k k k k k k k k

n n n n n n nk k k k k k k

rE u r E u r E u u u u u u u

r E u u u u u u t E u

ββ β

β σ

+− − − + + +

− + − + − +

= − + − − + + − +

+ + − + + + + ∆

( ) ( ) ( )( )

( ) ( ) ( ) ( )

221

3 3 2 2 1 1

22 222

3 3 2 2 1 1

49 491 1 2 27 27018 90 18

2 27 270 ,180

n n n n n n n n n nk k k k k k k k k k

n n n n n n nk k k k k k k

rE u r r E u u u E u u u E u u u

r E u u u u u u t E u

ββ β

β σ

+− + − + − +

− + − + − +

= − + − + − + + +

+ + − + + + + ∆

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284

since:

( ) 1

1 1,

2 1S s s

s

sE X Y k E X E Y k

s−

≤+ ≤ + = ≥

then:

( )( ) ( ) ( )

( )( ) ( ) ( ) ( )( )(

( )( ) ( )( ))

( )( )

2

3 3 2 2 1 12

2 2 2

3 3 2 2 1 1 3 3 2 22

3 3 1 1 2 2 1 1

2

32

1 2 27 270180

1 4 729 72900 54180

540 7290

1 4180

n n n n n nk k k k k k

n n n n n n n n n nk k k k k k k k k k

n n n n n n n nk k k k k k k k

nk

E u u u u u u

E u u u u u u u u u u

u u u u u u u u

E u

− + − + − +

− + − + − + − + − +

− + − + − + − +

−

+ − + + +

= + + + + + − + +

+ + + − + +

≤ ( )( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )(

( )) ( ) ( ) ( ) ( )( )

( )

2 2 2 2 22 23 2 2 1 1

3 2 3 2 3 2 3 2 3 1 3 1 3 1

3 1 2 1 2 1 2 1 2 1

2

729 72900

54 540

7290

1 4180

n n nk k k k k

n n n n n n n n n n n n n nk k k k k k k k k k k k k k

n n n n n n n n n nk k k k k k k k k k

k

u u u u u

u u u u u u u u u u u u u u

u u u u u u u u u u

E u

+ − + − +

− − − + + − + + − − − + + −

+ + − − − + + − + +

+ + + + +

− + + + + + +

+ − + + +

≤ ( ) ( )( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( )(

( ) ( )) ( ) ( ) ( )

2 2 2 2 2 2

3 3 2 2 1 1

3 2 3 2 3 2 3 2 3 1 3 1

3 1 3 1 2 1 2 1 2 1

729 72900

54 540

7290

n n n n n nk k k k k

n n n n n n n n n n n nk k k k k k k k k k k k

n n n n n n n n n nk k k k k k k k k k k

E u E u E u E u E u

E u u E u u E u u E u u E u u E u u

E u u E u u E u u E u u E u u E u

− + − + − +

− − − + + − + + − − − +

+ − + + − − − + + −

+ + + + +

− + + + + +

+ + − + + + ( )( )

( ) ( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( )(

( ) ( )) ( )

2 1

2 2 2 2 2 2

3 3 2 2 1 1

3 2 3 2 3 2 3 2 3 1 3 1

3 1 3 1 2 1 2

1 1 9 98100 8100 400 4

1 1600 60

940

n nk

n n n n n nk k k k k k

n n n n n n n n n n n nk k k k k k k k k k k k

n n n n n n nk k k k k k k

u

E u E u E u E u E u E u

E u u E u u E u u E u u E u u E u u

E u u E u u E u u E u

+ +

− + − + − +

− − − + + − + + − − − +

+ − + + − − −

= + + + + +

− + + + + +

+ + − + ( ) ( ) ( )( )1 2 1 2 1n n n n nk k k k ku E u u E u u+ + − + ++ +

then:

( ) ( )(( )) ( ) ( ) ( )

( ) ( )( ) ( ) ( )( )( )

22 21

3 3 2 2

2 221 1 3 3

2 2 2 2

2 2 1 1

3 2

49 491 1 2 2718 90 18

1 12708100 8100

9 9400 41

600

n n n n n n n n n nk k k k k k k k k k

n n n n n nk k k k k k

n n n nk k k k

n nk k

rE u r E u r E u u E u u E u u E u u

E u u E u u r E u E u

E u E u E u E u

E u u E

ββ β

β

+− + − +

− + − +

− + − +

− −

≤ − + − + − +

+ + + +

+ + + +

− +

( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( )(

( ) ( ) ( )) ( )

3 2 3 2 3 2

3 1 3 1 3 1 3 1 2 1

2222 1 2 1 2 1

22

1 960 40

49 49 1 118

s9 1

p8

u0

n n n n n nk k k k k k

n n n n n n n n n nk k k k k k k k k k

n n n n n n nk k k k k k k

n

kk

u u E u u E u u

E u u E u u E u u E u u E u u

E u u E u u E u u t E u

rr E u r

σ

ββ β

− + + − + +

− − − + + − + + − −

− + + − + +

+ +

+ + + + −

+ + + + ∆

≤ − + −

2 2sup su2 p

k k

n nk kE u E u +

W. W. Mohammed et al.

285

( )2 2 2 2 22

2 2 2 2 2

2 2 2

127 2708100

1 9 98100 400

sup sup sup sup sup

sup sup sup sup sup

sup sup sup

41 sup

600

n n n n nk k k k k

n n n n nk k k k k

n n n nk k k k

k k k k k

k k k k k

k k k k

E u E u E u E u r E u

E u E u E u E u E u

E u E u E u E u

β − + + + + + + + + +

− + + +

( )

