SQU Journal for Science, 2015, 20(2), 19-30 ©2015 Sultan Qaboos University 19 Numerical Wave Solutions for Nonlinear Coupled Equations using Sinc-Collocation Method Kamel Al-Khaled Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box: 36, PC 123, Al-Khod, Muscat, Sultanate of Oman. E-mail: kamel@squ.edu.om ABSTRACT: In this paper, numerical solutions for nonlinear coupled Korteweg-de Vries(abbreviated as KdV) equations are calculated by the Sinc-collocation method. This approach is based on a global collocation method using Sinc basis functions. The first step is to discretize time derivative of the KdV equations by a classic finite difference formula, while the space derivatives are approximated by a -weighted scheme. Sinc functions are used to solve these two equations. Soliton solutions are constructed to show the nature of the solution. The numerical results are shown to demonstrate the efficiency of the newly proposed method. Keywords: Nonlinear coupled KdV equations; Sinc-collocation method; Soliton solutions. التجميعية-حلول الموجة العذدية لمعادالت الذوال غير الخطية باستخذام طريقة سينك كامل الخالذ انتجًيعيح. -( تطزيقح سينكKdVتى في هذا انثحث إيجاد انحهىل انعذديح نًعادالخ انذوال غيز انخطيح وانًسًاج كىرتيج دو فزيس )يختصز ملخص: ح وفقا نصيغ KdVيستنذ هذا اننهج عهى طزيقح تجًيعيح شايهح تاستخذاو دوال انسينك األساسيح. تى في انخطىج األونى تقطيع يشتقح انشين نهًعادالخ و دوال سينك. كًا تى إيجاد افزق انًحذود انكالسيكيح، تينًا تى تقزية انًشتقاخ انفضائيح تطزيقح نظاو انًىاسنح. وتى حم هاتين انًعادنتين تاستخذ وحساب حهىل انًىجح انًنعشنح إلظهار طثيعح انحم. وأظهزخ اننتائج انعذديح كفاءج انطزيقح انًقتزحح. انتجًيعيح.-خطيح، حهىل انًىجح انًنعشنح، طزيقح سينكانانًقزونح غيز KdVيعادالخ :كلمات مفتاحية 1. Introduction onlinear partial differential equations appear in many branches of physics, engineering and applied mathematics. The coupled KdV equations, recently, have been an attractive research area for scientists, because of their many applications in scientific fields and many studies have been reported in the literature, see for example [1-5]. In this paper, we study the coupled KdV equations that were introduced by Hirota-Satsuma [6] = 6 2 , = 3 , t xxx x x t xxx x u u uu vv v v uv         (1) subject to the initial conditions ( , 0) = ( ), ( , 0) = ( ),u x f x v x g x a x b  (2) and boundary conditions 1 2 1 2 ( , ) = ( ), ( , ) = ( ), ( , ) = ( ), ( , ) = ( ),u a t f t u b t f t v a t g t v b t g t (3) where ( , )u x t and ( , )v x t are real-valued scalar functions, t is time, and x is a spatial variable. The equations (1)- (3) describe interactions of two long waves with different dispersion relations. Many powerful methods have been developed to find solutions (exact, or numerical) of such nonlinear evolution equations. These include the Adomian decomposition method [7] and the collocation method [8]. The soliton solutions for this system are constructed by Fan [9]. Mesh-free methods are the topic of recent research in many areas of computational science and approximation theory. Over the past several years mesh-free approximation methods have found their way into many different application areas ranging from engineering to the numerical solution of differential equations. A meshless method does not require a grid, and only makes use of a set of scattered collocation points. In [10,11], the authors propose a mesh- N KAMEL AL-KHALED 20 free collocation method and formulate simple classical radial basis functions for the numerical solution of the KdV equation, and coupled KdV equations. In [12], the authors employ a mesh-free technique for the computational solution of the two-dimensional coupled Burgers' equations. They combine the collocation method using the radial basis functions with first-order accurate forward difference approximation to obtain a mesh-free solution of the coupled Burgers' equations. A mesh-free technique was presented in [13] for the solution of the generalized regularized long wave equation, the collocation being founded using Sinc function basis. The purpose of this paper, and following [11], is to elaborate a special