Numerical Solution of Coupled System of Nonlinear Partial Differential Equations Using Laplace-Adomian Decomposition Method

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1 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c Numerical Solutio of Coupled Sytem of Noliear Partial Differetial Equatio Uig Laplace-Adomia Decompoitio Method ABSTRACT Mohamed S. M. Bahgat Departmet of Mathematic, Faculty of Sciece, Miia Uiverity, Egypt mohamed.bahagat@mu.edu.eg Or mmbahgat66@hotmail.com.aim of the paper i to ivetigate applicatio of Laplace Adomia Decompoitio Method (LADM) o oliear phyical problem. Some coupled ytem of o-liear partial differetial equatio (NLPDE) are coidered ad olved umerically uig LADM. The reult obtaied by LADM are compared with thoe obtaied by tadard ad modified Adomia Decompoitio Method. The behavior of the umerical olutio i how through graph. It i oberved that LADM i a effective method with high accuracy with le umber of compoet. Mathematic Subject Claificatio: 45J05, 45B05, 45D05 ad 45A05 Keyword: Laplace Adomia Decompoitio Method; Adomia' Polyomial; Coupled partial differetial equatio.. INTRODUCTION The partial differetial equatio (PDE) have o may eetial applicatio of ciece ad egieerig uch a wave propagatio, hallow water wave, fluid mechaic, thermodyamic, chemitry ad micro electro mechaic ytem, etc. It i difficult to hadle oliear part of thee ytem, although mot of the cietit applied umerical method to fid the olutio of thee ytem that baed o liearizatio, perturbed ad dicretizatio. Debath [] applied the characteritic method ad Loga [] ued the Riema ivariat method to hadle ytem of PDE. Wazwaz [3] ued the Adomia decompoitio method (ADM) to hadle the ytem of PDE. Laplace Decompoitio Method (LDM) i free of ay mall or large parameter ad ha advatage over other approximatio techique like perturbatio, LDM require o dicretizatio ad liearizatio, therefore, reult obtaied by LDM are more efficiet ad realitic. Thi method ha bee ued to obtai approximate olutio of a cla of oliear ordiary ad PDE [4-7]. I thi paper, we compute umerical olutio to oliear ytem of PDE by uig LADM. The umerical olutio become eaier ad higher accuracy tha the tadard Adomia Decompoitio Method (ADM).. LADM for NONLINEAR COUPLED of PDE Coider the geeral oliear coupled of PDE writte i a operator form (ee [8]) Lt ( u ) R( u) M ( u) N ( u, v ) f ( x, t ) L ( v ) R ( v ) M ( v ) N ( u, v ) f ( x, t ) t () Subject to the iitial coditio u ( x,0) g( x ), 0 x, v ( x,0) g ( x ), 0 x, () where ad are real cotat ad the otatio of Lt, R ad R ymbolized the liear patial t differetial operator, the otatio M, M, N ad N ymbolized the oliear differetial operator ad f ( x, t ), f ( x, t ) are give fuctio. The method coit of firt applyig the Laplace traform to both ide of equatio i ytem () ad the by uig iitial coditio, we have g( x) { u} { f ( x, t )} { R( u )} { M ( u )} { N ( u, v )} g( x) { v } { f ( x, t )} { R( v )} { M ( v )} { N ( u, v )} i the Laplace decompoitio method we aume the olutio i i a ifiite erie, give a follow (3) 6530 P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

2 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c u ( x, t ) u k ( x, t ), k 0 v ( x, t ) v k ( x, t ), k 0 where uk( x, t ) ad v k( x, t ) are to be recurively computed. Alo the oliear term M, M, N ad N are decompoed a a ifiite erie of Adomia polyomial (ee [9,0]), (5) k 0 k 0 M ( u ) A, N ( u, v ) B, For the oliear operator oliearity. They are defied by the followig relatio M (u), ad N (u, v) the Adomia' polyomial ca be geerated for all form of A ( u, u,, u ) M u ( x, t ), d k 0 0 k! d k 0 0 B ( u, u,, u ; v, v,, v ) N u ( x, t ), v ( x, t ). d k k k k! d k 0 k 0 0 Subtitutig (4) ad (6) ito (3) ad applyig the liearity of the Laplace traform, we get the followig recurively formula g( x) { u0( x, t )} { f ( x, t )} g( x) { v 0( x, t )} { f ( x, t )} ad { uk ( x, t )} R( uk ( x, t )) Ak B k, k 0 { v k ( x, t )} R( v k ( x, t ) C k Dk, k 0 Applyig the ivere Laplace traform, we ca evaluate u ( x, t ) ad v ( x, t ) 3. APPLICATIONS k I order to verify umerically whether the propoed methodology lead to the accurate olutio, we evaluate LADM uig the approximatio for ome example of o-liear ytem of PDE. To how the efficiecy of the preet method for our problem i compario with the exact olutio, we report the abolute error. The calculatio i thi paper have bee doe uig the Maple 8 oftware. The reult are lited i table - ad figure -4 below. Example : Coider the coupled ytem of oliear PDE of the form [] k (4) (6) (7) (8) pt v xw y v yw x - p v t w x p y pxw y v w t pxv y p yv x w with the followig iitial coditio (9) 653 P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

