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Numerical Solution of Nonlinear System of Partial Differential Equations by the Laplace Decomposition Method and the Pade Approximation

Numerical Solution of Nonlinear System of Partial Differential Equations by the Laplace... American Journal of Computational Mathematics, 2013, 3, 175-184 http://dx.doi.org/10.4236/ajcm.2013.33026 Published Online September 2013 (http://www.scirp.org/journal/ajcm) Numerical Solution of Nonlinear System of Partial Differential Equations by the Laplace Decomposition Method and the Pade Approximation 1 2 Magdy Ahmed Mohamed , Mohamed Shibl Torky Faculty of Science, Suez Canal University, Ismailia, Egypt The High Institute of Administration and Computer, Port Said University, Port Said, Egypt Email: [email protected], [email protected] Received March 7, 2013; revised April 28, 2013; accepted May 9, 2013 Copyright © 2013 Magdy Ahmed Mohamed, Mohamed Shibl Torky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions. Keywords: Nonlinear System of Partial Differential Equations; The Laplace Decomposition Method; The Pade Approximation; The Coupled System of the Approximate Equations for Long Water Waves; The Whitham Broer Kaup Shallow Water Model; The System of Hirota-Satsuma Coupled KdV 1. Introduction [6], uk u u vu, The Laplace decomposition method (LDM) is one of the txx x (3) efficient analytical techniques to solve linear and nonlin- vk v u vu, txx x ear equations [1-3]. LDM is free of any small or large with exact solution are given in [6] as parameters and has advantages over other approximation   techniques like perturbation. Unlike other analytical tech- Bc  4 k ux,2 t kc1 tanh tcx ,     niques, LDM requires no discretization and linearization.   (4) Therefore, results obtained by LDM are more efficient  Bc 44 k B c k and realistic. This method has been used to obtain ap- vx,2 t kctanh tcx,   proximate solutions of a class of nonlinear ordinary and 2 22kc  partial differential equations [1-4]. See for example, the and thesystem of Hirota-Satsuma coupled KdV [7]. Duffing equation [4] and the Klein-Gordon equation [3]. In this paper, the LDM is applied to, the Whitham-Broer- uu33 uu vw3wv, t xxx x x x Kaup shallow water model [5] (5) vv  3,uv uu u v u , t xxx x tx x xx (1) ww  3,uw t xxx x vv u uv  v  u , tx x xx xxx with exact solution are given in [7] as with exact solution are given in [5] as 22 2 ux,2 t k 2ktanh k x t,   ux,2 t  k tanh k xt  ,      22 2 2 22 2 44 kc k k  k   vx,2 t k  ò  (2)    vx,t t anh k xt,    2  3c 3c 1 1 22 2 2tkk ò,  anhxt   wx,t t  c canh kxt,   1 01  (6) and the coupled nonlinear reaction diffusion equations Copyright © 2013 SciRes. AJCM 176 M. A. MOHAMED, M. S. TORKY Pade approximation of a function is given by ratio of two we discuss how to solve Numerical solution of nonlinear polynomials. The coefficients of the polynomial in both system of parial differential equations by using LDM. The the numerator and the denominator are determined using results of the present technique have close agreement with the coefficients in the Taylor series expansion of the approximate solutions obtained with the help of the function. The Pade approximation of a function, symbol- Adomian decomposition method [8]. ized by [m/n], is a rational function defined by 2. Laplace Decomposition Method 2 m m aax a xa x  01 2 m  , (13)  2 n U n 1 bx b x  b x  12 n (7)  gxt,,  R U N U i1,2,,n, ii i t where we considered b = 1, and the numerator and de- where Uu  ,,u ,u ,   nominator have no common factors. In the LD-PA method 12 n with initial condition we use the method of Pade approximation as an after- treatment method to the solution obtained by the Laplace ux,0 f x , (8)     ii decomposition method. This after-treatment method im- proves the accuracy of the proposed method. the method consists of first applying the Laplace trans- formation to both