America Joural of Compuaioal Mahemaics, 04, 4, 80-88 Published Olie Sepember 04 i SciRes. hp://www.scirp.org/joural/ajcm hp://dx.doi.org/0.436/ajcm.04.4404 Mea Square Coverge Fiie Differece Scheme for Sochasic Parabolic PDEs W. W. Mohammed, M. A. Sohaly, A. H. El-Bassiouy, K. A. Elagar Deparme of Mahemaics, Faculy of Sciece, Masoura Uiversiy, Masoura, Egyp Email: wael.mohammed@mas.edu.eg, m_sa000@yahoo.com, el_bassiouy@mas.edu.eg Received 3 Jue 03; revised 8 July 04; acceped 3 Augus 04 Copyrigh 04 by auhors ad Scieific Research Publishig Ic. This wor is licesed uder he Creaive Commos Aribuio Ieraioal Licese (CC BY. hp://creaivecommos.org/liceses/by/4.0/ Absrac Sochasic parial differeial equaios (SPDEs describe he dyamics of sochasic processes depedig o space-ime coiuum. These equaios have bee widely used o model may applicaios i egieerig ad mahemaical scieces. I his paper we use hree fiie differece schemes i order o approximae he soluio of sochasic parabolic parial differeial equaios. The codiios of he mea square covergece of he umerical soluio are sudied. Some case sudies are discussed. Keywords Sochasic Parial Differeial Equaios, Mea Square Sese, Secod Order Radom Variable, Fiie Differece Scheme. Iroducio Sochasic parial differeial equaios (SPDEs frequely arise from applicaios i areas such as physics, egieerig ad fiace. However, i may cases i is difficul o derive a explici form of heir soluio. I rece years, some of he mai umerical mehods for solvig sochasic parial differeial equaios (SPDEs, lie fiie differece ad fiie eleme schemes, have bee cosidered []-[9], e.g. [0]-[], based o a fiie differece scheme i boh space ad ime. I is well ow ha explici ime discreizaio via sadard mehods (e.g., as he Euler-Maruyama mehod leads o a ime sep resricio due o he siffess origiaig from he discreizaio of he diffusio operaor (e.g. he Coura-Friedrichs-Lewy (CFL C( x, where Δ ad Δx are he ime ad space discreizaio, respecively. Mohammed [8] discussed sochasic fiie differece schemes by hree pois uder he followig codiio of sabiliy: 0 rβ = β, β > 0 ad five x ( How o cie his paper: Mohammed, W.W., Sohaly, M.A., El-Bassiouy, A.H. ad Elagar, K.A. (04 Mea Square Coverge Fiie Differece Scheme for Sochasic Parabolic PDEs. America Joural of Compuaioal Mahemaics, 4, 80-88. hp://dx.doi.org/0.436/ajcm.04.4404
pois uder his codiio of sabiliy: 0 rβ =, β > 0. Our aim of his paper is o use he so- β ( x 5 chasic fiie differece schemes by seve pois ha are srog covergeces o our problem ad much beer sabiliy properies ha hree ad five pois. This paper is orgaized as follows. I Secio, some impora prelimiaries are discussed. I Secio 3, he Fiie Differece Scheme wih seve pois for solvig sochasic parabolic parial differeial equaio is discussed. I Secio 4, some case sudies are discussed. The geeral coclusios are preseed i he las secio.. Prelimiaries I his secio we will sae some