3 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c xy p( x, y,0) e, xy v ( x, y,0) e yx w ( x, y,0) e. (0) The exact olutio of the ytem (9) i []. x y t p( x, y, t ) e ; x y t v ( x, y, t ) e ; y x t w ( x, y, t ) e Solutio: To olve the ytem of equatio (9)-(0) by mea of LADM, we cotruct a correctioal fuctioal which read { p} p( x, y,0) p v w v w { v } v ( x, y,0) v w x p y pxw y {w } w( x, y,0) w pxv y p yv x We ca defie the Adomia polyomial a follow x y y x () A v k ( x, y, t ) w k ( x, y, t ), k 0x y B v k ( x, y, t ) w k ( x, y, t ), k 0y x C w k ( x, y, t ) p k ( x, y, t ), k 0x y E p k ( x, y, t ) w k ( x, y, t ), k 0x y F p k ( x, y, t ) v k ( x, y, t ), k 0x y G p k ( x, y, t ) v k ( x, y, t ). k 0y x We defie a iterative cheme () 653 P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

4 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c xy { p0( x, y, t )} e xy { v 0( x, y, t )} e yx { w 0( x, y, t )} e { p ( x, y, t )} p ( x, y, t ) A B, 0 { v ( x, y, t )} v ( x, y, t ) C E, 0 { w ( x, y, t )} w ( x, y, t ) F G, 0 the Adomia' polyomial A, B, C, E, F ad Adomia' polyomial a follow A0, A t, A t t, A3 t t t B0, B t, B t t, B3 t t t y y y 4 C 0 e, C e ( t ), C e ( t 4t 4), 4 y 6 4 y y C 3 e ( t 9t 36t 36), E 0 e, E e ( t ), 36 y 4 y 6 4 E e ( t 4t 4), E 3 e ( t 9t 36t 36), 4 36 x x y 4 F0 e, F e ( t ), F e ( t 4t 4), 4 x 6 4 x x F3 e ( t 9t 36t 36), G0 e, G e ( t ), 36 x 4 x 6 4 G e ( t 4t 4), G3 e ( t 9t 36t 36), 4 36 Applyig the ivere Laplace traform, fially, Accordig to () the 0 th compoet x y x y y x p0( x, y, t ) e, v 0( x, y, t ) e ad w 0( x, y, t ) e x y x y y x p0( x, y, t ) e, v 0( x, y, t ) e, w 0( x, y, t ) e compoet: (3) G are geerated accordig to (6), we ca give the firt few writte a follow:, where, k 0 x y x y y x p ( x, y, t ) e t, v ( x, y, t ) e t, w ( x, y, t ) e t, p x y t v x y t w x y t t p x y t v x y t w x y t t x y x y y x (,, ) e t, (,, ) e t, (,, ) e,, x y 3 x y 3 y x 3 3(,, ) e t, 3(,, ) e t, 3(,, ) e. So, we get the followig 6533 P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

5 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c xy 30 p30( x, y, t ) e t, xy 30 v 30( x, y, t ) e t, yx 30 w 0( x, y, t ) e t, Similarly, we ca alo fid other compoet, ad the approximate olutio for calculatig 50 th. Uig (4), the erie olutio are therefore give by xy p( x, y, t ) e ( t 5t 0t 60t 0t 0 ), 0 xy v ( x, y, t ) e ( t 5t 0t 60t 0t 0 ), 0 yx v ( x, y, t ) e ( t 5t 0t 60t 0t 0 ). 0 t t By uig the Taylor expaio for e ad e we ca fid the exact olutio. The umerical behavior of approximate olutio of LADM of p( x, y, t ), v ( x, y, t ) ad w ( x, y, t ) with differet value of time are compared with the exact olutio are how through figure(a)-(c) p(x,y,t) LADM (a) p(x,y,t) exact v(x,y,t) LADM (b) v(x,y,t) exact 6534 P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