collocation method based on Sinc function basis for spatial derivatives, and classic finite difference formulae for time derivatives, to obtain numerically the traveling wave solutions of the KdV system in (1)- (3). The Sinc-collocation method will be used in the space direction. The main idea is to replace derivatives by their Sinc approximations. The ease of implementation coupled with the exponential convergence rate have demonstrated the viability of this method. Sinc functions are discussed in Stenger [14] and by Lund and Bowers [15]. The layout of the paper is as follows. In section2, we briefly review some general concepts of the Sinc function that are necessary for the formulation of the discrete system. In section 3, we discuss the mesh-free method together with the Sinc-collocation discretization of the coupled KdV equations. Section 4 is devoted to the stability of the method, by using a linearized stability analysis. Finally, numerical experiments are presented and some comparisons are made in section 5. Some concluding remarks are given in the final section 6. 2. Sinc-Collocation The goal of this section is to recall notations and definitions of the Sinc function that will be used in this paper. These are discussed in [14,10]. The Sinc function is defined on the whole real line R by sinc ( ) = sin( ) , 0 1, 0 x x x x x         (4) Recall that a radial basis function is a function whose value depends only on the distance of its input to a central point. For a series of nodes equally spaced h apart, the Sinc function can be written as a radial basis function: ( ) = sinc , = 0, 1, 2,... j x jh S x j h       ( 5) The Whittaker cardinal function ( , )C f h of a function f is defined as = ( , )( ) = ( ) ( ). j j C f h x f jh S x    Whenever this series converges, f is approximated by using the finite number of terms. So, for positive integer ,N define = ( , )( )= ( ) ( ). N N j j N C f h x f jh S x (6) Definition 2.1 Let > 0d , and let dD denote the region { = ,| |< }z x iy y d in the complex plane C , and  the conformal map of a simply connected domain D in the complex plane domain onto dD such that ( ) =a  and ( ) =b  , where a and b are the boundary points of .D Let  denote the inverse map of  , and let the arc  , with end points a and b ( ,a b  ), be given by = ( , ).   For > 0h , let the points kx on  be given by = ( ) , k x kh z Z  and ( ) = exp( ( ))z z  . Hence, the numerical process developed in the domain containing the whole real line can be carried over to infinite interval by the inverse map. The approximation of the th n derivatives of ( )f x by the Sinc expansion is given by ( ) = ( ) [ ( )]. nN n jn j N d f f jh S x dx   (7) The derivatives of Sinc functions evaluated at the nodes will be needed [14,15]. In particular, the following convenient notation will be useful in formulating the discrete system. (0) = = [ ( )] | 1, = = 0, jk j x x k S x j k j k     (8) NUMERICAL WAVE SOLUTIONS FOR NONLINEAR COUPLED EQUATIONS 21 (1) = = [ ( )] | = 0, = ( 1) , ( ) k j j x x k jk d S x dx j k j k h k j        (9) 3 (3) =3 2 2 3 = [ ( )] | = 0, = ( 1) [6 ( ) ] , ( ) k j j x x k jk d S x dx j k k j j k k j          (10) where the collocation points are = . k x x In practice, we need to use a finite number of terms in the series (7), say = ,..., ,j N N where N is the number of Sinc grid points. For a restricted class of functions known as the Paly- Weiner class, which are entire functions, the Sinc interpolation and quadratic formulas are exact [14]. A less restrictive class of functions which are analytic only on an infinite strip containing the real line and which allow specific growth restrictions have exponentially decaying absolute errors in the sinc approximation. In order to state the convergence theorem of the Sinc-collocation method, we introduce the following notation and definition. Definition 2.2 For all 0 < < 1 , let ( )dD  be defined by ( ) = { :| |< 1 / , | |< (1 )}. d D z C Rez Imz d    (11) Let ( )dB D be the Hardy space over the region dD , i.e., the set of all functions such that 0 | ( ) | | |< .lim ( )D d f z dz    (12) The properties of functions in ( )dB D and detailed discussions are given in [14]. We recall the following theorem for our convergence purposes. Theorem 2.1[14] Let ,  and d be positive constants. Assume that 1. ( )df B D 2. f decays exponentially on the real line, that is, | ( ) | exp( | |), .f x x x R    Then, we have ( ) ( ) ( 1)/ 2 1 = sup | ( ) ( ) | exp( ) N n n n j j j N f x f S x C N d N       for some constant 1C , where the mesh size h is taken as = / ( ).h d N  The above theorem states that if f is an analytic function on an infinite strip containing the real line, and satisfies some kind of decaying conditions, then the function f together with it derivatives can be approximated by Sinc function methodology with error of exponential order. Therefore, in order to approximate the solution of the KdV system (1) using Sinc basis, we should start with the assumption that the initial conditions in (3) belong to class ( ) d B D . The matrices (0) (1) (3) , ,I I I will appear in the final discrete system, and in order to study the stability of the Sinc-collocation method, we should find some bound for the eigenvalues of these matrices. The matrix (0) I is just the identity matrix which has eigenvalue1. For eigenvalue bounds for the Toeplitz matrices (1) (3) , ,I I we state the following theorem. Theorem 2.2[14] 1. The m m matrix (1) I is a singular skew-symmetric matrix, if its eigenvalues are denoted by (1) = { } N j j N i   , then (1) (1) ... . N N           2. The m m matrix (3) I is a singular skew-symmetric matrix, if its eigenvalues are denoted by (3) = { } N j j N i   , then 3 (3) (3) 3 ... . N N           KAMEL AL-KHALED 22 3. Construction of the Method Consider the nonlinear coupled KdV equations in (1), subject to the initial and boundary conditions (2)-(3). To implement the Sinc-collocation method, we discretize the time derivative of the nonlinear coupled KdV equations using the classic finite difference formula, and space derivatives by the - weighted (0 1)  scheme successive two time levels n and 1n  1 1 1 1 = ( (1 )( ) 6 ( ) 6 (1 )( ) 2 ( ) n n n n n n n xxx xxx x x x u u u u uu uu vv t                    (13) and, 1 1 = ( ) (1 )( ) 3 ( ) n n n n n xxx xxx x v v v v uv t            (14) where = ( , ) n n u u x t is the value of the solution at the th n time step, and 1= ,n nt t t  where t is a time step size. The nonlinear term 1 ( ) n x uu  must be linearized before continuing. This can be accomplished by using the following formula which is obtained by applying the Taylor expansion, as follows Thus, 1 2 ( ) ( ) [( ) ( ) ] ( ) n n n n n n x x t x xt uu uu t u u u u O t       (15) which can be simplified to 11 1 2( ) ( ) ( ) ( ) [ ] ( ) n nn n n n n n x x x x x u uu u uu uu t u u O t t t             (16) Finally, we arrive at the linearization 1 1 1 ( ) ( ) n n n n n n n x x x x uu u u u u u u       (17) so that equations (14)-(15) can be rewritten as 1 1 1 1 ( ) 6 ( ) = (1 ) 6 (1 ) 6 2 n n n n n n xxx x x n n n n n n n n xxx x x x u t u t u u u u u t u t u u t u u tv v                         (18) and 1 1 ( ) = (1 )( ) 3 n n n n n n xxx xxx x v t v v t v tu v           (19) where n u and n v are the th n iterates of the approximate solutions. Now the space variable is discretized upon the use of Sinc-collocation at the points 1 | | { = ,..., ( 1) ,..., = }, = . 1 i N b a x a x a i h x b h N      (20) The solution of equation (13) is interpolated and approximated by means of the Sinc functions as =1 =1 ( ) = ( ), ( ) = ( ) N N n n n n j j j j j j u x u S x v x v S x  (21) where ( 1) ( ) = sinc( ). j x j h a S x h    (22) The unknown parameters , j j u v in equation (21) are to be determined by the collocation method. Therefore, for each collocation point ix in (20), equation (22) can be written as =1 =1 ( ) = ( ), ( ) = ( ), = 1,..., . N N n n n n i j j i i j j i j j u x u S x v x v S x i N  (23) Substituting equation (23) into equations (18) and (19), we get 1 1 2 ( ) ( ) ( ) n n n n x x x x u u u u t O t t           NUMERICAL WAVE SOLUTIONS FOR NONLINEAR COUPLED EQUATIONS 23 1 1 1 =1 =1 =1 =1 1 =1 =1 =1 =1 =1 =1 =1 ( ) ( ) 6 [ ( ) ( ) ( ) ( )] = ( ) (1 ) ( ) 6 (1 ) ( ) ( ) 6 N N N N n n n n j j i j j i j j i j j i j j j j N N N N n n n n j j i j j i j j i j j i j j j j N N n n j j i j j i j j j u S x t u S x t u S x u S x u S x u S x u S x t u S x t u S x u S x t                                   =1 =1 =1 ( ) ( ) 2 ( ) ( ). N N n n j j i j j i j N N n n j j i j j i j j u S x u S x t v S x v S x       (24) and 1 1 =1 =1 =1 =1 =1 =1 ( ) ( ) = ( ) (1 ) ( ) 3 ( ) ( ) N N N n n n j j i j j i j j i j j j N N N n n n j j i j j i j j i j j j v S x t v S x v S x t v S x t u S x v S x                    (25) Equations (24) and (25) are used for all interior points = , = 2,..., 1ix x i N  , where primes in these equations denote differentiation with respect to the variable x . The boundary condition given by equation (17) for the boundary points = , = 1,ix x i N can be written as 1 1 1 1 1 2 =1 =1 ( ) = ( ), ( ) = ( ) N N n n n n j j i j j i j j u S x f t u S x f t       (26) and 1 1 1 1 1 2 =1 =1 ( ) = ( ), ( ) = ( ). N N n n n n j j i j j i j j v S x g t v S x g t       (27) To obtain matrix representation of the expression in equation (24) and (25), we introduce the following matrix and vector notations 1 2 (0) (1) (3) = [ , , ..., ] , = ( ), , = 1,..., , = ( ), , = 1,..., , = ( ), , = 1,..., . n n n n T N j i j i j i U u u n I S x i j N I S x i j N I S x i j N   (28) Note that the matrices (1) (3) ,I I are skew symmetric. The system of equations (24)-(27) can be solved for unknown parameters , j j u v in equation (21) simultaneously, and then the solutions for u and v can be obtained from equation (21). Equations (24)-(27) can be written in matrix form as (0) (3) (1) (0) 1 [ 6 [ ]] n n n x I t I t U I U I u          (0) (3) (1) (1) (1) 1 = [ (1 ) 6 (1 ) 6 2 ] n n n n n I tI tU I t U I tV I u F                    (29) and (0) (3) 1 (0) (3) (1) 1 [ ] = [ (1 ) 3 ] n n n n I t I v I tI tU I v G              (30) where  stands for component by component multiplication, and 1 1 1 1 2 = [ ( ), 0,..., 0, ( )] n n n T F f t f t    , and 1 1 1 1 2 = [ ( ), 0,..., 0, ( )] n n n T G g t g t    . Equations (29) and (30) can be written in a more compact form as 1 1 1 1 = n n n A u B u F    (31) and 1 1 2 2 = , n n n A v B v G    (32) where KAMEL AL-KHALED 24 (0) (3) (1) (0) 1 (0) (3) (1) (1) (1) 1 (0) (3) 2 (0) (3) (1) 2 = 6 [ ], = (1 ) 6 (1 ) 6 2 , = , = (1 ) 3 . n n x n n n n A I t I t U I U I B I tI tU I t U I tV I A I t I B I tI tU I                                       Equation (21) can written in matrix form as (0) (0) = , = . n n n n U I u V I v (33) From equation (31), we get 1 1 1 1 1 1 1 = . n n n u A B u A F      Combining the above equation together with equation (33), we arrive at 1 (0) 1 (0) 1 1 1 1 1 = . n n n U I A B u I A F      (34) Similarly, using equations (32) and (33) we arrive at 1 (0) 1 (0) 1 1 2 2 2 = . n n n V I A B v I A G      (35) We can obtain the coefficients of the approximate solution by solving the system in equations (34)-(35) using any iterative technique. For the convergence of the method, we state the following two theorems. Theorem 3.1 Let the function ( , )u x t be as in equation (1) with the initial condition as in equation (2), and let the matrix U be defined as in (34). Then for a sufficiently large ,N there exists a constant C independent of N such that   2 , || ( , ) || exp( ).sup n j n j x t u x t U CN d N    Theorem 3.2 Given a constant > 0R , there is a constant > 0T such that if 1 0 || || / 2U U R  , then the iterative scheme (34) converges to the unique solution. The proof of the above two theorems are immediate from [16,17]. We would like to mention here that for Theorem 3.1, we use the second part of Theorem 2.1 for = 1n and = 3n , which is a simple modification of Theorem 3.2 in [2]. For Theorem 3.2, we use contraction mapping of the iteration scheme given in equation (34) and apply fixed point theorem to prove convergence. Interested readers may follow [16] for a detailed analysis 4. Stability Analysis In this section, we present an analysis of the stability of the Sinc-collocation