A PSEUDOSPECTRAL METHOD FOR SOLVING THE TIME-FRACTIONAL GENERALIZED HIROTA SATSUMA COUPLED KORTEWEG DE VRIES SYSTEM

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1 A PSEUDOSPECTRAL METHOD FOR SOLVING THE TIME-FRACTIONAL GENERALIZED HIROTA SATSUMA COUPLED KORTEWEG DE VRIES SYSTEM M. A. SAKER,2, S. S. EZZ-ELDIEN 3, A. H. BHRAWY 4,5, Department of Basic Science, Modern Academy in Maadi, Cairo, Egypt 2 Department of Mathematics, Faculty of Science and Arts, Shaqraa University, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Assiut University, New Valley Branch, El-Kharja 725, Egypt 4 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 625, Egypt 5 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 2589, Saudi Arabia alibhrawy@yahoo.co.uk corresponding author Received September 9, 26 Abstract. In this paper, a new space-time spectral algorithm is constructed to solve the generalized Hirota-Satsuma coupled Korteweg-de Vries GHS-C-KdV system of time-fractional order. The present algorithm consists of applying the collocationspectral method in conjunction with the operational matrix of fractional derivative for the double Jacobi polynomials, which will be employed as a basis function for the spectral solution. The main characteristic behind this approach is that such problems will reduce to those of solving algebraic systems of equations that greatly simplifying the problem. For ensuring the accuracy and efficiency of the presented algorithm, we apply it to find the approximate solutions of two specific problems, namely, a homogeneous form of the GHS-C-KdV system and a inhomogeneous GHS-C-KdV system. Key words: Generalized Hirota-Satsuma coupled KdV system; spectral colocation method; fractional Caputo derivative; operational matrix; Jacobi-Gauss quadrature.. INTRODUCTION Finding effective and robust analytical and numerical techniques for solving fractional differential equations FDEs has become the focus of many research studies during the past years. Several techniques such as the Laplace transform method [, 2], the fractional sub-equation method [3], the Block pulse operational matrix method [4], the iterative method [5], the finite element method [6], the homotopy analysis method [7], the implicit compact difference scheme [8], collocation method [9 8] and other effective methods have been put forward [9 3]. Spectral methods [3 38] have been proposed and developed to numerically solve fractional ordinary and partial differential equations, see [39 44]. In addition to this, the spectral methods have been employed in conjunction with operational matrices techniques of fractional derivatives to spectrally solve various types of fractional Romanian Journal of Physics 62, 5 27 v.2.*27.3.3#85329b

2 Article no. 5 M. A. Saker, S. S. Ezz-Eldien, A. H. Bhrawy 2 differential equations [45 49]. On the other hand, the operational matrix of fractional integrals is constructed and used coupling with spectral methods to solve FDEs [5]. Bhrawy [5] proposed a new operational matrix for the pseudospectral method for numerically solve the one-dimensional fractional sub-diffusion equations based on double Jacobi polynomials; moreover, the method was developed based on the triple Jacobi polynomials to handle the two dimensional version of such problem. However, in Ref. [52] Bhrawy proposed and developed a new pseudospectral technique for achieving a high accurate approximate solution of time-space fractional partial differential equation PDE, in which, the Legendre Gauss-Lobatto collocation approximation is adapted for spatial discretization and the Chebyshev Gauss-Radau collocation approximation is employed for temporal discretization. The generalized Hirota-Satsuma coupled Korteweg-de Vries GHS-C-KdV system has been firstly introduced by Wu et al. [53]. This equation can be used to describe the interaction of two long waves with different dispersion relations. Ganji et al. [54] introduced the GHS-C-KdV system with time-fractional order and applied the homotopy perturpation method for introducing solitary wave solutions for it, while in [55], the variational iteration method was applied for the same purpose. Recently, the authors of Ref. [56] applied the improved fractional sub-equation method to find an analytical solution of the GHS-C-KdV system with time-fractional order. The objective of this paper is to introduce a new approximate solution of the GHS-C-KdV system with time-fractional order ν ux,t t ν = 3 ux,t 2 3 3ux,t ux,t + 3 vx, twx, t + qx,t, ν vx,t t ν = 3 vx,t 3 + 3ux,t vx,t + rx,t, ν wx,t t ν = 3 wx,t 3 + 3ux,t wx,t + sx,t, subject to the initial conditions: ux, = ψ x, vx, = ψ 2 x, wx, = ψ 3 x, < x L, 2 and boundary conditions: u,t = f t, v,t = f 2 t, w,t = f 3 t, u x L,t = g t, v x L,t = g 2 t, w x L,t = g 3 t, u xx L,t = h t, v xx L,t = h 2 t, w xx L,t = h 3 t, < t τ, 3 where ν, < ν is a real constant, and qx,t, rx,t and sx,t are the source functions. The introduced technique consists of expanding the functions ux,t, vx,t, c RJP 62, id:5-27 v.2.*27.3.3#85329b