6 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c w(x,y,t) LADM (c) w(x,y,t) exact Figure. (a) Exact ad umerical olutio of p(x,y, t), 5 x 5, 5 y 5 for differet value of t; (b) Exact ad umerical olutio of v(x,y, t), 5 x 5, 5 y 5 for differet value of t; (c) Exact ad umerical olutio of w(x,y, t), 5 x 5, 5 y 5 for differet value of t Example : The coupled Burger ytem u u uu ( uv ) =0 t xx x x v v vv +( uv ) =0 t xx x x (4) where q q( x, t ), q = q( x, t ), with iitial coditio u( x,0) i x, v ( x,0) i x. The exact olutio of the ytem (4) i u( x,t) v ( x, t ) i x e t (ee [] ad [] ). Solutio: To olve the ytem of equatio (4) by mea of LADM, we cotruct a correctioal fuctioal which read { u} u ( x,0) u uu ( uv ) { v } v ( x,0) v vv ( uv ) xx x x xx x x We ca defie the Adomia polyomial a follow A C E k k k k 0x k 0 x k k k k 0x k 0x k 0 u ( x, t ), B u ( x, t ) u ( x, t ), u ( x, t ) v ( x, t ), F v ( x, t ) v k ( x, t ) v k ( x, t ), x We defie a iterative cheme { u0( x, t )} i x { v 0( x, t )} i x { u ( x, t )} A B C, 0 { v ( x, t )} F E C, 0. Applyig the ivere Laplace traform, fially, the Adomia' polyomial A, B, C, E ad accordig to (6), we ca give the firt few compoet of the Adomia' polyomial repectively a follow 6535 P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06 (5) (6) (7) F are geerated

7 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c A i x, B i x co x, C i x co x, E i x co x, F i x, A i x ( t ), B 4t i x co x, C ( t ) i x co x, E 4t i x co x, F i x ( t ), A i x t it i x (3t t (( t 3t 3) co x 3)) 6 t co x i x ( t 3t 3), B i x ( co x (3t t (( t 3t 3) co x 3)) 6 t i x ( t 3t 3)) t i x co x i x (3t t (( t 3t 3) co x 3 3 3)) cox, C ( co x (3t t (( t 3t 3) co x 3)) t i x ( t 3t 3)) ix(3t t (( t 3t 3) co x 3)) t ix co x i x co x, E i x ( co x (3t t (( t 3t 3) co x 3)) t i x ( t 3t 3)) 6 3 t i x ( t 3t 3)) t ix co x i x (3t t (( t 3t 3) co x 3 3 3)) co x, F i x t it i x (3t t (( t 3t 3) co x 3)) 6 c t o x i x ( t 3t 3) Accordig to (7), the 0 th compoet u 0 ( x, t ) ad v 0 ( x, t ) writte a follow:, where, 0. So, we get the followig compoet: u ( x, t ) t i x, v ( x, t ) t i x u ( x, t ) i x (3t t (( t 3t 3)cox 3)) 6 v ( x, t ) i x (3t t (( t 3t 3)cox 3)) u3( x, t ) t i x (60 70t 84t 55t 40 t ((t 4t t 63t 4) co 4)) ( 630 co(3 t 35t 80t 60t 60) t (8co x (t 4t 4t 63t 4) ) 4 t (co x (5t 6) 3 ( t 5t 30) 45)) t i x v 3( x, t ) t i x (60 70t 84t 55t 40 t ((t 4t t 63t 4) co 4)) ( 630 co(3 t 35t 80t 60t 60) t (8co x (t 4t 4t 63t 4) ) 4 t (co x (5t 6) 3 ( t 5t 30) 45)) t i x Similarly, we ca alo fid other compoet, ad the approximate olutio for calculatig 5 th. Uig (4), the erie olutio are therefore give by u ( x, t ) i x t i x i x (3t t (( t 3t 3) co x 3 )) 6 v ( x, t ) ix t ix i x (3t t (( t 3t 3) cox 3 )) 6 ad Figure. (a) ad (b) how the exact ad umerical olutio of ytem (4) with 0 th term by LADM P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

