NUMERICAL TREATMENT OF COUPLED NONLINEAR HYPERBOLIC KLEIN-GORDON EQUATIONS

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1 NUMERICAL TREATMENT OF COUPLED NONLINEAR HYPERBOLIC KLEIN-GORDON EQUATIONS E.H. DOHA 1,a, A.H. BHRAWY,3,b, D. BALEANU 4,5,6,c, M.A. ABDELKAWY 3,d 1 Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt, a : eiddoha@frcu.eun.eg Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 1589, Saudi Arabia, b : alibhrawy@yahoo.co.u 3 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 6511, Egypt, d : melawy@yahoo.com 4 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 1589, Saudi Arabia 5 Department of Mathematics and Computer Sciences, Canaya University, Esisehir Yolu 9.m, Anara, Turey 6 Institute of Space Sciences, P.O. BOX, MG-3, RO 07715, Magurele-Bucharest, Romania, c : dumitru@canaya.edu.tr Received October 10, 013 A semi-analytical solution based on a Jacobi-Gauss-Lobatto collocation (J-GL- C) method is proposed and developed for the numerical solution of the spatial variable for two nonlinear coupled Klein-Gordon (KG) partial differential equations. The general Jacobi-Gauss-Lobatto points are used as collocation nodes in this approach. The main characteristic behind the J-GL-C approach is that it reduces such problems to solve a system of ordinary differential equations (SODEs) in time. This system is solved by diagonally-implicit Runge-Kutta-Nyström scheme. Numerical results show that the proposed algorithm is efficient, accurate, and compare favorably with the analytical solutions. Key words: Nonlinear coupled hyperbolic Klein-Gordon equations; Nonlinear phenomena; Jacobi collocation method; Jacobi-Gauss-Lobatto quadrature. PACS: 0.30.Gp, 0.30.Hq, 0.30.Jr, 0.30.Mv, x. 1. INTRODUCTION Spectral collocation method is very easy to implement and adaptable to various problems, including variable coefficients and nonlinear differential equations [1 3], integral equations [4, 5], integro-differential equations [6, 7], fractional orders differential equations [8, 9] function approximation and variational problems [10]. The one-dimensional linear or nonlinear KG equation is given in the form: u tt + αu xx + G(u) = H(x,t), (1) RJP Rom. 59(Nos. Journ. Phys., 3 4), Vol , Nos. 3 4, (014) P , (c) Bucharest, 014

2 48 E.H. Doha et al. where u(x,t) and G(u) represent the wave displacement at position x and time t, and the nonlinear force, respectively. The KG equation appears in many types of nonlinearities, and it has a wide range of applications in many scientific fields such as solid state physics, propagation of fluxons in Josephson unctions [11] between two superconductors, motion of rigid pendula attached to a stretched wire, nonlinear optics and optical solitons [1 15], condensed matter physics [16], interaction of solitons in a collisionless plasma and the recurrence of initial states, and quantum field theory [17]. Porsezian and Alagesan [18] used Painlevé analysis to establish the integrability properties of coupled KG equations. Alagesan et al. [19] introduced the traveling wave solution and the bilinear form of coupled nonlinear KG equations and they discussed the integrability of two-coupled nonlinear KG equations. Moreover, Yusufoğlu and Beir [0] obtained the soliton solutions of coupled KG equations using tanh method. Some new exact solutions for nonlinear KG equations have been obtained by Wazwaz in Ref. [1] and some solutions of the Boussinesq and the KG equations have been obtained in Ref. []. There are also numerous results on studying the solitary and periodic wave solutions for several types of KG equations (see, for instance [3 8]). There are a lot of studies on the approximate and analytical solutions of initialboundary problems of partial differential equations [9 33]. In [34], Deeba and Khuri applied the decomposition method for solving the nonlinear KG equation. Yusufoğlu [35] presented and developed the variational iteration technique for solving such equation. Application of differential transform scheme was extended for both linear and nonlinear KG equation in Ref. [36]. Recently, Iqbal et al. [37] investigated the homotopy asymptotic technique for the optimal solution of nonlinear KG equations. In the direction of numerical solutions, a spline collocation scheme has been proposed by Khuri and Sayfy [38] for solving the generalized nonlinear KG equation. In [39], finite difference scheme is applied to numerically solve KG equation. The pseudo-spectral methods are also used in [40] and [41] for approximating the solution of KG equations. Moreover in [4] the authors proposed the collocation method based on expanding the solution in terms of thin plate splines and radial basis functions for solving the nonlinear KG equation. A cubic B-spline collocation scheme has been investigated by Rashidinia et al. in [43] to numerically solve the nonlinear KG equation. In this paper, we propose the J-GL-C method for the numerical solution of two coupled nonlinear KG equations based on Jacobi polynomials. The Jacobi polynomials are the eigenfunctions of the singular Sturm-Liouville problem. Moreover, the Jacobi polynomials which depend on two general parameters θ and ϑ satisfy the orthogonality condition on the unit interval with respect to the weight function (1 x) θ (1 + x) ϑ. There are several particular cases of Jacobi polynomials such as