3 3 Pseudospectral method for generalized Hirota-Satsuma coupled KdV system Article no. 5 wx,t, qx,t, rx,t, and sx,t with the shifted Jacobi polynomials and the operational matrices of fractional derivatives described in the Caputo sense. Then, we apply the pseudospectral technique to reduce the problem to an algebraic system of equations. Several illustrative examples are presented to ensure the high accuracy and the power of the present method. This article is organized as follows. In Sec. 2, some definitions and notations are presented. In Sec. 3, the operational matrix and the pseudo-spectral method are implemented to propose a new approximate solution for the time-fractional GHS-C- KdV system -3. In Sec. 4, the high accuracy and validity of the present scheme are reported through two illustrative examples. The conclusions are given in Sec PRELIMINARIES AND NOTATION This Section reports two fractional definitions and an overview of Jacobi polynomials. Definition.. The integral of order γ fractional according to Riemann- Liouville is given by where I γ fx = Γγ I fx = fx, Γγ = x x t γ ftdt, γ >, x >, x γ e x dx is gamma function. Definition.2. The γ order fractional derivative in the Caputo sense is defined as D γ fx = Γm γ x x t m γ d m 4 ftdt, m < γ m, x >, 5 dtm where m is the ceiling function of γ. The Jacobi polynomial of degree j, denoted by P α,β j z; α, β and defined on the interval [, ], constitutes an orthogonal system with respect to the weight function ω α,β z = z α + z β, i.e., P α,β j where δ jk is the Kronecker function and zp α,β k zω α,β zdz = δ jk γ α,β k, γ α,β k = 2α+β+ Γk + α + Γk + β + 2k + α + β + k!γk + α + β +. c RJP 62, id:5-27 v.2.*27.3.3#85329b

4 Article no. 5 M. A. Saker, S. S. Ezz-Eldien, A. H. Bhrawy 4 The shifted Jacobi polynomial of degree j, denoted by P α,β x; α, β and defined on the interval [,L], is generated by introducing the change of variable z = 2x L where and α,β, i.e., P j 2x L L P α,β α,β P x. Then xp α,β L,k xω α,β L L,k = L 2 α+β+ γ α,β j, xdx = δ jk L,k, 6 ω α,β L x = x β L x α. According to the properties of Jacobi polynomials, we get the following recurrence relation for the shifted Jacobi polynomials with P α,β + x = µ jx ξ j P α,β x ζ j P α,β x, j, P α,β L, x =, P α,β L, x = α + β + 2x β +, L where µ j, ξ j and ζ j are given in Bhrawy et al. [5]. The shifted Jacobi polynomials P α,β x can be obtained from P α,β x = Accordingly, we get j k= j k Γj + β + Γj + k + α + β + Γk + β + Γj + α + β + j k!k!l k xk. 7 P α,β L,i = i Γi + β + Γβ + i!, D q P α,β L,i = i q Γi + β + i + α + β + q L q Γi q + Γq + β +, q i, P α,β L,i L = Γi + α + Γα + i!, D q P α,β L,i L = Γi + α + i + α + β + q L q Γi q + Γq + α +, q i. If yx L 2, then it can be expressed as ω α,β L x yx = a j P α,β x, j= c RJP 62, id:5-27 v.2.*27.3.3#85329b