defiiio from [9] as follow: X,, o Defiiio. A sequece of r.v s { > } coverges i mea square (. lim 0.. ms. X X = ie X X ms o a radom variable X if: Defiiio. A sochasic differece scheme Lu = G approximaig SPDE Lv = G is cosise i Φ=Φ x,, we have i mea square: mea square a ime = (, if for ay differeiable fucio ( ( ( ( E LΦG LΦ x, G 0 ( ( As, x 0, 0 ad x, x,. Defiiio 3. A sochasic differece scheme is sable i mea square if here exis some posiive cosas ε, δ ad cosas, b such ha: ( b 0 e Eu Eu For all 0 =, 0 x ε ad 0 δ. Defiiio 4. A sochasic differece scheme Lu = G approximaig SPDE Lv = G is coverge i =, if: mea square a ime ( Eu u 0 ( ( As,, x 0, 0 ad x, x,. 3. Sochasic Parabolic Parial Differeial Equaio (SPPDE I his secio he sochasic fiie differece mehod is used for solvig he SPPDE. Cosider he followig sochasic parabolic parial differeial equaio i he form: (, β (, σ (, d ( u x = u x u x W ( xx (,0 (, [ 0, ], [ 0, ] u x = u x x X T ( 0 ( ( where W( is a whie oise sochasic process ad βσ, are cosas. 3.. Sochasic Differece Scheme (wih Seve Pois For he Equaios (-(3 he differece scheme is: u0, = u X, = 0 (3 rβ u = u ( u3 7u 70u 490u 70u 7u u 3 σu ( W W, (4 80 0 ( u = u x (5 0, u0 = u x = 0, (6 8
where r = ad i ca be wrie i he form: x ( 49 r u β = rβ u ( u 3 7u 70u 70u 7u u 3 σ u ( W W. 8 80 (7 3... Cosisecy I his subsecio we sudy he cosisecy i mea square sese of he Equaio (7. Theorem 3.. The sochasic differece scheme (7 is cosise i mea square sese. Φ x, be a smooh fucio he: Proof. Assume ha ( he we have: ( ( ( ( ( β xx ( σ ( ( ( L Φ =Φ x, Φ x, Φ x, s d s Φ x, s d W s, rβ L Φ=Φ( x, ( Φ( x, ( Φ( ( 3 x, 7Φ( ( x, 80 70Φ, 490 Φ, 70Φ, 7Φ, (( x ( x (( x (( x ( x σ x s W( W( ( ( ( ( Φ 3, Φ,, ( ( ( β xx ( ( ( EL Φ L Φ = E Φ xs, d s σ Φ xs, dw s sice: he: rβ ( Φ 3, 7Φ, 70Φ 3, 80 (( x (( x (( x ( (( (( ( ( (, (( ( ( 490 Φ x, 70Φ x, 7Φ x, Φ 3 x, σ Φ xs W W = E β ( ( xs, ds (( 3 x, 7 (( x, (( x ( x (( x (( x (( x ( r Φxx ( Φ Φ 80 70Φ, 490 Φ, 70Φ, 7Φ, Φ 3, σ ( ( xs, d W( s ( xs, W( ( W( Φ Φ ( for s S s s s EX Y EX EY =, s ( EL( Φ L Φ β E xx ( xs, ds ( 3 x, Φ Φ 80 x (( (( x (( x ( x 7Φ, 70Φ, 490 Φ, (( x (( x (( x 70Φ, 7Φ, Φ 3, ad from he iequaliy ( σ E ( xs, ( xs, d W( s. Φ Φ 8
Applyig i he par ( s ( 0 ( ( E f sw, dw E f sw, d. s 0 0 ( ( ( ( ( ( ( ( ( E xs, xs, dw s E xs, xs, ds Φ Φ Φ Φ ad ( x, Φ is deermiisic fucio we have: as ime ( ( ( ( = E xs, xs, Φ Φ d, s ( EL( Φ L Φ β E xx ( xs, ds ( 3 x, Φ Φ 80 x =, 0, x 0 (( (( x (( x ( x 7Φ, 70Φ, 490 Φ, 70Φ (( x, 7Φ (( x, Φ (( 3 x, ( σ (, (, d E Φ xs Φ xs s, ad ( x, ( x,, he: ( ( EL Φ LΦ x, 0, hece he sochasic differece scheme (7 is cosise i mea square sese. 3... Sabiliy Theorem 3.. The sochasic differece scheme (7 is sable i mea square sese. Proof. Sice 49 r u β = rβ u ( u 3 7u 70u 70u 7u u 3 σ u ( W W, 8 80 he: 49 r E u β = E rβ u u 7u 70u 70u 7u u u W W. 8 80 Sice { W(., W(., s } ( ( 3 3 σ is ormally disribued wih mea zero ad variace s icremes of he wier process are idepede u ad The we have: ( W W ( W( x, ( W( x, =. 49 rβ 49 = β β 7 70 70 7 ( 3 3 Eu r Eu r Eu u u u u u u 8 90 8 rβ E( u 3 u 3 7( u u 70( u u σ ( Eu 80 ( ( 3 3 ( ( 49 rβ 49 = β β 7 70 Eu r r Eu u u Eu u u Eu u u 8 90 8 rβ E( u 3 u 3 7( u u 70 ( u u σ ( Eu, 80 83