3 3 Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations 49 Legendre, Chebyshev and Gegenbauer polynomials [44 46]. It would be very useful to carry out a systematic study on J-GL-C method with general indexes (θ,ϑ > 1). The coupled nonlinear KG types equation will be collocated only for the space variable at the nodes of the Jacobi-Gauss-Lobatto interpolation which depends upon the two general parameters (θ, ϑ > 1); these equations together with the boundary conditions constitute SODEs in time. This system can be solved by diagonally-implicit Runge-Kutta-Nyström (DIRKN) method. The organization of the paper is as follows: we present some properties of Jacobi polynomials in the next section. Section 3 is devoted to developing the numerical algorithm of the Jacobi collocation method for solving the coupled nonlinear KG types equations. Two test problems are presented in Section 4 to demonstrate the accuracy of our method. We present some conclusions in the last section.. JACOBI POLYNOMIALS We collect in this section some basic nowledge of Jacobi polynomials that are most relevant to spectral approximations [47]. Jacobi polynomials include Legendre and Chebyshev polynomials as two special cases, so it is worthwhile to wor with general Jacobi polynomials. A basic property of the Jacobi polynomials is that they are the eigenfunctions to a singular Sturm-Liouville problem: (1 x )ϕ (x) + [ϑ θ + (θ + ϑ + )x]ϕ (x) + n(n + θ + ϑ + 1)ϕ(x) = 0. () The following recurrence relation generate the Jacobi polynomials: where +1 (x) = (a(θ,ϑ) x b (θ,ϑ) ) (x) c (θ,ϑ) 1 0 (x) = 1, 1 (x) = 1 (θ + ϑ + )x + 1 (θ ϑ), a (θ,ϑ) = ( + θ + ϑ + 1)( + θ + ϑ + ), ( + 1)( + θ + ϑ + 1) (x), 1, a (θ,ϑ) (ϑ θ )( + θ + ϑ + 1) = ( + 1)( + θ + ϑ + 1)( + θ + ϑ), c (θ,ϑ) ( + θ)( + ϑ)( + θ + ϑ + ) = ( + 1)( + θ + ϑ + 1)( + θ + ϑ). The Jacobi polynomials are satisfying the following identities ( x) = ( 1) (x), ( 1) = ( 1) Γ( + ϑ + 1). (3)!Γ(ϑ + 1)