8 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c LADM ad exact for u(x,t) (a) LADM ad exact for v(x,t) (b) Figure. (a) how the exact ad LADM umerical olutio u(x, t) of example, 5 x 5, - t ; (b) how the exact ad LADM umerical olutio v (x, t) of example, 5 x 5, - t. Example 3: we coider the oliear ytem [3] u uu vu t x y v vv uv t y x, with the iitial coditio u( x, y,0) x y ; v(x, y,0) x y, 0 x, y. x y x y The exact olutio give a u ( x, y, t ) ; v(x, y, t ), 0 t T. t t Solutio: Takig the Laplace traform o both ide of Eq. (0) the, by uig the differetiatio property of Laplace traform ad iitial coditio, a the ame procedure i the above example whe we uig equatio (9)-(0) for the ytem (0) ad accordig to (3) we have the followig 0 th compoet ;, ad the recurive relatio ca be writte a follow: ; ; ; (8) ; ; ; ; ; Similarly, we ca alo fid other compoet, ad the approximate olutio for calculatig more 0 th. Uig (3), the erie olutio are therefore give by u ( x, y, t ) v ( x, y,t) (x y) (x y) t 4(x y) t 8(x y) t 6(x y) t ( x y )[ t 4t 8t 6 t ] x y ( x y )( t ) t x y x y that coverge to the exact olutio u ( x, y, t ) ; v ( x, y, t ), t t ad Figure 3 eure that the previou obtaied reult of ytem (8) with 30 th term by LADM it coverge to the exact olutio. Table how the abolute error betwee the exact olutio ad the reult obtaied from the LADM olutio 6537 P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

9 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c ad compario with the reult obtaied by the tadard algorithm for Adomia' polyomial ADM ad modified ADM olutio of ytem of equatio (8) (ee [4] ad [6]). u(x,y,t) LADM v(x,y,t) ) LADM u(x,y,t) ad v(x,y,t) exact Figure 3. how the exact ad LADM umerical olutio of u(x, t) ad v (x, t) for example 3, 5 x 5, - y ; t=0.. Table : The umerical reult i compario with the aalytical olutio for variou value of x, y ad t for example 3 The abolute error t y x Exact(u )=Exact(v ) LADM AMD [6] Modified AMD[4] e e e e e e e e e e e e e-05.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

10 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c We ote from the above reult, the abolute error obtaied by the propoed algorithm LADM a compared with the abolute error of the tadard algorithm for Adomia' polyomial ADM ad modified ADM give reult more accurate. Example 4: The mathematical model o may pheomea i applied ciece lead to o-liear PDE uch a the homogeeou form of the ytem of two dimeioal Burger' equatio which i propoed a mathematical model of free turbulece [5, 6] ut uu x vu y ( u xx + u yy ), R v t vv y uv x ( v xx + v yy ), R ubject to the iitial coditio 3 ux (, y,0), 4 R ( y x )/8 4 e 3 ux (, y,0), 4 R ( y x )/8 4 e (9) (0) the exact olutio [8]: 3 u ( x, y, t ), 4 R ( t4y 4 x )/3 4 e 3 v ( x, y, t ), 4 R ( t4y 4 x )/3 4 e where R i Reyold umber. A the ame procedure i the above example whe we ue (3)-(6) for the ytem (9)-(0) ad accordig to (7) (8), we have the followig 0 th compoet: 3 u0 ( x, y, t ), 4 R ( y x )/8 4 e 3 v 0 ( x, y, t ), 4 R ( y x )/8 4 e () ad the recurive relatio ca be writte a follow: u ( x, y, t )} F A B, v ( x, y, t )} G C E, ad the Adomia polyomial A, B, C, E, F ad where 0 G are defid a follow () ; 6539 P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

11 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c ; (3). We ca give the firt few Adomia' polyomial of repectively to get the followig compoet: 6540 P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

12 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c ad the recurive relatio ca be writte a follow: 654 P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

13 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c. Similarly, we ca alo fid other compoet, ad Uig (3), the erie olutio are therefore give by ad Figure 4. (a)-(d) how the exact ad umerical olutio of ytem (9) with 3 th term by LADM alo table how the abolute error betwee the exact olutio ad the reult obtaied from the LADM olutio ad compario with the reult obtaied by the tadard algorithm ADM (ee [6]). u(x,y,t) LADM u(x,y,t) exact LADM ad exact for u(x,y,t) (a) v(x,y,t) LADM v(x,y,t) exact LADM ad exact for v(x,y,t) (b) Figure 4. (a) ad (b) how the exact ad LADM of u(x,y, t) ad v (x, y,t) for example 4, 0. x 0.5; 0. y 0.5 ; t=0.; R= P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

14 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c u(x,y,t) LADM u(x,y,t) exact LADM ad exact for u(x,y,t) (c) v(x,y,t) LADM v(x,y,t) exact LADM ad exact for v(x,y,t) (d) Figure 4. (c) ad (d) how the exact ad LADM of u(x,y, t) ad v (x, y,t) for example 4, 0. x 0.5; 0. y 0.5 ; t=0.5; R=00 Table : The umerical reult i compario with the aalytical olutio for variou value of x ad t for example 4 whe y =, R = 50 ad R =00 are repectively The abolute error t x Exact(u ) Exact(v ) LADM(u ) AMD(u ) [6] LADM(v ) AMD(v )[6] R=50 R=00 R=50 R=50 R=00 R= e e e e e e e e e e-003.8e e e e e e e e e e e e e e e e e e e e-003.8e-005.9e e e e e e e-00.84e e-003 The umerical reult how that uig of LADM by 3- compoet give reult more accurate tha the reult uig 5- compoet by tadard ADM which i preeted i [6]. 4. CONCLUSION LADM i a powerful tool which i capable of hadlig coupled ytem of oliear of partial differetial equatio. I thi paper the LADM ha bee uccefully applied to fid approximate olutio for the homogeeou form of the ytem of two dimeioal Burger' equatio without ay dicretizatio. The method preet a ueful way to develop a aalytic 6543 P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06

15 I S S N V o l u m e N u m b e r 0 8 J o u r a l o f A d v a c e i M a t h e m a t i c treatmet for thee ytem. The propoed cheme ca be applied for ytem more tha two liear ad oliear partial differetial equatio. REFERENCES [] L. Debath, Noliear Partial Differetial Equatio for Scietit ad Egieer, Birkhauer, Boto, 997. [] J. D. Loga, A Itroductio to Noliear Partial Differetial Equatio, Wiley, New York, 994. [3] A. M. Wazwaz, The decompoitio method applied to ytem of partial differetial equatio ad to the reactiodiffuio bruelator model, Appl. Math. Comput., 0 (000), pp [4] S. A. Khuri, A Laplace Decompoitio Algorithm Applied to Cla of Noliear Differetial Equatio, Jour- al of Applied Mathematic, Vol., No. 4, 00, pp [5] H. Hoeizadeh, H. Jafari ad M. Roohai, Applicatio of Laplace Decompoitio Method for Solvig Klei- Gordo Equatio, World Applied Sciece Joural, Vol.8, No. 7, 00, pp [6] M. Kha, M. Huai, H. Jafari ad Y.Kha, Applicatio of Laplace Decompoitio Method to Solve Noliear Coupled Partial Differetial Equatio, World Applied Sciece Joural, Vol. 9, No., 00, pp [7] E. Yuufoglu, Numerical Solutio of Duffig Equatio by the Laplace Decompoitio Algorithm, Applied Mathematic ad Computatio, Vol. 77, No., 006, pp doi:0.06/j.amc [8] D. Kaya ad I.E. Ia, Exact ad Numerical Travelig Wave Solutio for Noliear Coupled Equatio Uig Symbolic Computatio, Appl. Math.ad Comput. 5, (004) [9] H. Jafari ad V. Daftardar-Gejji, Solvig Liear ad Noliear Fractioal Diffutio ad Wave Equatio by Adomia Decompoitio, Applied Mathematic ad Computatio, Vol. 80, No., 006, pp doi:0.06/j.amc [0] F. Abdelwahid, A mathematical Model of Adomia Polyomial, Applied Mathematic ad Computatio, Vol. 4, No. -3, 003, pp doi:0.06/s (0) [] Mahmoud S. Rawahdeh ad Shehu Maitama, SOLVING COUPLED SYSTEM OF NONLINEAR PDE S USING THE NATURAL DECOMPOSITION METHOD. Iteratioal Joural of Pure ad Applied Mathematic, Volume 9 No. 5 04, [] A.S. Ravi Kath ad K. Arua, Differetial traform method for olvig liear ad o-liear ytem of partial differetial equatio, Phy. Lett. A., 37, (008). [3] R. Bellma, B. G. Kahef ad J. Cati, Differetial Quadrature Techique for The Rapid Solutio of Noliear Partial Differetial equatio, J. of Comput. Phyic, 0, (97), No., [4] H. O. AL-Humedi et.al, Modified Algorithm to Compute Adomia' Polyomial for Solvig No-Liear Sytem of Partial Differetial Equatio, It. J. Cotemp. Math. Sciece, Vol. 5, 00, o. 5, [5] M. Dehgha, H. Agar ad S. Mohammad, The Solutio of Coupled Burger' Equatio Uig Adomia Pade techique, Appl. Math. ad Comput. 89, (008),No, [6] S.M. El- Sayed ad D. Kaya, O The Numerical Solutio of The Sytem of Two-dimeioal Burger' Equatio by The Decompoitio Method, Appl. Math. ad Comput. 58, (004), P a g e c o u c i l f o r I o v a t i v e R e e a r c h S e p t e m b e r 06