sice: he: he: ( 80 ( 80 ( 80 (, s S s s s EX Y EX EY = s ( 3 3 ( ( E u u 7 u u 70 u u ( = E 4 u u 79 u u 7900 u u 54 u u u u ( 3 3 ( ( ( 3 3( ( u 3 u 3( u u ( u u ( u u 540 790 ( u 3 79( ( u ( u 7900( ( u ( u E 4 ( u 3 ( 54( ( u3u ( u 3u ( u 3u ( u 3u 540( ( u3u ( u 3u ( u 3u ( u 3u 790( ( uu ( u u ( u u ( u u 4 E u ( 3 E( u 3 79 E u E u 7900 E u E u ( 80 ( ( ( ( ( ( ( 54( E( u3u E( u 3u E( u 3u E( u 3u 540( E( u3u E( u 3u E( u 3u E( u 3u 790( E( uu E( u u E( u u E( u u ( ( ( ( 9 9 = E( u E( u E( u E( u E u E u 800 800 400 4 600 60 3 3 ( E( u3u E( u 3u E( u 3u E( u 3u ( E( u3u E( u 3u 9 E( u 3u E( u 3u ( E( uu E( uu ( ( E u u E u u 40 ( ( 3 3 ( 49 rβ 49 β β 7 Eu r Eu r Euu Euu Euu Euu 8 90 8 ( Euu Euu ( ( ( rβ E u E u 3 3 70 800 800 9 9 ( E( u E( u ( E( u E( u 400 4 ( E( u3u E( u 3u E( u 3u E( u 3u 600 9 E( u u E( u u E( u u E( u u ( E u u 60 40 E( u u E( u u E( u u σ ( E u ( 3 3 3 3 ( 49 rβ 49 β sup r Eu rβ sup Eu sup Eu 8 90 8 84
Now wih: he: Hece: 7 sup sup 70 sup sup ( β sup 800 9 9 sup Eu sup Eu sup Eu sup Eu sup Eu 800 400 4 sup Eu sup Eu sup Eu sup Eu 600 sup Eu sup Eu sup Eu sup Eu 60 9 sup Eu sup Eu sup Eu sup Eu σ ( sup Eu 40 49 rβ 49 rβ sup Eu rβ 490 supeu 8 90 8 40 ( rβ sup Eu σ ( sup Eu 648 Eu Eu Eu Eu r Eu 49 49 49 = rβ sup Eu rβ rβ Eu 8 9 8 40 ( rβ sup Eu σ sup Eu 648 sup 49 49 49 40 = rβ rβ rβ ( rβ sup Eu σ ( sup Eu 8 9 8 648 49 49 49 40 = rβ rβ rβ ( rβ sup Eu 8 9 8 34 40 ( rβ sup Eu σ ( sup Eu. 34 0 rβ =, 8 β ( x 49 49 49 rβ = rβ. 8 8 49 49 40 β β ( β σ Eu r r 8 8 Eu r Eu Eu 34 40 = Eu ( rβ Eu σ ( Eu 34 40 = rβ σ ( Eu 34 I is eough o selec λ such ha: ( 40 34 rβ σ λ for all he we pu = o ge: λ 0 λ 0 = Eu Eu e Eu. (8 85
Hece he scheme is codiioally sable wih = ad 0 rβ = β for β > 0. 8 ( x 49 3..3. Covergece Theorem 3.3. The radom differece scheme (7 is coverge i mea square sese Proof. Sice he scheme is cosise he we have as 0, x 0 ad ( x, ( x, b = λ i mea square sese wih he codiio: ( ( = Eu u E L Lu Lu L u ms Lu he we obai ( ( E L Lu Lu 0, ad sice he scheme is sable he ( Eu u 0 as 0, x 0, he he radom differece scheme (7 is coverge i mea square sese. 4. Applicaios Cosider he liear hea equaio wih muliplicaive oise i his form The SFDS wih seve pois is: (, = 0.00 (, d (, [ 0, ], [ 0,] u x u u x w x xx 0 ( = ( ( = ( = u x x x u 0, u, 0. L is bouded hece: 49 rβ u = rβ u ( u3 7u 70u 70u 7u u 3 u ( W W. 8 80 Sice he sabiliy codiio from Theorem 3. is: le M = 50 he he we have: x = herefore: 50 I Theorem 3.8, we assumed ha: 0 rβ = β, β = 0.00 8 ( x 49 50 8 0 0.00 49 4 0 N 6 45 40 ( 34 rβ σ λ. Therefore, for differe amou N, we ca derive low boudary of λ (see Table. 86
