Access the full text.
Sign up today, get DeepDyve free for 14 days.
References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.
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 vu, The Laplace decomposition method (LDM) is one of the txx x (3) efficient analytical techniques to solve linear and nonlin- vk v u vu, 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 tcx , 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 tcx, 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- uu33 uu vw3wv, 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 xt , 22 2 2 22 2 44 kc k k k vx,2 t k ò (2) vx,t t anh k xt, 2 3c 3c 1 1 22 2 2tkk ò, anhxt wx,t t c canh kxt, 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 aax a xa x 01 2 m , (13) 2 n U n 1 bx b x b x 12 n (7) gxt,, R U N U i1,2,,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 i1,2,,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 i1,2,,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 i1,2,,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 ni x Here we will investigate the construction of the Pade ii 00 i0 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 , n1 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 Cv 3u , n1 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 1616tanh 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 coshx 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 £2uuv10u, xx x 11 1 ss £0 uu £ u 3uu3vw3, wv (19) xxx x x x 11 ss 2 £0 vv £2vuv10u , 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 i0 we can define the Adomian polynomial as follows: we define an iterative scheme nn n Au u ,, B vw C wv , nin i n i ni x ni x ni x ££uu2 A10u, ii 00 i0 nn 1 xx n n (25) nn (21) Du v ,, E uw nini ni x n i x ££vv2 A10u, nn 1 xx n n ii 00 we define an iterative scheme applying the inverse Laplace transform, finally we get ££ uu3A3B3C, n1 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 coshxx 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 tanhx , v x,t tanhx , 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 , coshxx 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 coshxx cosh vx,, t w x,t , coshxx 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 coshx 3 sinhx 8 approximant of the infinite series solution (18), (23), and (28) which gives the rational fraction approximation to the coshxx 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) coshxx 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 tanhx 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 4.7610 3.1710 1.1910 35 35 35 0.4 7.9510 5.3010 1.9810 34 35 35 0.6 1.0010 6.7210 2.5210 34 35 35 0.8 1.1510 7.6810 2.8810 34 35 35 1.0 1.2410 8.3210 3.1210 34 35 35 1.2 1.3110 8.7510 3.2810 34 35 35 1.4 1.3510 9.0310 3.3910 34 35 35 1.6 1.3810 9.2310 3.4610 34 35 35 1.8 1.4010 9.3610 3.5010 34 35 35 2.0 1.4110 9.4410 3.5410 (b) uu vv ww ex LD – PA eLD –PA ex LD – PA 0 0 0 41 41 41 0.2 5.9310 3.9710 1.2010 39 39 40 0.4 2.7810 1.8510 6.9710 38 38 39 0.6 2.3510 1.5710 5.8910 38 38 38 0.8 9.9610 6.6410 2.4910 37 37 38 1.0 2.8910 1.9210 7.2210 37 37 37 1.2 6.6310 4.4210 1.6510 36 37 37 1.4 1.3010 8.6610 3.2510 36 36 37 1.6 2.2710 1.5110 5.6810 36 36 37 1.8 3.6410 2.4310 9.1110 36 36 36 2.0 5.4710 3.6510 1.3610 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
American Journal of Computational Mathematics – Unpaywall
Published: Jan 1, 2013
You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.