sides of (7) 4. Application  U ££gx,t   In this section, we demonstrate the analysis of our nu-   t  (9) merical methods by applying methods to the system of partial differential Equations (1), (3) and (5). A com- £,  RU N U i1,2,,n,    ii  parison of all methods is also given in the form of graphs and tables, presented here. using the formulas of the Laplace transform, we get sU££  f x  g x,t      4.1. The Laplace Decomposition Method ii (10) Exampe 1. The Whitham-Broer-Kaup model [5] £,  RU N U i1,2,,n,     ii To solve the system of Equation (1) by means of Laplace decomposition method, and for simplicity, we in the Laplace decomposition method we assume the   solution as an infinite series, given as follows  ,2 k , 3,1,1,0 take , we con-    UU  , (11) struct a correctional functional which reads £0 uu    where the terms U are to be recursively computed. Also the linear and nonlinear terms R and Ni,1  ,2,,n is decomposed as an infinite series of £, uu v u  xx xx Adomian polynomials (see [8,9]). Applying the inverse (14) Laplace transform, finally we get £0 vv    1 Uf ££ x g x,t ii  (12) £3 vu uv v  u,  xx xx xxx £,  RU N U i1,2,,n, s ii  we can define the Adomian polynomial as follows: 3. The Pade Approximant nn n Au u ,, B vu C uv , (15)   ni ni x n i ni x ni ni x    Here we will investigate the construction of the Pade ii 00 i0 approximates [10] for the functions studied. The main we define an iterative scheme advantage of Pade approximation over the Taylor series approximation is that the Taylor series approximation ££uA  v u ,   n1 n nx nxx can exhibit oscillati which may produce an approxima- (16) tion error bound. Moreover, Taylor series approxima- tions can never blow-up in a fin region. To overcome ££vB  Cv 3u ,  n1 n n nxx nxxx these demerits we use the Pade approximations. The Copyright © 2013 SciRes. AJCM M. A. MOHAMED, M. S. TORKY 177 applying the inverse Laplace transform, finally we get terms as Equation (18), and Figures 1(a) and (b) show th Equations (17). Similarly, we can also find other com- the exact and numerical solution of system (1) with 16 th ponents, and the approximate solution for calculating 16 terms by (LDM). (a) (b) Figure 1. (a) Exact and numerical solution of u(x, t), −10 ≤ x ≤ 10, -1 ≤ t ≤ 1; (b) Exact and numerical solution of v (x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1. 32tx sinh 2 18 2 t ux,8 t tanh 2x, v x,t 1616tanh 2x , u x,t  , v x,t  ,         00 1 1 cosh 2xx cosh 2 (17) 16tx 8 cosh  8 coshx 1    8stx inh 2  ux,, t  v x,t  , 38 6 4 2 cosh 2xx 1 16 cosh 32 coshx 24 coshx 8 coshx       88tx cosh  8coshx 1       8stx inh 2 18t  ux,8 t tanh2x       23 8 cosh 2xx cosh 2 3(1 16 coshx    (18) 16tx 8 cosh 8 coshx 1 32 sinh 2  tx vx,1 t 6 16tanh 2x  ,    cosh 2xx cosh 2   Copyright © 2013 SciRes. AJCM 178 M. A. MOHAMED, M. S. TORKY Example 2. coupled nonlinear RDEs [6] Example 3. Hirota-Satsuma coupled KdV System To solve the system of Equation (3) by means of [7] Laplace decomposition method, and for simplicity, we To solve the system of Equation (5) by means of take kc 2, 1, 10 , we construct a correctional Laplace decomposition method, and for simplicity, we   functional which reads take kc  c 1 , we construct a correctional   functional which reads  £0 uu £2uuv10u,   xx x  11 1   ss £0 uu £ u 3uu3vw3, wv (19)   xxx x x x   11 ss 2    £0 vv £2vuv10u ,   xx x  ss 11 £0 vv £v 3uv, (24)    xxx x ss we can define the Adomian polynomial as follows: £0 ww £w 3uw,    xxx x Au  v , (20)  ss ni ni x  i0 we can define the Adomian polynomial as follows: we define an iterative scheme nn n Au u ,, B vw C wv , nin i n i  ni x  ni x  ni x ££uu2 A10u,   ii 00 i0 nn 1 xx n n (25) nn (21) Du v ,, E uw nini  ni x  n i x ££vv2 A10u,   nn 1 xx n n ii 00 we define an iterative scheme applying the inverse Laplace transform, finally we get   ££ uu3A3B3C, n1 nxxx n n n   ux,2 t 2tanh x, v x,t  2tanh x,       00 s 