5 5 Pseudospectral method for generalized Hirota-Satsuma coupled KdV system Article no. 5 where a j are obtained from a j = L ω α,β L xyxp α,β xdx, j =,,. 8 If we approximate yx by the first N + -terms, then we can write y N x N j= which alternatively may be written in the matrix form: with a j P α,β x, 9 y N x A T L,N x, A T [a,a,,a N ], with and y ij = L,N x [P α,β L, x,p α,β L, x,,p α,β L,N x]t. Consider yx,t defined on < x L and < t τ. Then we have y M,N x,t = M N i= j= τ L y ij P α,β tp α,β x = T τ,mty L,N x, 2 Y = [y ij ] R M N, yx,tp α,β i =,,,M, t P α,β x ω α,β τ j =,,,N. t ω α,β L xdxdt, The fractional differentiation of order ν of L,N x can be expressed as 3 D ν L,N x D ν L,N x, 4 where D ν is the N + N + Jacobi operational matrix of differentiation and can be obtained from: D ν = [d ν ij ] RM N, where {, i < ν, d ν ij = Ψ α,β ν i,j, i ν, c RJP 62, id:5-27 v.2.*27.3.3#85329b

6 Article no. 5 M. A. Saker, S. S. Ezz-Eldien, A. H. Bhrawy 6 and Ψ α,β ν i,j = i i k L α+β ν+ Γi + k + α + β + Γj + β + Γi + β + k= ν j l= Γk + β + Γj + α + β + Γk ν + i k!γi + α + β + j l Γl + k + β ν + Γα + Γj + l + α + β +. Γl + k + α + β ν + 2 Γl + β + j l! l! Note that in D ν, the first ν rows are all zero. see [49] for a proof. 3. THE NUMERICAL SCHEME In the current Section, we apply the collocation-spectral method in combination with the operational matrix of fractional derivative with the shifted Jacobi polynomials as basis functions to approximate the solution of the GHS-C-KdV system of time-fractional order with the initial conditions 2 and the boundary conditions 3. Making use of the initial conditions 2, the problem -3 may be rewritten as in the form ν ux,t t ν + ux, ψ x = 3 ux,t ν vx,t t ν + vx, ψ 2 x = 3 vx,t 3 ν wx,t t ν + wx, ψ 3 x = 3 wx,t 3 3ux,t ux,t vx, twx, t + qx,t, + 3ux,t vx,t + 3ux,t wx,t + rx,t, + sx,t, subject to the conditions 3. Now, we approximate ux,t, vx,t, wx,t, qx,t, rx,t, and sx,t by the shifted Jacobi polynomials as u M,N x,t T τ,mtu L,N x, w M,N x,t T τ,mtw L,N x, r M,N x,t T τ,mtr L,N x, v M,N x,t T τ,mtv L,N x, q M,N x,t T τ,mtq L,N x, s M,N x,t T τ,mts L,N x, where the unknown coefficients U, V, and W, which are M + N + matrices will be evaluated to determine the approximate solutions, while Q, R, and S are known matrices and may be obtained from Q = [q ij ] R M N, R = [r ij ] R M N, S = [s ij ] R M N, 5 6 c RJP 62, id:5-27 v.2.*27.3.3#85329b