4 50 E.H. Doha et al. 4 Moreover, the q derivative of Jacobi polynomials of degree ( (x), can be obtained from: D (q) Γ( + θ + ϑ + q + 1) (x) = q Γ( + θ + ϑ + 1) J (θ+q,ϑ+q) q (x). (4) Let w (θ,ϑ) (x) = (1 x) θ (1+x) ϑ, then we define the weighted space L as usual. w (θ,ϑ) The inner product and the norm of L with respect to the weight function are w (θ,ϑ) defined as follows: (u,v) w (θ,ϑ) = 1 1 u(x)v(x)w (θ,ϑ) (x)dx, u w (θ,ϑ) = (u,u) 1 w (θ,ϑ). (5) The set of Jacobi polynomials forms a complete L w (θ,ϑ) -orthogonal system, and θ+ϑ+1 Γ( + θ + 1)Γ( + ϑ + 1) w (θ,ϑ) = h = ( + θ + ϑ + 1)Γ( + 1)Γ( + θ + ϑ + 1). (6) 3. JACOBI SPECTRAL COLLOCATION METHOD The main obective of this section is to develop the J-GL-C method to numerically solve the coupled nonlinear KG types equations in the following form: D t u(y,t) = D yu(y,t) u(y,t) + u 3 (y,t) + u(y,t)v(y,t), D t v(y,t) + 4u(y,t)D t u(y,t) = D y v(y,t), (y,t) [A,B] [0,T ] with the initial-boundary conditions u(a,t) = g 1 (t), u(b,t) = g (t), v(a,t) = g 3 (t), t [0,T ], u(y,0) = f 1 (y), v(y,0) = f (y), D t u(y,0) = f 3 (y), y [A,B]. Now, suppose the change of variables x = B A A B, w(x,t) = u(y,t), z(x,t) = v(y,t), which will be used to transform problem (7)-(8) into another one in the classical interval, [ 1,1], for the space variable, to directly implement collocation method based on Jacobi family defined on [ 1,1], y+ A+B Dt w(x,t) = ( B A ) Dxw(x,t) w(x,t) + w 3 (x,t) + w(x,t)z(x,t), D t z(x,t) + 4w(x,t)D t w(x,t) = ( B A )D xz(x,t), (x,t) [ 1,1] [0,T ], with the initial-boundary conditions w(a,t) = g 4 (t), w(b,t) = g 5 (t), z(a,t) = g 6 (t), t [0,T ], w(x,0) = f 4 (x), z(x,0) = f 5 (x), D t w(x,0) = f 6 (x), x [ 1,1]. (7) (8) (9) (10)

5 5 Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations 51 The collocation nodes are the set of points where the spatial variable values are approximated in a specified domain. The Jacobi collocation points are the roots of the Jacobi polynomials in which the distribution of these nodes can be tuned by the Jacobi parameters, θ and ϑ. This choice of collocation points gives accurate approximations for the spatial variable in the pseudo-spectral methods. The aim of this wor is to demonstrate the advantage of using the Jacobi collocation method for approximating the spatial variable values for the nonlinear KG equation, in a specified domain, [ 1,1]. Now, we outline the main step of the J- GL-C method for solving coupled nonlinear KG types equation. Let us expand the dependent variable in a Jacobi series, w(x,t) = =0 and in virtue of (5)-(6), we deduce that a (t) (x), z(x, t) = a (t) = 1 h b (t) = 1 h =0 w(x,t)w (θ,ϑ) (x) (x)dx, z(x,t)w (θ,ϑ) (x) (x)dx. b (t) (x), (11) To evaluate the previous integrals accurately, we present the Jacobi-Gauss-Lobatto quadrature. For any ϕ S N+1 [ 1,1], 1 1 w (θ,ϑ) (x)ϕ(x)dx = =0 ϖ (θ,ϑ) N, ϕ(x(θ,ϑ) N, (1) ), (13) where S N [ 1,1] is the set of polynomials of degree less than or equal to N, x (θ,ϑ) N, (0 N) and ϖ (θ,ϑ) N, (0 N) are the nodes and the corresponding Christoffel numbers of the Jacobi-Gauss-Lobatto quadrature formula on the interval [ 1, 1], respectively. In accordance to (5) the coefficients a (t) in terms of the solution at the collocation points can be approximated by a (t) = 1 h b (t) = 1 h N N )ϖ (θ,ϑ) )ϖ (θ,ϑ) w(x (α,β),t), z(x (α,β),t). (14)