method for solving the coupled KdV system using spectral radius matrices. Following the method outlined in [11], let ,U V be the exact, and ,U V be the numerical solutions of the coupled system in (1). Define the error vectors = , n u U U  and = . n v V V  Using equations (34) and (35) the errors , n n u v   can be written as 1 1 1 (0) 1 1 1 1 (0) 1 1 1 2 2 = = , = = . n n n n n n n n u u v v U U I A B V V I A B              (36) For stability of the method, we need 0 n u   , and 0 n v   for large values of n . Therefore, the scheme is considered numerically stable if (0) 1 1 1 ( ) < 1I A B  , and (0) 1 2 2 ( ) < 1I A B  , where (.) denotes the spectral radius. The Sinc matrices (1) (3) ,I I are contained in the matrices 1 2 1, ,A A B and 2B , where the bounds for the eigenvalues of these matrices are given in Theorem 2.2. Therefore, stability is assured if 3 1 2 3 1 1 1 < 1 1 6 [ ] 1 (1 ) 6 (1 ) 6 2 N N N N N t t t t t t                            (37) and NUMERICAL WAVE SOLUTIONS FOR NONLINEAR COUPLED EQUATIONS 25 3 3 1 1 < 1 1 (1 ) 3 N t t t             (38) where we have used the numbers 1 , 1 3 1 2, , ,N N    being eigenvalues of the matrices (0) (1) (3) (1) (0) , , , , n n I I I U I U I  respectively. Using the upper bound for the eigenvalues in Theorem 2.2, together with the fact that 1 1 = | |i  , 3 3 = | |i  , and if 1 2 , N N   are complex, then after algebraic manipulation (see, [13,10,11]), the conditions (37) and (38) must hold for all eigenvalues of the respective matrices. Having 1/ 2 < 1 is necessary, but not sufficient, to guarantee the stability of the Sinc collocation method. 5. Numerical Results Choosing examples with known solutions allows for a more complete error analysis. In order to assess the advantages of the proposed method, in terms of accuracy and efficiency for solving nonlinear coupled KdV equations, the following examples are presented in this section. Example 5.1 In this example, we have to apply our scheme to solve equation (1) with = 1, 1/ 2   , and > 0 , with the initial conditions 2 2 2 2 1 ( , 0) = 4 , ( , 0) = 3 6 (1 ) (1 ) kx kx kx kx e Me u x k k v x e e         (39) Where 224 =M k   , and k is an arbitrary constant. The exact solution is given by [7] ( ) ( ) 2 2 ( ) 2 ( ) 2 1 ( , ) = 4 , ( , ) = 3 6 (1 ) (1 ) k x ct k x ct k x ct k x ct e Me u x t k k v x t e e              (40) The computations associated with the example were performed using Mathematica. In our computational work, we take = 1.5, = 0.1, = 0.1, = 0.1c k  , and two different time step sizes = 0.1t and = 0.05t through the interval [ 30, 30] and = 160N for the set of collocation points as in equation (20). The accuracy of the scheme is measured by using the following two error norms [1, 11] 2 2 2 1=1 =|| || = | | , =|| || = | |max N j j j j j Nj L u u h u u L u u u u         where u and u represent the exact and approximate solutions, respectively, and h is the minimum distance between any two points in equation (21), similarly for the v solution. The pointwise rate of convergence in time is also calculated by using the following formula [1,11] 10 1 110 (|| || / || ||)log Order = ( / )log exact t exact t j j j j u u u u t t         where exactu is the exact solution, and t j u  is the numerical solution with time step size j t . In Table 1, the number of collocation points is = 160N , and the time step size = 0.8, 0.4, 0.2, 0.1, 0.05, 0.025, 0.01t is varied to compute the time rate of convergence when = 1/ 2 . It can be noted from Table 1 that the method has order of convergence 2 . The accuracy of the proposed method is demonstrated for the absolute errors for solution of (1) with their exact solutions. Table 2 reports the supremum norm error between the exact solution (40) and our approximate solution compared with the results in [11]. A clear conclusion can be drawn from the numerical results in Figures 1, 2 and Table 2 that Sinc methodology provides highly accurate numerical solutions. KAMEL AL-KHALED 26 Figure 1. The exact and approximate solutions for ( , )u x t in Example 5.1. Figure 2. The exact and approximate