7 7 Pseudospectral method for generalized Hirota-Satsuma coupled KdV system Article no. 5 where the coefficients q ij, r ij and s ij ; i =,,,M, j =,,,N can be evaluated respectively by q ij = r ij = s ij = τ L τ L τ L qx,tp α,β rx,tp α,β sx,tp α,β i =,,,M, t P α,β t P α,β x ω α,β τ t ω α,β L x dxdt, x ω α,β t ω α,β x dxdt, τ t P α,β x ω α,β t ω α,β x dxdt, τ j =,,,N. L L In general, the exact evaluation of the integrals in the functions qx,t, rx,t, and sx,t are not easy. Therefore, we empoly one of the most accurate methods for evaluating integrals, which is named the Jacobi-Gauss quadrature method. Accordingly, the coefficients q ij, r ij, and s ij are approximated as q ij = r ij = s ij = M N qx α,β L,N,ɛ,tα,β τ,m,δ δ= ɛ= M i =,,,M, N rx α,β L,N,ɛ,tα,β τ,m,δ δ= ɛ= M N sx α,β L,N,ɛ,tα,β τ,m,δ δ= ɛ= j =,,,N, α,β P t α,β α,β P t α,β α,β P t α,β τ,m,δ P α,β τ,m,δ P α,β τ,m,δ P α,β x α,β L,N,ɛ ϖα,β τ,m,δ ϖα,β L,N,ɛ, x α,β L,N,ɛ ϖα,β τ,m,δ ϖα,β L,N,ɛ, x α,β L,N,ɛ ϖα,β τ,m,δ ϖα,β L,N,ɛ, where x α,β L,N,ɛ, ɛ N are the zeros of Jacobi-Gauss quadrature in the interval,l, with ϖ α,β L,N,ɛ, ɛ N are the corresponding Christoffel numbers and t α,β τ,m,δ, δ M are the zeros of Jacobi-Gauss quadrature in the interval,τ, with ϖ α,β τ,m,δ, δ M are the corresponding Christoffel numbers. c RJP 62, id:5-27 v.2.*27.3.3#85329b

8 Article no. 5 M. A. Saker, S. S. Ezz-Eldien, A. H. Bhrawy 8 Using Eqs. 4 and 6, we can write ν ux,t t ν T τ,mtd T ν U L,Nx; ux,t T τ,mtud L,N x; 3 ux,t 3 T τ,mtud 3 L,N x; ux, T τ,mu L,N x; ν vx,t t ν T τ,mtd T ν V L,Nx; vx,t T τ,mtvd L,N x; 3 vx,t 3 T τ,mtvd 3 L,N x; vx, T τ,mv L,N x; ν wx,t t ν T τ,mtd T ν W L,Nx; wx,t T τ,mtwd L,N x; 3 wx,t 3 T τ,mtwd 3 L,N x; wx, T τ,mw L,N x. By substituting 6 and 7 in 5, we get 7 T τ,mtd T ν U L,Nx 2 T τ,mtud 3 L,N x + 3 T τ,mtu L,N x T τ,mtud L,N x 3 T τ,mtv L,N x T τ,mtwd L,N x 3 T τ,mtw L,N x T τ,mtvd L,N x + T τ,mu L,N x ψ x = T τ,mtq L,N x, T τ,mtd T ν V L,Nx + T τ,mtvd 3 L,N x 3 T τ,mtu L,N x T τ,mtvd L,N x 8 + T τ,mv L,N x ψ 2 x = T τ,mtr L,N x, T τ,mtd T ν W L,Nx + T τ,mtwd 3 L,N x 3 T τ,mtu L,N x T τ,mtwd L,N x + T τ,mw L,N x ψ 3 x = T τ,mts L,N x. We collocate 8 at M + N points, as T τ,mt i D T ν U L,Nx j 2 T τ,mt i UD 3 L,N x j + 3 T τ,mt i U L,N x j T τ,mt i UD L,N x j 3 T τ,mt i V L,N x j T τ,mt i WD L,N x j c RJP 62, id:5-27 v.2.*27.3.3#85329b