2   22 tt ££vv  3D, (26)   ux,, t  v x,t  , nn 1 xxx n   coshxx cosh ££ww  3E,   2stx inh 2stx inh nn 1 xxx n   (22) ux,, t  v x,t  ,   coshx coshx   applying the inverse Laplace transform, finally we get tx  32cosh  2 ux ,t   2 tanhx , v x,t  tanhx , ux,, t  00  cosh x  4stx inh  3 wx,1 t tanh x, u x,t  ,      tx  3 2 cosh cosh 3   cosh x  vx,, t  , cosh cosh x  8tt vx,, t w x,t ,   coshxx cosh similarly, we can also find other components, and the th approximate solution for calculating 16 terms as fol- 2 22cosh 3 tx   lows: ux,, t   2 cosh x  2stx inh 2t   ux,2 t 2tanh x   22    tx sinh tx sinh 8   coshxx cosh vx,, t  w x,t  ,   coshxx cosh tx  32cosh    tx cosh  3 sinhx    cosh x  ux,, t  (23) cosh x  2stx inh 92t vx,2 t tanh x   3   tx 2cosh  3   cosh x cosh x vx,, t   2 9 3 cosh x tx  32cosh   , 4 tx 2cosh  3   3 1 cosh x  wx,, t  cosh x  and Figures 2(a) and (b) show the exact and numerical th (27) solution of system (3) with 16 terms by (LDM). Copyright © 2013 SciRes. AJCM M. A. MOHAMED, M. S. TORKY 179 (a) (b) Figure 2. (a) Exact and numerical solution of u(x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1; (b) Exact and numerical solution of v(x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1. similarly, we can also find other components, and the ap- and Figures 3(a)-(c) show the exact and numerical so- th th proximate solution for calculating 16 terms as follows: lution of system (5) with 16 terms by (LDM). 4stx inh 1  4.2. The Pade Approximation ux,2 t  tanh x  cosh x  In this section we use Maple to calculate the [3/2] the Pade 2tx 2 cosh 3 t coshx 3 sinhx   8 approximant of the infinite series solution (18), (23), and   (28) which gives the rational fraction approximation to the coshxx cosh solution, and Figures 4(a)-(c) show the results obtained 4stx inh 1  vx,2 t  tanh x  by the Pade approximant (LD-PA) solution of systems (1), cosh x  (3) and (5), and Figures 5(a)-(c) show comparison be- tween the exact solution, LDM solution and the Pade 2tx 2 cosh 3 t coshx 3 sinhx        approximant (LD-PA) solution of systems (1), (3) and (5) coshxx cosh at, x = 5, −1 ≤ t ≤ 1. Tables 1-3 show the absolute error tx sinh t  between the exact solution and the results obtained from wx,1 t tanhx  the, LDM solution and the Pade approximant (LD-PA) coshxx cosh   solution of systems (1)-(3). tx 2cosh  3    , 5. Conclusion cosh x (28) The Laplace decomposition method is a powerful tool Copyright © 2013 SciRes. AJCM 180 M. A. MOHAMED, M. S. TORKY (a) (b) (c) Figure 3. (a) Exact and numerical solution of u(x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1; (b) Exact and numerical solution of v(x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1; (c) Exact and numerical solution of w(x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1. Copyright © 2013 SciRes. AJCM M. A. MOHAMED, M. S. TORKY 181 (a) (b) (c) Figure 4. (a) The Pade approximant (LD-PA) solution of u(x, t) and v(x, t) of example 1, −10 ≤ x ≤ 10, −1 ≤ t ≤ 1; (b) The Pade approximant (LD-PA) solution of u(x, t) and v(x, t) of example 2, −10 ≤ x ≤ 10, −1 ≤ t ≤ 1; (c) The Pade approximant (LD-PA) solution of u(x, t) and v(x, t) and w(x, t) of example 3, −10 ≤ x ≤ 10, −1 ≤ t ≤ 1. Copyright © 2013 SciRes. AJCM 182 M. A. MOHAMED, M. S. TORKY (a) (b) (c) Figure 5. (a) The exact, (LDM)and the Pade approximant (LD-PA) solution of u(x, t) and v(x, t) of example 1, x = 5, −1 ≤ t ≤ 1; (b) The exact, (LDM)and the Pade approximant (LD-PA) solution of u(x, t) and v(x, t) of example 2, x = 5, −1 ≤ t ≤ 1; (c) The exact, (LDM)and the Pade approximant (LD-PA) solution of u(x, t), v(x, t) and w(x, t) of example 3, x = 5, −1 ≤ t ≤ 1. which is capable of handling nonlinear system of partial proximate solutions for,the Whitham-Broer-Kaup shal- differential equations. In this paper the (LDM) and Pade low water model, the coupled nonlinear reaction diffu- approximant has been successfully