6 5 E.H. Doha et al. 6 Therefore, (11) can be rewritten as ( N 1 w(x,t) = h z(x,t) = =0 ( N 1 h =0 ) (x)ϖ (θ,ϑ) ) (x)ϖ (θ,ϑ) ) ) w z Furthermore, if we differentiate (15) once, and evaluate it at all J-GL-C points, it is easy to compute the first spatial partial derivative in terms of the values at theses collocation points as where D x w N N,n,t) = A ni w,t), A ni = B ni = D x z N N,n,t) = B ni z,t), n = 0,1,,N, =0 =0 + θ + ϑ + 1 h + θ + ϑ + 1 h )J (θ+1,ϑ+1) 1 )J (θ+1,ϑ+1) 1,t),,t). N,n )ϖ(θ,ϑ), N,n )ϖ(θ,ϑ), Similar steps can be applied to the second spatial partial derivative to get where D ni = =0 E ni = =0 Dxw(x (θ,ϑ) N N,n,t) = D ni w,t), Dxz(x (θ,ϑ) N N,n,t) = E ni z,t),n = 0,1,,N, ( + θ + ϑ + )( + θ + ϑ + 1) 4h ( + θ + ϑ + )( + θ + ϑ + 1) 4h )J (θ+,ϑ+) )J (θ+,ϑ+) N,n )ϖ(θ,ϑ), N,n )ϖ(θ,ϑ). In the proposed J-GL-C method the residual of (7) is set to zero at N 1 of Jacobi- Gauss-Lobatto points, moreover, the boundary conditions (8) will be enforced at the (15) (16) (17) (18) (19)

7 7 Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations 53 two collocation points 1 and 1. Therefore, adopting (7)-(10), enable one to write (9)-(10) in the form: ẅ n (t) = δ 1 ( B A ) D ni w i (t) w n (t) + (w n (t)) 3 + w n (t)z n (t), ż n (t) + 4w n (t)ẇ n (t) = ( B A ) B ni z i (t), where w (t) = w N,,t), (0) z (t) = z N,,t), = 1,,N 1, n = 1,,N 1. This provides a (N ) system of second order ordinary differential equations in the expansion coefficients a (t), b (t), where w 0, w N, z 0 are nown from the boundary conditions. This mean that problem (0) is transformed to the following SODEs ẅ n (t) = ( B A ) E ni w i (t) w n (t) + (w n (t)) 3 + w n (t)z n (t), N ż n (t) + 4w n (t)ẇ n (t) = ( B A ) B ni z i (t), subect to the initial values w n (0) = f 4 N,n ); z n(0) = f 5 N,n ); ẇ n(0) = f 6 N,n ); n = 1,,N 1. () Finally, (1)-() can be rewritten into a matrix form of N ordinary differential equations with their vectors of initial values: where Ẅ(t) = F(t,w(t),z(t)), Ż(t) + 4W(t)Ẇ(t) = G(t,z(t)), W(0) = f 4, Z(0) = f 5, Ẇ(0) = f 6, Ẅ(t) = [ẅ 1 (t),ẅ (t),...,ẅ N 1 (t)] T, Ẇ(t) = [ẇ 1 (t),ẇ (t),...,ẇ N 1 (t)] T, Ż(t) = [ż 1 (t),ż (t),...,ż N 1 (t)] T, f 4 = [f 4 N,1 ),f 4 N, ),...,f 4 N,N 1 )]T, f 5 = [f 5 N,1 ),f 5 N, ),...,f 5 N,N 1 )]T, f 6 = [f 6 N,1 ),f 6 N, ),...,f 6 N,N 1 )]T, F(t,w(t),z(t)) = [F 1 (t,w(t),z(t)),f 1 (t,w(t),z(t)),...,f N 1 (t,w(t),z(t))] (1) (3)

8 54 E.H. Doha et al. 8 and G(t,z(t)) = [G 1 (t,z(t)),g 1 (t,z(t)),...,g N 1 (t,z(t))], where F n (t,w(t),z(t))=( B A ) E ni w i (t) w n (t) + (w n (t)) 3 + w n (t)z n (t), G n (t,z(t)) = ( B A ) B ni z i (t). The SODEs (3) can be solved by using the DIRKN method. This method is one of the suitable methods for solving SODEs of second order. The DIRKN methods have excellent stability properties, which allow to reduce computational costs. Note: As stated in the previous two subsections, the presented algorithm can also solve the coupled nonlinear KG equations in the form D t u(y,t) = D yu(y,t) u(y,t) + v(y,t), D t v(y,t) = D yv(y,t) + u(y,t) v(y,t), (y,t) [A,B] [0,T ] with the initial-boundary conditions u(a,t) = g 1 (t), u(b,t) = g (t), v(a,t) = g 3 (t), u(y,0) = f 1 (y), v(y,0) = f (y), D t u(y,0) = f 3 (y) v(b,t) = g 4 (t), D t v(y,0) = f 4 (y), (y,t) [A,B] [0,T ]. These coupled nonlinear KG equations (5) describe the long-wave dynamics of two coupled one-dimensional periodic chains of particles [0]. (4) (5) (6) 4. TEST PROBLEMS Two examples are presented in this section to demonstrate the applicability of the proposed method and its performance. Comparison of the results obtained by various choices of Jacobi parameters θ and ϑ reveal that the present method is very effective and convenient for all choices of θ and ϑ. We consider the following two examples TEST PROBLEM 1 First, we tested the nonlinear KG equations coupled with a scalar field v, Dt u(y,t) = Dyu(y,t) u(y,t) + u 3 (y,t) + u(y,t)v(y,t), D t v(y,t) + 4u(y,t)D t u(y,t) = D y v(y,t), (y,t) [A,B] [0,T ] (7)