9 9 Pseudospectral method for generalized Hirota-Satsuma coupled KdV system Article no. 5 3 T τ,mt i W L,N x j T τ,mt i VD L,N x j + T τ,mu L,N x j ψ x j = T τ,mt i Q L,N x j, T τ,mt i D T ν V L,Nx j + T τ,mt i VD 3 L,N x j 3 T τ,mt i U L,N x j T τ,mt i VD L,N x j + T τ,mv L,N x j ψ 2 x j = T τ,mt i R L,N x j, T τ,mt i D T ν W L,Nx j + T τ,mt i WD 3 L,N x j 3 T τ,mt i U L,N x j T τ,mt i WD L,N x j 9 + T τ,mw L,N x j ψ 3 x j = T τ,mt i S L,N x j, where t i, i =,,,M and x j, j =,,,N 3 are the roots of P α,β τ,m+ t and P α,β L,N 2 x, respectively. This in turn generate three nonlinear algebraic systems of M + N 2. The rest of these systems is given from 3, as T τ,mt i U L,N = f t i, T τ,mt i UD L,N L = g t i, T τ,mt i UD 2 L,N L = h t i, T τ,mt i V L,N = f 2 t i, T τ,mt i VD L,N L = g 2 t i, T τ,mt i VD 2 L,N L = h 2 t i, T τ,mt i W L,N = f 3 t i, T τ,mt i WD L,N L = g 3 t i, T τ,mt i WD 2 L,N L = h 3 t i, i =,,,M. Finally, the three M + N 2 algebraic systems 9 together with the 9M + system 2 constitute a nonlinear algebraic system of order 3M +N +. Accordingly, u ij, v ij and w ij, can be evaluated by Newton s iterative scheme. Consequently, approximate solutions u M,N x,t, v M,N x,t, and w M,N x,t presented in 6 can be evaluated NUMERICAL RESULTS This Section reports two illustrative numerical examples to ensure the high order accuracy and efficiency of the present numerical scheme. The first example considers a homogeneous form of the GHS-C-KDdV system of time-fractional order. We investigate the convergence and efficiency of our method and compare our results with those obtained using the homotopy perturbation method HPM [54] and the fractional iteration method FIM [55]. The second example aims to examine the numerical results of our method for an inhomogeneous GHS-C-KdV system of c RJP 62, id:5-27 v.2.*27.3.3#85329b

10 Article no. 5 M. A. Saker, S. S. Ezz-Eldien, A. H. Bhrawy time-fractional order. We shall investigate the approximation solution and convergence order with different orders by using the introduced method. Let us define the convergence order CO by CO = logerrorn /errorn 2, 2 logn 2 /N where errorn denotes the error corresponding to polynomial degree N. In this section the CPU time is performed on a HP laptop with the following configuration: Intel Core, i7-374 CPU, 2.7 GHz, and 8 Gb RAM. 4.. HOMOGENEOUS PROBLEM As the first example, we consider the following homogeneous problem: ν ux,t t ν = 3 ux,t 2 3 3ux,t ux,t + 3 vx, twx, t, ν vx,t t ν = 3 vx,t 3 + 3ux,t vx,t, ν wx,t t ν = 3 wx,t 3 + 3ux,t wx,t, subject to the initial conditions: 22 ux, = γ 2k2 + 2k 2 tanh 2 kx, 3 vx, = 4k2 c γ + k 2 3c 2 + 4k2 γ + k 2 tanhkx, 3c wx, = c + c tanhkx, where γ, k, c, and c are arbitrary constants. The exact solution at ν = is ux,t = γ 2k2 3 vx,t = 4k2 c γ + k 2 3c 2 + 2k 2 tanh 2 kx + γt, wx,t = c + c tanhkx + γt. + 4k2 γ + k 2 tanhkx + γt, 3c Ganji et al. [54] introduced this problem and applied the HPM for approximating its solution, while in Ref. [55], the authors applied the FIM for the same purpose. In order to show that our technique is more accurate than HPM [54] and FIM [55], we compare our results with those obtained in Refs. [54] and [55]. Taking k = c =., γ = c = 3 2, in Tables -6, we compare the numerical values of ux,t, vx,t, wx,t at α = β =, N = M = with those obtained by HPM and FIM at c RJP 62, id:5-27 v.2.*27.3.3#85329b