applied to find ap- sion equations and thesystem of Hirota-Satsuma coupled Copyright © 2013 SciRes. AJCM M. A. MOHAMED, M. S. TORKY 183 Table 1. The absolute error of u(x, t) and v(x, t) of example 1, x = 40. uu  vv  uu  vv  ex LDM ex LDM ex LD – PA ex LD – PA 0 0 0 0 0 −69 −69 −75 −74 0.2 2.56 × 10 1.02 × 10 4.16 × 10 1.66 × 10 −69 −69 −73 −72 0.4 6.38 × 10 2.55 × 10 3.79 × 10 1.51 × 10 −68 −69 −72 −71 0.6 1.20 × 10 4.83 × 10 6.24 × 10 2.49 × 10 −68 −69 −71 −70 0.8 2.06 × 10 8.24 × 10 5.14 × 10 2.05 × 10 −68 −67 −70 −69 1.0 3.32 × 10 1.33 × 10 2.90 × 10 1.16 × 10 −68 −67 −69 −69 1.2 5.22 × 10 2.08 × 10 1.29 × 10 5.16 × 10 −68 −67 −69 −68 1.4 8.05 × 10 3.21 × 10 4.81 × 10 1.92 × 10 −67 −67 −68 −68 1.6 1.22 × 10 4.90 × 10 1.54 × 10 6.19 × 10 −67 −67 −68 −67 1.8 1.85 × 10 7.42 × 10 4.33 × 10 1.73 × 10 −67 −66 −67 −67 2.0 2.79 × 10 1.11 × 10 1.05 × 10 4.22 × 10 Table 2. The absolute error of u(x, t) and v(x, t) of example 2, x = 40. uu  vv  uu  vv  ex LDM ex LDM ex LD – PA ex LD – PA 0 0 0 0 0 −45 −45 −41 −41 5.42 × 10 0.2 5.42 × 10 5.76 × 10 5.76 × 10 −45 −45 −39 −39 0.4 3.96 × 10 3.96 × 10 5.25 × 10 5.25 × 10 −45 −45 −38 −38 0.6 1.41 × 10 1.41 × 10 8.64 × 10 8.64 × 10 −45 −45 −37 −37 0.8 5.95 × 10 5.95 × 10 7.12 × 10 7.12 × 10 −44 −44 −36 −36 1.0 3.28 × 10 3.28 × 10 4.02 × 10 4.02 × 10 −43 −43 −35 −35 1.2 6.81 × 10 6.81 × 10 1.78 × 10 1.78 × 10 −42 −42 −35 −35 1.4 9.59 × 10 9.59 × 10 6.66 × 10 6.66 × 10 −41 −41 −34 −34 1.6 9.52 × 10 9.52 × 10 2.14 × 10 2.14 × 10 −40 −40 −34 −34 1.8 7.24 × 10 7.24 × 10 6.00 × 10 6.00 × 10 −39 −39 −33 −33 2.0 4.46 × 10 4.46 × 10 1.46 × 10 1.46 × 10 Table 3. (a) The absolute error of u(x, t), v(x, t) and w(x, t) of example 3, x = 40; (b) The absolute error of u(x, t), v(x, t) and w(x, t) of example 3, x = 40. (a) uu  vv  ww  ex LDM exact LDM exact LDM 0 0 0 35 35 35 0.2 4.7610 3.1710 1.1910 35 35 35 0.4 7.9510 5.3010 1.9810 34 35 35 0.6 1.0010 6.7210 2.5210 34 35 35 0.8 1.1510 7.6810 2.8810 34 35 35 1.0 1.2410 8.3210 3.1210 34 35 35 1.2 1.3110 8.7510 3.2810 34 35 35 1.4 1.3510 9.0310 3.3910 34 35 35 1.6 1.3810 9.2310 3.4610 34 35 35 1.8 1.4010 9.3610 3.5010 34 35 35 2.0 1.4110 9.4410 3.5410 (b) uu  vv  ww  ex LD – PA eLD –PA ex LD – PA 0 0 0 41 41 41 0.2 5.9310 3.9710 1.2010 39 39 40 0.4 2.7810 1.8510 6.9710 38 38 39 0.6 2.3510 1.5710 5.8910 38 38 38 0.8 9.9610 6.6410 2.4910 37 37 38 1.0 2.8910 1.9210 7.2210 37 37 37 1.2 6.6310 4.4210 1.6510 36 37 37 1.4 1.3010 8.6610 3.2510 36 36 37 1.6 2.2710 1.5110 5.6810 36 36 37 1.8 3.6410 2.4310 9.1110 36 36 36 2.0 5.4710 3.6510 1.3610 Copyright © 2013 SciRes. AJCM 184 M. A. MOHAMED, M. S. TORKY putation,” Physics Letters A, Vol. 369, No. 5-6, 2007, pp. KdV. It was noted that the scheme found the solutions 458-463. doi:10.1016/j.physleta.2007.05.047 without any discretization or restrictive assumption, and [6] A. A. Solimana and M. A. Abdoub, “Numerical Solutions it was free from round-off errors and therefore reduced of Nonlinear Evolution Equations Using Variational It- the numerical computations to a great extent. eration Method,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 111-120. doi:10.1016/j.cam.2006.07.016 REFERENCES [7] E. Fan, “Soliton Solutions for a Generalized Hirota-Sa- [1] S. A. Khuri, “A Laplace Decomposition Algorithm Ap- tsuma Coupled KdV Equation and a Coupled MKdV plied to Class of Nonlinear Differential Equations,” Jour- Equation,” Physics Letters A, Vol. 2852, No. 1-2, 2001, nal of Applied Mathematics, Vol. 1, No. 4, 2001, pp. 141- pp. 18-22. doi:10.1016/S0375-9601(01)00161-X [8] H. Jafari and V. Daftardar-Gejji, “Solving Linear and [2] H. Hosseinzadeh, H. Jafari and M. Roohani, “Application Nonlinear Fractional Diffution and Wave Equations by of Laplace Decomposition Method for Solving Klein- Adomian Decomposition,” Applied Mathematics and Com- Gordon Equation,” World Applied Sciences Journal, Vol. putation, Vol. 180, No. 2, 2006, pp. 488-497. 8, No. 7, 2010, pp. 809-813. doi:10.1016/j.amc.2005.12.031 [3] M. Khan, M. Hussain, H. Jafari and Y. Khan, “Applica- [9] F. Abdelwahid, “A Mathematical Model of Adomian tion of Laplace Decomposition Method to Solve Nonlin- Polynomials,” Applied Mathematics and Computation, ear Coupled Partial Differential Equations,” World Ap- Vol. 141, No. 2-3, 2003, pp. 447-453. plied Sciences Journal, Vol. 9, No. 1, 2010, pp. 13-19. doi:10.1016/S0096-3003(02)00266-7 [4] E. Yusufoglu (Aghadjanov), “Numerical Solution of Duff- [10] T. A. Abassya, M. A. El-Tawil and H. El-Zoheiry, “Exact ing Equation by the Laplace Decomposition Algorithm,” Solutions of Some Nonlinear Partial Differential Equa- Applied Mathematics and Computation, Vol. 177, No. 2, tions Using the Variational Iteration Method Linked with 2006, pp. 572-580. doi:10.1016/j.amc.2005.07.072 Laplace Transforms and the Pade Technique,” Computers [5] T. Xu, J. Li and H.-Q. Zhang, “New Extension of the and Mathematics with Applications, Vol. 54, No. 7-8, Tanh-Function Method and Application to the Whitham- 2007, pp. 940-954. doi:10.1016/j.camwa.2006.12.067 Broer-Kaup Shallow Water Model with Symbolic Com- Copyright © 2013 SciRes. AJCM http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png American Journal of Computational Mathematics Unpaywall

Numerical Solution of Nonlinear System of Partial Differential Equations by the Laplace Decomposition Method and the Pade Approximation

American Journal of Computational MathematicsJan 1, 2013

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American Journal of Computational Mathematics, 2013, 3, 175-184 http://dx.doi.org/10.4236/ajcm.2013.33026 Published Online September 2013 (http://www.scirp.org/journal/ajcm) Numerical Solution of Nonlinear System of Partial Differential Equations by the Laplace Decomposition Method and the Pade Approximation 1 2 Magdy Ahmed Mohamed , Mohamed Shibl Torky Faculty of Science, Suez Canal University, Ismailia, Egypt The High Institute of Administration and Computer, Port Said University, Port Said, Egypt Email: [email protected], [email protected] Received March 7, 2013; revised April 28, 2013; accepted May 9, 2013 Copyright © 2013 Magdy Ahmed Mohamed, Mohamed Shibl Torky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions. Keywords: Nonlinear System of Partial Differential Equations; The Laplace Decomposition Method; The Pade Approximation; The Coupled System of the Approximate Equations for Long Water Waves; The Whitham Broer Kaup Shallow Water Model; The System of Hirota-Satsuma Coupled KdV 1. Introduction [6], uk u u vu, The Laplace decomposition method (LDM) is one of the txx x (3) efficient analytical techniques to solve linear and nonlin- vk v u vu, txx x ear equations [1-3]. LDM is free of any small or large with exact solution are given in [6] as parameters and has advantages over other approximation   techniques like perturbation. Unlike other analytical tech- Bc  4 k ux,2 t kc1 tanh tcx ,     niques, LDM requires no discretization and linearization.   (4) Therefore, results obtained by LDM are more efficient  Bc 44 k B c k and realistic. This method has been used to obtain ap- vx,2 t kctanh tcx,   proximate solutions of a class of nonlinear ordinary and 2 22kc  partial differential equations [1-4]. See for example, the and thesystem of Hirota-Satsuma coupled KdV [7]. Duffing equation [4] and the Klein-Gordon equation [3]. In this paper, the LDM is applied to, the Whitham-Broer- uu33 uu vw3wv, t xxx x x x Kaup shallow water model [5] (5) vv  3,uv uu u v u , t xxx x tx x xx (1) ww  3,uw t xxx x vv u uv  v  u , tx x xx xxx with exact solution are given in [7] as with exact solution are given in [5] as 22 2 ux,2 t k 2ktanh k x t,   ux,2 t  k tanh k xt  ,      22 2 2 22 2 44 kc k k  k   vx,2 t k  ò  (2)    vx,t t anh k xt,    2  3c 3c 1 1 22 2 2tkk ò,  anhxt   wx,t t  c canh kxt,   1 01  (6) and the coupled