solutions for ( , )v x t in Example 5.1. Table 1. Time rate of convergence solution of equation (1): = 16, = 1/ 2, = 160, 30 30t N x     t L  Order 2 L Order 0.8 33.10524 10    2 1.10518 10    0.4 31.22201 10   1.96166 3 3.22148 10   1.96453 0.2 45.35209 10   1.99086 4 8.35020 10   1.99486 0.1 41.49743 10   1.99748 4 4.49278 10   1.99782 0.05 58.66014 10   1.99835 4 1.65082 10   1.99892 0.025 56.84252 10   1.99809 5 5.82611 10   1.99212 0.010 52.04701 10   1.98971 5 1.02058 10   1.98775 Table 2. Numerical results for Example 5.1when = 0.001.t x | |u u | |u u as in [13] | |v v | |v v as in [13] 4 82.039 10   7 1.766 10   8 9.672 10   8 1.239 10   3 81.565 10   6 1.368 10   8 7.423 10   8 9.832 10   2 81.060 10   5 1.056 10   8 5.031 10   7 8.829 10   1 95.356 10   5 2.966 10   8 2.540 10   5 1.023 10   0 132.485 10   4 1.367 10   12 1.179 10   6 2.001 10   1 95.353 10   5 3.897 10   8 2.540 10   5 1.195 10   2 81.060 10   5 1.241 10   8 5.031 10   6 1.252 10   3 81.565 10   5 1.666 10   8 7.423 10   6 1.182 10   4 82.039 10   6 7.380 10   8 9.672 10   7 5.219 10   NUMERICAL WAVE SOLUTIONS FOR NONLINEAR COUPLED EQUATIONS 27 Example 5.2 In this example, we solve the problem = 2 , = t xxx x x t xxx x u u uu vv v v uv     (41) subject to the initial conditions 2 2 3 1 1 3 1 1 1 ( , 0) = s [ ], ( , 0) = [ ],tanh 4 2 2 4 2 2 2 u x ech x v x x a x b  (42) and boundary conditions ( , ) = ( , ) = 0, ( , ) = ( , ) = 0u a t u b t v a t v b t as a and b both approaches  . The exact solution is known as [7] 2 2 3 1 1 1 3 1 1 1 1 ( , ) = s [ ( )], ( , ) = [ ( )].tanh 4 2 2 2 4 2 2 2 2 u x t ech x t v x t x t  (43) The numerical solutions are shown in Figures 3 9 . These solutions are the bell-shaped waves, which agree with the results of [7]. From Figures 3 and 5, we understand that in both cases of u and ,v the solutions are a solitary wave pattern. Also, from Figures 4 and 6 , we see that for 0.5t  , the solution starts to bifurcate into three waves. Figures 7, 8 show both the approximate and exact solutions for both ( , )u x t and ( , )v x t , while Figure 9 shows the absolute error when finding the solution of ( , )u x t . Figure 3.Three profile solutions for ( , )v x t when 160,N  t =0.1, 0.2, 0.3 and 0.05t  . Figure 4. Two profile solutions for ( , )u x t when 160,N  t =0.5, 0.6 and 0.05t  . 5 5 x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 5 5 x 0.2 0.4 0.6 0.8 1.0 KAMEL AL-KHALED 28 Figure 5. Three profile solutions for ( , )v x t when 160,N  t =0.1, 0.2, 0.3 and 0.05t  . Figure 6. Two profile solutions for ( , )v x t when 160,N  t =0.5, 0.6 and 0.01t  . Figure 7. The exact and approximate solutions of ( , )u x t in Example 5.2. 5 5 x 0.1 0.2 0.3 0.4 0.5 4 2 2 4 x 0.2 0.4 0.6 0.8 NUMERICAL WAVE SOLUTIONS FOR NONLINEAR COUPLED EQUATIONS 29 Figure 8. The exact and approximate solutions of ( , )v x t in Example 5.2. Figure 9. The error in our approximate solution for ( , )u x t in Example 5.2. Table 3. The absolute error in finding both ( , )u x t and ( , )v x t in Example 5.2 for t = 0.1. x | |u u | |v v 5 62.6530 10   4 2.8135 10   4 51.7980 10   3 1.0097 10   3 59.9222 10   3 3.1410 10   2 42.6240 10   3 7.1015 10   1 41.6752 10   3 7.6237 10   0 44.8000 10   4 1.8600 10   1 53.9810 10   3 7.1700 10   2 41.7891 10   3 6.5813 10   3 59.5560 10   3 3.3004 10   4 52.5442 10   3 1.1990 10   5 64.7766 10   4 3.6223 10   KAMEL AL-KHALED 30 6. Conclusions The fundamental goal of this paper wasto propose an efficient algorithm for the solution of coupled KdV equations. The Sinc-collocation method was described in detail, and implemented to compute a numerical solution to the system in (1). A brief stability analysis was provided, which produced a necessary condition for stability of the method. The efficiency of the method wastested on one example of soliton type, and the accuracy examined in terms of the 2,L L error norms. 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Sinc numerical solution for solitons and solitary waves. J. Comput. Appl. Math., 2001, 130(1-2), 283-292. Received 28 August 2014 Accepted 24 February 2015