Galerkin method for the numerical solution of the RLW equation using quintic B-splines

Transcription

1 Journal of Computational and Applied Mathematics 19 (26) Galerkin method for the numerical solution of the RLW equation using quintic B-splines İdris Dağ a,, Bülent Saka b, Dursun Irk b a Computer Engineering Department, Osmangazi University, 2648 Eskişehir, Türkiye b Mathematics Department, