Application of Laplace Adomian Decomposition Method on Linear and Nonlinear System of PDEs

Transcription

1 Applied Mathematical Science, Vol. 5, 2011, no. 27, Application of Laplace Adomian Decompoition Method on Linear and Nonlinear Sytem of PDE Jaem Fadaei Mathematic Department, Shahid Bahonar Univerity of Kerman, Kerman, Iran Abtract In thi paper, a numerical Laplace tranform algorithm which i baed on the Adomian decompoition method (LADM) i introduced for the approximate olution of the linear and nonlinear ytem of partial differential equation. Illutrative example are included to demontrate the high accuracy and fat convergence of our method. Mathematic Subject Claification: 45A05, 45B05, 45D05 and 45J05 Keyword: Laplace Adomian decompoition method, Sytem of PDE, Adomian polynomial 1 Introduction Sytem of partial differential equation, have attracted much attention in a variety of applied cience. The general idea and the eential feature of thee ytem are of wide applicability. Thee ytem were formally derived to decribe wave propagation [1 5], to control the hallow water wave [1 5], and to examine the chemical reaction-diffuion model of Bruelator [4 6]. The method of characteritic, the Riemann invariant, and Adomian method [6] were the commonly ued method. In thi work, we will ue Laplace decompoition method introduced by Khuri [7, 8]. Agadjanov [9] olved Duffing equation with the help of thi method. Thi numerical technique baically illutrate how the Laplace tranform may be ued to approximate the olution of the nonlinear partial differential equation by manipulating the decompoition method. Elgaery [10] applied Laplace decompoition method for the olution of Falkner-Skan equation. Here Laplace- Adomian decompoition i implemented to three ytem of partial differential equation [11, 12]. The modification of Laplace decompoition method introduced by Huain and Majid Khan [13].

2 1308 J. Fadaei 2 Laplace Adomian decompoition method In thi ection, we preent a Laplace-Adomian decompoition method for olving of partial differential equation written in an operator form L t u + R 1 (u, v)+n 1 (u, v) =g 1, L t v + R 2 (u, v)+n 2 (u, v) =g 2, with initial data u(x, 0) = f 1 (x), (1) v(x, 0) = f 2 (x), (2) where L t i conidered a firt-order partial differential operator, R 1,R 2 and N 1,N 2 are linear and nonlinear operator, repectively. and g 1 and g 2 are ource term. The method conit of firt applying the Laplace tranform to both ide of equation in ytem (1) and then by uing initial condition (2), we have: [L t u]+ [R 1 (u, v)] + [N 1 (u, v)] = [g 1 ], [L t v]+ [R 2 (u, v)] + [N 2 (u, v)] = [g 2 ]. Uing the differentiation property of Laplace tranform, we get [u] = f 1(x) + 1 [g 1] 1 [R 1(u, v)] 1 [N 1(u, v)], (3) [v] = f 2(x) + 1 [g 2] 1 [R 2(u, v)] 1 [N 2(u, v)]. The LADM define the olution u(x, t) and v(x, t) by the infinite erie u(x, t) = u n, v(x, t) = v n. (5) The nonlinear term N 1,N 2 are uually repreented by an infinite erie of the o-called Adomian polynomial [14] N 1 (u, v) = A n, N 2 (u, v) = Ã n. (6) The Adomian polynomial can be generated for all form of nonlinearity. They are determined by the following relation: A n = 1 [ [ d n ]] N n! dλ n 1 (λ i u i ), i=0 λ=0 (7) (4)