11 Pseudospectral method for generalized Hirota-Satsuma coupled KdV system Article no error t.4 x Fig. Error function for ux,t at N = M =, ν = and α = β = for problem 22. error t.4 x Fig. 2 Error function for vx,t at N = M =, ν = and α = β = for problem 22. c RJP 62, id:5-27 v.2.*27.3.3#85329b

12 Article no. 5 M. A. Saker, S. S. Ezz-Eldien, A. H. Bhrawy 2 error t.4 x Fig. 3 Error function for wx,t at N = M =, ν = and α = β = for problem 22. ν =.5,.75,, while in Figs. -3, we plot the error functions of ux,t, vx,t, and wx,t at α = β =, N = M =, and ν =. It is observed from these Tables that our technique is more accurate than both HPM and FIM. Table Numerical values of ux,t at k = c =., γ = c = 3 2, and t =.2 for problem 22. x ν =.5 ν =.75 Our method FIM [55] HPM [54] Our method FIM [55] HPM [54] Table 2 Numerical values of ux,t at k = c =., γ = c = 3 2, and ν = for problem 22. t x HPM [54] FIM [55] Our method Exact c RJP 62, id:5-27 v.2.*27.3.3#85329b

13 3 Pseudospectral method for generalized Hirota-Satsuma coupled KdV system Article no. 5 Table 3 Numerical values of vx,t at k = c =., γ = c = 3 2, and t =.4 for problem 22. x ν =.5 ν =.75 Our method FIM [55] HPM [54] Our method FIM [55] HPM [54] Table 4 Numerical values of vx,t at k = c =., γ = c = 3 2, and ν = for problem 22. t x HPM [54] FIM [55] Our method Exact Table 5 Numerical values of wx,t at k = c =., γ = c = 3 2, and t =.6 for problem 22. x ν =.5 ν =.75 Our method FIM [55] HPM [54] Our method FIM [55] HPM [54] Table 6 Numerical values of wx,t at k = c =., γ = c = 3 2, and ν = for problem 22. t x HPM [54] FIM [55] Our method Exact c RJP 62, id:5-27 v.2.*27.3.3#85329b

14 Article no. 5 M. A. Saker, S. S. Ezz-Eldien, A. H. Bhrawy 4 with 4.2. INHOMOGENEOUS PROBLEM Consider the inhomogeneous GHS-C-KdV system of time-fractional order initial conditions: ν ux,t t ν = 3 ux,t 2 3 ν vx,t t ν ν wx,t t ν qx,t = rx,t = sx,t = and boundary conditions u π 2,t 2 u π 2,t 2 The exact solution is + 3 = 3 vx,t 3 = 3 wx,t 3 3ux,t ux,t vx, twx, t + qx,t, + 3ux,t vx,t + 3ux,t wx,t + rx,t, + sx,t, 2t2 ν Γ3 ν cosx 2 t2 sinx t4 sin2x, 2t2 ν Γ3 ν cosx + t2 sinx t4 sin2x, 2t2 ν Γ3 ν cosx + t2 sinx t4 sin2x, ux, = vx, = wx, =, u,t = v,t = w,t = t 2, = v π 2,t = 2 v π 2,t 2 = w π 2,t = t 2, = 2 w π 2,t 2 =. ux,t = vx,t = wx,t = t 2 cosx. In order to investigate the convergence order of the space-time spectral method, in case of α = β =, we list the L 2 L 2 -error with two different fractional orders ω α,β L and the convergence order CO of ux,t, vx,t, and wx,t, in Tables 7, 8, and 9, respectively. In addition, we add the CPU time in Table 7. Figs. 4 and 5 plot the error functions of ux,t, vx,t at α = β =, ν =.5,.7 with N = M =, while in Fig. 6, we plot the error function of wx,t at α = β =,ν =.3 with N = M =. This example illustrates that the space-time spectral method is highly accurate and efficient for solving the inhomogeneous time-fractional GHS-C-KdV system. 23 c RJP 62, id:5-27 v.2.*27.3.3#85329b