9 9 Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations 55 Fig. 1 The approximate solution ũ(x,t) for problem (7) where θ = ϑ = 0 and N = 0 in the interval [ π,π]. Exacte and approximate solutions u x,0.1 u x,0.1 u x,0.5 u x,0.5 u x,0.9 Exacte and approximate solutions u x,0.1 u x,0.1 u x,0.5 u x,0.5 u x, x u x, x u x,0.9 a b Fig. The approximate solution ũ(x,t) and the exact solution u(x,t) for t = 0.1, 0.5, and 0.9 for problem (7) where θ = ϑ = 0 and N = 4 in the interval [ π,π].

10 56 E.H. Doha et al. 10 Fig. 3 The approximate solution ṽ(x,t) for problem (7) where θ = ϑ = 0 and N = 0 in the interval [ π,π] Exacte and approximate solutions v x,0.1 v x,0.1 v x,0.5 v x,0.5 v x,0.9 Exacte and approximate solutions v x,0.1 v x,0.1 v x,0.5 v x,0.5 v x, v x, v x, x x a b Fig. 4 The approximate solution ṽ(x,t) and the exact solution v(x,t) for t = 0.1, 0.5 and 0.9 for problem (7) where θ = ϑ = 0 and N = 4 in the interval [ π,π].

11 11 Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations 57 Exacte and approximate solutions u x,0.1 u x,0.1 u x,0.5 u x,0.5 u x, x u x,0.9 Fig. 5 The approximate solution ũ(x,t) and the exact solution u(x,t) for t = 0.1, 0.5, and 0.9 for problem (3) where θ = ϑ = 1 and N = Exacte and approximate solutions t u 0,t u 0,t u 0.1,t u 0.1,t u 0.,t u 0.,t Fig. 6 The approximate solution ũ(x,t) and the exact solution u(x,t) for x = 0.0, 0.1, and 0. for problem (3) where θ = ϑ = 1 and N = 0.

12 58 E.H. Doha et al Exacte and approximate solutions x v x,0.1 v x,0.1 v x,0.5 v x,0.5 v x,0.9 v x,0.9 Fig. 7 The approximate solution ṽ(x,t) and the exact solution v(x,t) for t = 0.1, 0.5, and 0.9 for problem (3) where θ = ϑ = 1 and N = Exacte and approximate solutions t v 0,t v 0,t v 0.1,t v 0.1,t v 0.,t v 0.,t Fig. 8 The approximate solution ṽ(x,t) and the exact solution v(x,t) for x = 0.0, 0.1, and 0. for problem (3) where θ = ϑ = 1 and N = 0.

13 13 Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations 59 Fig. 9 The approximate solution ũ(x,t) for problem (3) where θ = ϑ = 1 and N = 0. Fig. 10 The approximate solution ṽ(x,t) for problem (3) where θ = ϑ = 1 and N = 0.