3 Sytem of PDE 1309 Ã n = 1 [ [ d n ]] N n! dλ n 2 (λ i v i ). (8) i=0 λ=0 Subtituting (5) and (6) into (4), give [ u n ]= f 1(x) + 1 [g 1] 1 [R 1([ u n ], [ v n ])] 1 [ A n ], [ v n ]= f 2(x) + 1 [g 2] 1 [R 2([ u n ], [ v n ])] 1 [ ] Ã n. (9) Applying the linearity of the Laplace tranform, we define the following recurively formula: [u 0 ]= f 1(x) + 1 [g 1], [v 0 ]= f 2(x) + 1 [g 2], [u 1 ]= 1 [R 1(u 0,v 0 )] 1 [A 0], (10) [v 1 ]= 1 [R 2(u 0,v 0 )] 1 [Ã0]. In general, for k 1, the recurive relation are given by [u k+1 ]= 1 [R 1(u k,v k )] 1 [A k], (11) [v k+1 ]= 1 [R 2(u k,v k )] 1 [Ãk]. (12) Applying the invere Laplace tranform, we can evaluate u k and v k (k 0). In ome cae the exact olution in the cloed form may alo be obtained. 3 Application In thi ection, we ue the LADM to olve homogeneou and inhomogeneou linear ytem of partial differential equation and homogeneou and inhomogeneou nonlinear ytem of partial differential equation. The algorithm are performed by Matlab The homogeneou linear ytem Conider the homogeneou linear ytem of PDE: u t v x +(u + v) =0, v t u x +(u + v) =0, (13)

4 1310 J. Fadaei with initial condition u(x, 0) = inhx, v(x, 0) = cohx. (14) Taking the Laplace tranform on both ide of Eq. (13) then, by uing the differentiation property of Laplace tranform and initial condition (14) give [u] = 1inhx + 1 [v x] 1 [u + v], [v] = 1 cohx + 1 [u x] 1 [u + v]. (15) The LADM define the olution u(x, t),v(x, t) by the erie u(x, t) = u n, v(x, t) = v n, (16) and the term v x and u x by an infinite erie u x (x, t) = u nx, v x (x, t) = v nx. (17) Subtituting erie in (16) and (17) into both ide of Eq. (15) yield [ u n ]= 1inhx + 1 [ v nx ] 1 [ u n + v n ], [ v n ]= 1 cohx + 1 [ u nx ] 1 [ u n + v n ]. Now we define the following recurively formula: [u 0 ]= 1inhx, (18) [v 0 ]= 1 cohx, (19) [u n+1 ]= 1 [v n x ] 1 [u n + v n ], n 0, [v n+1 ]= 1 [u n x ] 1 [u n + v n ], n 0. (20) Taking the invere Laplace tranform of both ide of the (19) and (20), we have u 0 = inhx, v 0 = cohx, (21)

5 Sytem of PDE 1311 u 1 = tcohx, v 1 = tinhx, u 2 = t2 2! inhx, (22) v 2 = t2 cohx, 2! (23) and o on for other component. Uing (16), the erie olution are therefore given by u(x, t) =inhx ( 1+ t2 2! + t4 4! ) cohx ( t + t3 3! + t5 5! ), v(x, t) =cohx ( 1+ t2 + ) t4 inhx ( t + t3 + ) t5, 2! 4! 3! 5! (24) uing the Taylor expanion for inht and coht, we can find the exact olution u(x, t) =inh(x t), v(x, t) =coh(x t). 3.2 The inhomogeneou linear ytem Conider the inhomogeneou linear ytem u t v x (u v) = 2, v t + u x (u v) = 2, with initial condition (25) u(x, 0) = 1 + e x, v(x, 0) = 1+e x. (26) Taking the Laplace tranform on both ide of Eq. (25) then, by uing the differentiation property of Laplace tranform and initial condition (26) give [u] = 1 + ex [v 2 x]+ 1 [u v], [v] = 1 + ex 2 1 [u 2 x]+ 1 [u v]. (27) Uing the decompoition erie (16) and (17) for the linear term u(x, t), v(x, t) and u x, v x, we obtain [ u n ]= 1 + ex [ v nx ]+ 1 [ u n v n ], [ v n ]= 1 + ex [ u nx ]+ 1 [ u n v n ]. (28)