( )

2

2 2 2 2

2 2 2 2 222

22 2

2

160940

49 49 1 1 4901

sup sup sup sup

sup sup sup sup sup

sup8 90 1

s p

24

u8

n n n nk k k k

n n n

k k k k

k k k k k

k k

n nk k k k k

n nk k

E u E u E u E u

E u E u E u E u t E u

rr E u r E u

r

σ

ββ β

β

+ + + +

− + + + + ∆

≤ − + −

+ ( )

( )

( )

2 222

22 2

2 22 2

22

01648

49 49 49 1 118 9 18

24

sup sup

sup sup

sup s01648

49 49 49 2401 1 118 9 18 48

up

6

k k

k k

k

n nk k

n nk k

n nk k

k

E u t E u

r E u r r E u

r E u t E u

r r r r

σ

β β β

β σ

β β β β

+ ∆

= − + −

+ + ∆

= − + − +

( )

( )

( ) ( )

2 222

222

2 22 22

49 49 49 2401 1 118 9 18 324

2401 .3

sup sup

sup

sup su24

p

k k

k

k k

n nk k

nk

n nk k

E u t E u

r r r r E u

r E u t E u

σ

β β β β

β σ

+ ∆

= − + − +

+ + ∆

Now with:

( )2

180 ,49

trx

β β∆≤ = ≤

∆

then: 49 491 1 .18 18

r rβ β− = −

Hence:

( )

( ) ( )

( )

22 2 2 221 2

2 2 22 22

22 22

49 49 2401118 18 324

2401324

24011324

n n n nk k k k

n n nk k k

nk

E u r r E u r E u tE u

E u r E u t E u

r t E u

β β β σ

β σ

β σ

+ ≤ − + + + ∆

= + + ∆

= + + ∆

It is enough to select λ such that: ( )2 22 22401324

r t tβ σ λ+ ∆ ≤ ∆ for all k then we put 1

ttn

∆ =+

to get:

2122 2 21 0 01 e .

1

nn tk k k

tE u E u E un

λλ+

+ ≤ + = +

(8)

W. W. Mohammed et al.

286

Hence the scheme is conditionally stable with 1k = and 2b λ= in mean square sense with the condition:

( )2

180 for 0.49

trx

β β β∆≤ = ≤ >

∆

3.1.3. Convergence Theorem 3.3. The random difference scheme (7) is convergent in mean square sense

Proof.

( ) ( )212.n n n n n

k k k k kE u u E L L u L u−

− = −

Since the scheme is consistent then we have msn nk kL u Lu→ then we obtain

( ) ( )21

0,n n n nk k k kE L L u L u

−− →

as 0t∆ → , 0x∆ → and ( ) ( ), ,k x n t x t∆ ∆ → and since the scheme is stable then ( ) 1nkL

− is bounded hence:

20 as 0, 0,n

kE u u t x− → ∆ → ∆ →

then the random difference scheme (7) is convergent in mean square sense.

4. Applications Consider the linear heat equation with multiplicative noise in this form

( ) ( ) ( ) [ ] [ ]( ) ( )( ) ( )

220

, 0.001 , d , 0,1 , 0,1

1

0, 1, 0

t xxu x t u u x t w t t x

u x x x

u t u t

= − ∈ ∈

= −

= =

The SFDS with seven points is:

( ) ( )1 2 2 2 13 2 1 1 2 3

491 2 27 270 270 27 2 .18 180

n n n n n n n nk k k k k k k k k k k

ru r u u u u u u u tu W Wββ+ +− − − + + +

= − + − + + − + − ∆ −

Since the stability condition from Theorem 3.2 is:

( )2180 , 0.00149

trx

β β β∆≤ = ≤ =

∆

let 150M = then 1150

x∆ = therefore:

2180 0.001491

150

t∆≤ ≤

then we have:

40 62245

t N≤ ∆ ≤ ⇒ ≥

In Theorem 3.8, we assumed that:

( )2 22 22401 .324

r t tβ σ λ+ ∆ ≤ ∆

Therefore, for different amount N, we can derive low boundary of 2λ (see Table 1).

W. W. Mohammed et al.

287

Table 1. λ2 for stability.

N 62 200 500 1000 1500 6000

2λ 60.52520161 18.7628125 7.505125 3.7525625 2.501708333 0.62542

Figure 1. Stability.

W. W. Mohammed et al.

288

On the other hand, in Equation (8), we had

2 2 2

2 21 12 21 0

2 20 0e e e , ,

1

n nk kn t t t

k k

k k

E u E u tE u E u y tnE u E u

λ λ λ+ +

+ ≤ ⇒ ≤ ⇒ = ≤ ∆ =+

and for stability, we represent y for different N (see Figure 1).

5. Conclusion and Future Works The stochastic parabolic partial differential equations can be solved numerically using the stochastic difference (with seven points) method in mean square sense. More complicated problems in linear stochastic parabolic par-tial differential equations can be studied using finite difference method in mean square sense. The techniques of solving nonlinear stochastic partial differential equations using the finite difference method with the aid of mean square calculus can enhance greatly the treatment of stochastic partial differential equations in mean square sense.

References [1] Barth, A. (2009) Stochastic Partial Differential Equations: Approximations and Applications. Ph.D. Thesis, University

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