Table. λ for sabiliy. N 6 00 500 000 500 6000 λ 60.5506 8.7685 7.5055 3.75565.50708333 0.654 Figure. Sabiliy. 87
O he oher had, i Equaio (8, we had λ 0 λ λ 0 0 Eu Eu e Eu e Eu e, Eu Eu y= =, ad for sabiliy, we represe y for differe N (see Figure. 5. Coclusio ad Fuure Wors The sochasic parabolic parial differeial equaios ca be solved umerically usig he sochasic differece (wih seve pois mehod i mea square sese. More complicaed problems i liear sochasic parabolic parial differeial equaios ca be sudied usig fiie differece mehod i mea square sese. The echiques of solvig oliear sochasic parial differeial equaios usig he fiie differece mehod wih he aid of mea square calculus ca ehace grealy he reame of sochasic parial differeial equaios i mea square sese. Refereces [] Barh, A. (009 Sochasic Parial Differeial Equaios: Approximaios ad Applicaios. Ph.D. Thesis, Uiversiy of Oslo, Oslo. [] Barh, A. (00 A Fiie Eleme Mehod for Marigale-Sochasic Parial Differeial Equaios. Commuicaios o Sochasic Aalysis, 4, 355-375. [3] Cores, J.C. (007 Compuig Mea Square Approximaios of Radom Differeial Models. Mahemaics ad Compuers i Simulaio, 76, 44-48. hp://dx.doi.org/0.06/j.macom.007.0.00 [4] Corés, J.C., Jódar, L., Villaueva, R.-J. ad Villafuere, L. (00 Mea Square Coverge Numerical Mehods for Noliear Radom Differeial Equaios. Trasacios o Compuaioal Sciece, 7, -. [5] Corés, J.C., Jódar, L., Villafuere, L. ad Rafael, R.J. (007 Compuig Mea Square Approximaios of Radom Diffusio Models wih Source Term. Mahemaics ad Compuers i Simulaio, 76, 44-48. [6] Gardo, A. (004 The Order of Approximaio for Soluios of Iô-Type Sochasic Differeial Equaios wih Jumps. Sochasic Aalysis ad Applicaios,, 679-699. [7] Kha, I.R., Ohba, R. ad Hozumi, N. (003 Mahemaical Proof of Closed Form Expressios for Fiie Differece Approximaios Based o Taylor Series. Joural of Compuaioal ad Applied Mahemaics, 50, 303-309. hp://dx.doi.org/0.06/s0377-047(000667- [8] Mohammed, A.S. (04 Mea Square Coverge Three ad Five Pois Fiie Differece Scheme for Sochasic Parabolic Parial Differeial Equaios. Elecroic Joural of Mahemaical Aalysis ad Applicaios,, 64-7. [9] Soog, T.T. (973 Radom Differeial Equaios i Sciece ad Egieerig. Academic Press, New Yor. [0] Alaber, A. ad Gyogy, I. (006 O Numerical Approximaio of Sochasic Burgers Equaio. I: Kabaov, Y., Lipser, R. ad Soyaov, J., Eds., From Sochasic Calculus o Mahemaical Fiace, Spriger, Berli, -5. hp://dx.doi.org/0.007/978-3-540-30788-4_ [] Davie, A.M. ad Gaies, J.G. (00 Covergece of Numerical Schemes for he Soluio of Parabolic Sochasic Parial Differeial Equaios. Mahemaics of Compuaio, 70, -34. hp://dx.doi.org/0.090/s005-578-00-04- [] Priems, J. (00 O he Discreizaio i Time of Parabolic Sochasic Parial Differeial Equaios. ESAIM: Mahemaical Modellig ad Numerical Aalysis, 35, 055-078. hp://dx.doi.org/0.05/ma:0048 88