14 60 E.H. Doha et al. 14 Fig. 11 The absolute error E 1 (y,t) for problem (3) where θ = ϑ = 0 and N = 16 in the interval [0,1]. with the boundary-initial conditions 1 + c u(a, t)= u(b,t) = A ct sech( 1 c ); D tu(y,0)= 1 c 1+c c 1 c sech( )tanh( 1 c 1 c y y ) 1 c 1 + c B ct 1 + c sech( 1 c ); u(y,0) = 1 c 1 c sech( y ) (8) 1 c c sech( a ct v(a,t) = 1 c ) 1 c ; v(y,0) = c sech( y 1 c ). 1 c The difference between the measured or inferred value of approximate solution and its actual value (absolute error), is given by E 1 (x,t) = u(x,t) ũ(x,t), E (x,t) = v(x,t) ṽ(x,t), (9) where u(x,t)( v(x,t) ) and ũ(x,t) (ṽ(x,t) ) are the exact and approximate solutions at the point (x,t), respectively. Moreover, the maximum absolute error is given by M 1 = Max{E 1 (x,t) : (x,t) D [0,T ]}, M = Max{E (x,t) : (x,t) [A,B] [0,T ]}. (30)

15 15 Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations 61 Table 1 Maximum absolute errors with various choices of N, θ and ϑ for Example 1. N θ ϑ M 1 M θ ϑ M 1 M Maximum absolute errors of u(x,t) and v(x,t) related to (7) are presented in Table 1 using J-GL-C method with various choices of N, θ and ϑ in the interval [0,1], compared with the exact solution u(y,t) = 1 + c y ct sech( 1 c ), 1 c sech( y ct v(y,t) = c 1 c ), (31) 1 c and we see that the results are very accurate for small choice of N. The computed approximations, ũ(x,t) and ṽ(x,t), for problem (7) at θ = ϑ = 0 and N = 0 are plotted in Figs. 1 and 3, respectively, in the interval [ π, π]. Meanwhile, in Fig., we see that the computed approximations, ũ(x,t) and the exact solution, u(x,t), coincide for different values of t; the corresponding curves are clearly seen in (b). Again, in Fig. 4, we see that the computed approximations, ṽ(x,t) and the exact solution, v(x, t), coincide for different values of t; the corresponding curves are clearly seen in 4(b). 4.. TEST PROBLEM In this subsection, we introduce the coupled nonlinear KG equations in the form Dt u(y,t) = Dyu(y,t) u(y,t) + v(y,t), Dt v(y,t) = Dyv(y,t) (3) + u(y,t) v(y,t), (y,t) [A,B] [0,T ], with the boundary-initial conditions u(a,t) = sech(a t), u(b,t) = sech(b t), v(a,t) = sech(a t), v(b,t) = sech(b t), u(y,0) = sech(y), v(y,0) = sech(y) (33), D t u(y,0) = sech(y) tanh(y), D t v(y,0) = sech(y) tanh(y).

16 6 E.H. Doha et al. 16 Table Maximum absolute errors with various choices of N, θ and ϑ for Example. N θ ϑ M 1 M θ ϑ M 1 M Maximum absolute errors of u(x,t) and v(x,t) related to (3) are presented in Table using J-GL-C method with various choices of N, θ and ϑ in the interval [0,1], compared with the exact solution u(y,t) = sech(y t), v(y,t) = sech(y t), (34) and we see that the results are very accurate for relatively small values of N. In interval [ π,π], we see that the approximate solutions ũ(x,t) and ṽ(x,t), and the exact solutions u(x,t) and v(x,t) coincide for different values of t or x in Figs. 5, 6, 7, and 8, respectively, with values of parameters listed in their captions. The approximate solutions ũ(x,t) and ṽ(x,t) for problem (3) where θ = ϑ = 1, and N = 0 are displayed in Figs. 9 and 10, respectively. While, in Fig. 11, we present the absolute errors E 1 (x,t) with N = 16 and θ = ϑ = 0 in the interval [0,1]. 5. CONCLUSION This wor was concerned with an efficient numerical scheme based on the J- GL-C method for solving the two coupled nonlinear hyperbolic KG partial differential equations. The problems are transformed to the numerical solution of SODEs. Two test examples were presented to demonstrate the applicability and the accuracy of the method. The results show that the proposed numerical scheme is accurate. Indeed by providing limited collocation nodes, excellent results are demonstrated. REFERENCES 1. K. Parand, M. Shahini, M. Dehghan, J. Comput. Phys. 8, 8830 (009).. M. Dehghan, F.F. Izadi, Math. Comput. Modell. 53, 1865 (011). 3. E.H. Doha, A.H. Bhrawy, D. Baleanu, R.M. Hafez, Rep. Math. Phys. 53, 19 (013).

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