15 5 Pseudospectral method for generalized Hirota-Satsuma coupled KdV system Article no error t x.2.5 Fig. 4 Error function for ux,t at N = M =, ν =.5, and α = β = for problem error t x.2.5 Fig. 5 Error function for vx,t at N = M =, ν =.7, and α = β = for problem 23. c RJP 62, id:5-27 v.2.*27.3.3#85329b

16 Article no. 5 M. A. Saker, S. S. Ezz-Eldien, A. H. Bhrawy 6 - error t x.2.5 Fig. 6 Error function for wx,t at N = M =, ν =.3, and α = β = for problem 23. Table 7 L 2 -error, convergence order of ux,t at α = β =, and CPU time for problem 23. N ν =.2 ν =.7 L 2 -error CO L 2 -error CO CPU time sec Table 8 L 2 -error and convergence order of vx,t at α = β =, for problem 23. N ν =.2 ν =.7 L 2 -error CO L 2 -error CO c RJP 62, id:5-27 v.2.*27.3.3#85329b

17 7 Pseudospectral method for generalized Hirota-Satsuma coupled KdV system Article no. 5 Table 9 L 2 -error and convergence order of wx,t at α = β =, for problem 23. N ν =.2 ν =.7 L 2 -error CO L 2 -error CO CONCLUSION In this paper, a new space-time spectral algorithm is investigated and applied to numerically solve the time-fractional GHS-C-KdV system. Introducing the shifted Jacobi polynomials as basis functions, we used the operational matrix of the basis functions in conjunction with the pseudospectral technique together with the Gauss quadrature formula to approximate the solution of the time-fractional GHS-C-KdV system. The obtained numerical results demonstrated the high accuracy of this new technique. The present method can be extended to handle both two- and threedimensional fractional partial differential equations. REFERENCES. J. Singh, D. Kumar, and S. Kumar, 4, D. W. Brzezinski and P. Ostalczyk, Nonlinear Dyn. 84, S. Zhang and H-Q. Zhang, Phys. Lett. A 375, M. Yi, J. Huang, and J. Wei, Appl. Math. Comput. 22, V. K. Baranwal, R. M. Pandey, M. A. Tripathi, and O. P. Singh, Commun. Nonlinear Sci. Numer. Simulat. 7, Z. Zhao and C. Li, Appl. Math. Comput. 29, M. G. Sakar and F. Erdogan, Appl. Math. Modell. 37, X. Hu and L. Zhang, Appl. Math. Modell. 36, A. H. Bhrawy, M. A. Zaky, and D. Baleanu, Rom. Rep. Phys. 67, A. H. Bhrawy and M. A. Abdelkawy, J. Comput. Phys. 294, H. Khalil, M. Al-Smadi, K. Moaddy, R. A. Khan, and I. Hashim, Disc. Dyn. Nature. Soc. 26, M. A. Abdelkawy, M. A. Zaky, A. H. Bhrawy, and D. Baleanu, Rom. Rep. Phys. 67, R. Salehi, Numer. Algorithms, DOI:.7/s z S. Esmaeili, Math. Meth. Appl. Sci., DOI:.2/mma A. H. Bhrawy, A. A. Al-Zahrani, Y. A. Alhamed, and D. Baleanu, Rom. J. Phys. 59, M.A. Abdelkawy, E. A. Ahmed, and R. T. Alqahtani, Open Phys. 4, M. R. Hooshmandasl, M. H. Heydari, and C. Cattani, Eur. Phys. J. Plus 3, A. H. Bhrawy, M. A. Abdelkawy, A. A. Alzahrani, D. Baleanu, and E. O. Alzahrani, Proc. Romanian Acad. A 6, c RJP 62, id:5-27 v.2.*27.3.3#85329b

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