nonlinear reaction diffusion equations Copyright © 2013 SciRes. AJCM 176 M. A. MOHAMED, M. S. TORKY Pade approximation of a function is given by ratio of two we discuss how to solve Numerical solution of nonlinear polynomials. The coefficients of the polynomial in both system of parial differential equations by using LDM. The the numerator and the denominator are determined using results of the present technique have close agreement with the coefficients in the Taylor series expansion of the approximate solutions obtained with the help of the function. The Pade approximation of a function, symbol- Adomian decomposition method [8]. ized by [m/n], is a rational function defined by 2. Laplace Decomposition Method 2 m m aax a xa x  01 2 m  , (13)  2 n U n 1 bx b x  b x  12 n (7)  gxt,,  R U N U i1,2,,n, ii i t where we considered b = 1, and the numerator and de- where Uu  ,,u ,u ,   nominator have no common factors. In the LD-PA method 12 n with initial condition we use the method of Pade approximation as an after- treatment method to the solution obtained by the Laplace ux,0 f x , (8)     ii decomposition method. This after-treatment method im- proves the accuracy of the proposed method. the method consists of first applying the Laplace trans- formation to both sides of (7) 4. Application  U ££gx,t   In this section, we demonstrate the analysis of our nu-   t  (9) merical methods by applying methods to the system of partial differential Equations (1), (3) and (5). A com- £,  RU N U i1,2,,n,    ii  parison of all methods is also given in the form of graphs and tables, presented here. using the formulas of the Laplace transform, we get sU££  f x  g x,t      4.1. The Laplace Decomposition Method ii (10) Exampe 1. The Whitham-Broer-Kaup model [5] £,  RU N U i1,2,,n,     ii To solve the system of Equation (1) by means of Laplace decomposition method, and for simplicity, we in the Laplace decomposition method we assume the   solution as an infinite series, given as follows  ,2 k , 3,1,1,0 take , we con-    UU  , (11) struct a correctional functional which reads £0 uu    where the terms U are to be recursively computed. Also the linear and nonlinear terms R and Ni,1  ,2,,n is decomposed as an infinite series of £, uu v u  xx xx Adomian polynomials (see [8,9]). Applying the inverse (14) Laplace transform, finally we get £0 vv    1 Uf ££ x g x,t ii  (12) £3 vu uv v  u,  xx xx xxx £,  RU N U i1,2,,n, s ii  we can define the Adomian polynomial as follows: 3. The Pade Approximant nn n Au u ,, B vu C uv , (15)   ni ni x n i ni x ni ni x    Here we will investigate the construction of the Pade ii 00 i0 approximates [10] for the functions studied. The main we define an iterative scheme advantage of Pade approximation over the Taylor series approximation is that the Taylor series approximation ££uA  v u ,   n1 n nx nxx can exhibit oscillati which may produce an approxima- (16) tion error bound. Moreover, Taylor series approxima- tions can never blow-up in a fin region. To overcome ££vB  Cv 3u ,  n1 n n nxx nxxx these demerits we use the Pade approximations. The Copyright © 2013 SciRes. AJCM M. A. MOHAMED, M. S. TORKY 177 applying the inverse Laplace transform, finally we get terms as Equation (18), and Figures 1(a) and (b) show th Equations (17). Similarly, we can also find other com- the exact and numerical solution of system (1) with 16 th ponents, and the approximate solution for calculating 16 terms by (LDM). (a) (b) Figure 1. (a) Exact and numerical solution of u(x, t), −10 ≤ x ≤ 10, -1 ≤ t ≤ 1; (b) Exact and numerical solution of v (x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1. 