6 1312 J. Fadaei The LADM preent the recurive relation [u 0 ]= 1 + ex 2, 2 [v 0 ]= 1 + ex 2 2, [u n+1 ]= 1 [v n x ]+ 1 [u n v n ], n 0, (29) [v n+1 ]= 1 [u n x ]+ 1 [u n v n ]. n 0. (30) Taking the invere Laplace tranform of both ide of the (29) and (30), we have u 0 =1+e x 2t, v 0 = 1+e x 2t, u 1 = te x +2t, v 1 = te x +2t, u 2 = t2 2! ex, (31) (32) v 2 = t2 2! ex, (33) and o on for other component. Uing (16), the erie olution are therefore given by u(x, t) =1+e x ( 1+t + t2 2! + t3 3! ), v(x, t) = 1+e x ( 1 t + t2 2! t3 3! ), that converge to the exact olution u(x, t) =1+e x+t, v(x, t) = 1+e x t. 3.3 The inhomogeneou nonlinear ytem Conider the ytem of inhomogeneou partial differential equation u t + vu x + u =1, (34) v t uv x v =1, (35)

7 Sytem of PDE 1313 with initial condition u(x, 0) = e x, v(x, 0) = e x. (36) Taking the Laplace tranform on both ide of Eq. (35) then, by uing the differentiation property of Laplace tranform and initial condition (36) give [u] = ex [vu 2 x] 1 [u], [v] = e x [uv 2 x]+ 1 [v]. (37) We repreent u(x, t) and v(x, t) by the infinite erie (16) then, inerting thee erie into both ide of Eq. (37) yield [ u n ]= ex [ A n ] 1 [ u n ], [ v n ]= e x [ ] Ã 2 n + 1 [ v n ]. (38) Where A n and Ãn are the o-called Adomian polynomial by (12) that repreent the nonlinear term vu x and uv x, repectively. We have a few term of the Adomian polynomial for vu x and uv x, which are given by A 0 = v 0 u 0x, A 1 = v 1 u 0x + v 0 u 1x, A 2 = v 2 u 0x + v 1 u 1x + v 0 u 2x, A 3 = v 3 u 0x + v 2 u 1x + v 1 u 2x + v 0 u 3x, and A 0 = u 0 v 0x, A 1 = u 1 v 0x + u 0 v 1x, A 2 = u 2 v 0x + u 1 v 1x + u 0 v 2x, A 3 = u 3 v 0x + u 2 v 1x + u 1 v 2x + u 0 v 3x,.... Now we define the following recurively formula: [u 0 ]= 1 + ex, [v 0 ]= 1 + e x, (39) [u n+1 ]= [A n] 1 [u n], [v n+1 ]= [Ãn]+ 1 [v n]. (40)

8 1314 J. Fadaei Operating with Laplace invere on both ide of (39) and (40) give u 0 = t + e x, v 0 = t + e x, u 1 = t t2 2! tex t2 2! ex, v 1 = t t2 2! + te x t2 2! e x, u 2 = t2 2! + t2 e x +..., (41) (42) v 2 = t2 + 2! t2 e x +... (43) Similarly, we can find other component. Uing (16), the erie olution are therefore given by u(x, t) =e x ( 1 t + t2 2! t3 3! ), v(x, t) =e ( x 1+t + t2 + ) t3. 2! 3! (44) By uing the Taylor expanion for e t and e t we can find the exact olution u(x, t) =e x t, v(x, t) =e x+t. 4 Concluion In thi work, a numerical Laplace tranform algorithm which i baed on the Adomian decompoition method i ued to for olving the linear and nonlinear ytem of partial differential equation. The method preent a ueful way to develop an analytic treatment for thee ytem. The propoed cheme can be applied for ytem more than two linear and nonlinear partial differential equation. ACKNOWLEDGEMENTS. The author i thankful to Profeor M. Moheni Moghadam for hi endle upport and guidance. Thi work ha been partially upported by the Mahani Mathematical Reearch Center and the Center of Excellent in Linear Algebra and Optimization of Shahid Bahonar Univerity of Kerman.

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