32tx sinh 2 18 2 t ux,8 t tanh 2x, v x,t 1616tanh 2x , u x,t  , v x,t  ,         00 1 1 cosh 2xx cosh 2 (17) 16tx 8 cosh  8 coshx 1    8stx inh 2  ux,, t  v x,t  , 38 6 4 2 cosh 2xx 1 16 cosh 32 coshx 24 coshx 8 coshx       88tx cosh  8coshx 1       8stx inh 2 18t  ux,8 t tanh2x       23 8 cosh 2xx cosh 2 3(1 16 coshx    (18) 16tx 8 cosh 8 coshx 1 32 sinh 2  tx vx,1 t 6 16tanh 2x  ,    cosh 2xx cosh 2   Copyright © 2013 SciRes. AJCM 178 M. A. MOHAMED, M. S. TORKY Example 2. coupled nonlinear RDEs [6] Example 3. Hirota-Satsuma coupled KdV System To solve the system of Equation (3) by means of [7] Laplace decomposition method, and for simplicity, we To solve the system of Equation (5) by means of take kc 2, 1, 10 , we construct a correctional Laplace decomposition method, and for simplicity, we   functional which reads take kc  c 1 , we construct a correctional   functional which reads  £0 uu £2uuv10u,   xx x  11 1   ss £0 uu £ u 3uu3vw3, wv (19)   xxx x x x   11 ss 2    £0 vv £2vuv10u ,   xx x  ss 11 £0 vv £v 3uv, (24)    xxx x ss we can define the Adomian polynomial as follows: £0 ww £w 3uw,    xxx x Au  v , (20)  ss ni ni x  i0 we can define the Adomian polynomial as follows: we define an iterative scheme nn n Au u ,, B vw C wv , nin i n i  ni x  ni x  ni x ££uu2 A10u,   ii 00 i0 nn 1 xx n n (25) nn (21) Du v ,, E uw nini  ni x  n i x ££vv2 A10u,   nn 1 xx n n ii 00 we define an iterative scheme applying the inverse Laplace transform, finally we get   ££ uu3A3B3C, n1 nxxx n n n   ux,2 t 2tanh x, v x,t  2tanh x,       00 s 2   22 tt ££vv  3D, (26)   ux,, t  v x,t  , nn 1 xxx n   coshxx cosh ££ww  3E,   2stx inh 2stx inh nn 1 xxx n   (22) ux,, t  v x,t  ,   coshx coshx   applying the inverse Laplace transform, finally we get tx  32cosh  2 ux ,t   2 tanhx , v x,t  tanhx , ux,, t  00  cosh x  4stx inh  3 wx,1 t tanh x, u x,t  ,      tx  3 2 cosh cosh 3   cosh x  vx,, t  , cosh cosh x  8tt vx,, t w x,t ,   coshxx cosh similarly, we can also find other components, and the th approximate solution for calculating 16 terms as fol- 2 22cosh 3 tx   lows: ux,, t   2 cosh x  2stx inh 2t   ux,2 t 2tanh x   22    tx sinh tx sinh 8   coshxx cosh vx,, t  w x,t  ,   coshxx cosh tx  32cosh    tx cosh  3 sinhx    cosh x  ux,, t  (23) cosh x  2stx inh 92t vx,2 t tanh x   3   tx 2cosh  3   cosh x cosh x vx,, t   2 9 3 cosh x tx  32cosh   , 4 tx 2cosh  3   3 1 cosh x  wx,, t  cosh x  and Figures 2(a) and (b) show the exact and numerical th (27) solution of system (3) with 16 terms by (LDM). Copyright © 2013 SciRes. AJCM M. A. MOHAMED, M. S. TORKY 179 (a) (b) Figure 2. (a) Exact and numerical solution of u(x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1; (b) Exact and numerical solution of v(x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1. similarly, we can also find other components, and the ap- and Figures 3(a)-(c) show the exact and numerical so- th th proximate solution for calculating 16 terms as follows: lution of system (5) with 16 terms by (LDM). 4stx inh 1  4.2. The Pade Approximation ux,2 t  tanh x  cosh x  In this section we use Maple to calculate the [3/2] the Pade 2tx 2 cosh 3 t coshx 3 sinhx   8 approximant of the infinite series solution (18), (23), and   (28) which gives the rational fraction approximation to the coshxx cosh solution, and Figures 4(a)-(c) show the results obtained 4stx inh 1  vx,2 t  tanh x  by the Pade approximant (LD-PA) solution of systems (1), cosh x  (3) and (5), and Figures 5(a)-(c) show comparison be- tween the exact solution, LDM solution and the Pade 2tx 2 cosh 3 t coshx 3 sinhx        approximant (LD-PA) solution of systems (1), (3) and (5) coshxx cosh at, x = 5, −1 ≤ t ≤ 1. Tables 1-3 show the absolute error tx sinh t  between the exact solution and the results obtained from wx,1 t tanhx  the, LDM solution and the Pade approximant (LD-PA) coshxx cosh   solution of systems (1)-(3). tx 2cosh  3    , 5. Conclusion cosh x (28) The Laplace decomposition method is a powerful tool Copyright © 2013 SciRes. AJCM 180 M. A. MOHAMED, M. S. TORKY (a) (b) (c) Figure 3. (a) Exact and numerical solution of u(x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1; (b) Exact and numerical solution of v(x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1; (c) Exact and numerical solution of w(x, t), −10 ≤ x ≤ 10, −1 ≤ t ≤ 1. Copyright © 2013 SciRes. AJCM M. A. MOHAMED, M. S. TORKY 181 (a) (b) (c) Figure 4. (a) The Pade approximant (LD-PA) solution of u(x, t) and v(x, t) of example 1, −10 ≤ x ≤ 10, −1 ≤ t ≤ 1; (b) The Pade approximant (LD-PA) solution of u(x, t) and v(x, t) of example 2, −10 ≤ x ≤ 10, −1 ≤ t ≤ 1; (c) The Pade approximant (LD-PA) solution of u(x, t) and v(x, t) and w(x, t) of example 3, −10 ≤ x ≤ 10, −1 ≤ t ≤ 1. Copyright © 2013 SciRes. AJCM 182 M. A. MOHAMED, M. S. TORKY (a) (b) (c) Figure 5. (a) The exact, (LDM)and the Pade approximant (LD-PA) solution of u(x, t) and v(x, t) of example 1, x = 5, −1 ≤ t ≤ 1; (b) The exact, (LDM)and the Pade approximant (LD-PA) solution of u(x, t) and v(x, t) of example 2, x = 5, −1 ≤ t ≤ 1; (c) The exact, (LDM)and the Pade approximant (LD-PA) solution of u(x, t), v(x, t) and w(x, t) of example 3, x = 5, −1 ≤ t ≤ 1. which is capable of handling nonlinear system of partial proximate solutions for,the Whitham-Broer-Kaup shal- differential equations. In this paper the (LDM) and Pade low water model, the coupled nonlinear reaction diffu- approximant has been successfully applied to find ap- sion equations and thesystem of Hirota-Satsuma coupled Copyright © 2013 SciRes. AJCM M. A. MOHAMED, M. S. TORKY 183 Table 1. The absolute error of u(x, t) and v(x, t) of example 1, x = 40. uu  vv  uu  vv  ex LDM ex LDM ex LD – PA ex LD – PA 0 0 0 0 0 −69 −69 −75 −74 0.2 2.56 × 10 1.02 × 10 4.16 × 10 1.66 × 10 −69 −69 −73 −72 0.4 6.38 × 10 2.55 × 10 3.79 × 10 1.51 × 10 −68 −69 −72 −71 0.6 1.20 × 10 4.83 × 10 6.24 × 10 2.49 × 10 −68 −69 −71 −70 0.8 2.06 × 10 8.24 × 10 5.14 × 10 2.05 × 10 −68 −67 −70 −69 1.0 3.32 × 10 1.33 × 10 2.90 × 10 1.16 × 10 −68 −67 −69 −69 1.2 5.22 × 10 2.08 × 10 1.29 × 10 5.16 × 10 −68 −67 −69 −68 1.4 8.05 × 10 3.21 × 10 4.81 × 10 1.92 × 10 −67 −67 −68 −68 1.6 1.22 × 10 4.90 × 10 1.54 × 10 6.19 × 10 −67 −67 −68 −67 1.8 1.85 × 10 7.42 × 10 4.33 × 10 1.73 × 10 −67 −66 −67 −67 2.0 2.79 × 10 1.11 × 10 1.05 × 10 4.22 × 10 Table 2. The absolute error of u(x, t) and v(x, t) of example 2, x = 40. uu  vv  uu  vv  ex LDM ex LDM ex LD – PA ex LD – PA 0 0 0 0 0 −45 −45 −41 −41 5.42 × 10 0.2 5.42 × 10 5.76 × 10 5.76 × 10 −45 −45 −39 −39 0.4 3.96 × 10 3.96 × 10 5.25 × 10 5.25 × 10 −45 −45 −38 −38 0.6 1.41 × 10 1.41 × 10 8.64 × 10 8.64 × 10 −45 −45 −37 −37 0.8 5.95 × 10 5.95 × 10 7.12 × 10 7.12 × 10 −44 −44 −36 −36 1.0 3.28 × 10 3.28 × 10 4.02 × 10 4.02 × 10 −43 −43 −35 −35 1.2 6.81 × 10 6.81 × 10 1.78 × 10 1.78 × 10 −42 −42 −35 −35 1.4 9.59 × 10 9.59 × 10 6.66 × 10 6.66 × 10 −41 −41 −34 −34 1.6 9.52 × 10 9.52 × 10 2.14 × 10 2.14 × 10 −40 −40 −34 −34 1.8 7.24 × 10 7.24 × 10 6.00 × 10 6.00 × 10 −39 −39 −33 −33 2.0 4.46 × 10 4.46 × 10 1.46 × 10 1.46 × 10 Table 3. (a) The absolute error of u(x, t), v(x, t) and w(x, t) of example 3, x = 40; (b) The absolute error of u(x, t), v(x, t) and w(x, t) of example 3, x = 40. (a) uu  vv  ww  ex LDM exact LDM exact LDM 0 0 0 35 35 35 0.2 4.7610 3.1710 1.1910 35 35 35 0.4 7.9510 5.3010 1.9810 34 35 35 0.6 1.0010 6.7210 2.5210 34 35 35 0.8 1.1510 7.6810 2.8810 34 35 35 1.0 1.2410 8.3210 3.1210 34 35 35 1.2 1.3110 8.7510 3.2810 34 35 35 1.4 1.3510 9.0310 3.3910 34 35 35 1.6 1.3810 9.2310 3.4610 34 35 35 1.8 1.4010 9.3610 3.5010 34 35 35 2.0 1.4110 9.4410 3.5410 (b) uu  vv  ww  ex LD – PA eLD –PA ex LD – PA 0 0 0 41 41 41 0.2 5.9310 3.9710 1.2010 39 39 40 0.4 2.7810 1.8510 6.9710 38 38 39 0.6 2.3510 1.5710 5.8910 38 38 38 0.8 9.9610 6.6410 2.4910 37 37 38 1.0 2.8910 1.9210 7.2210 37 37 37 1.2 6.6310 4.4210 1.6510 36 37 37 1.4 1.3010 8.6610 3.2510 36 36 37 1.6 2.2710 1.5110 5.6810 36 36 37 1.8 3.6410 2.4310 9.1110 36 36 36 2.0 5.4710 3.6510 1.3610 Copyright © 2013 SciRes. AJCM 184 M. A. MOHAMED, M. S. 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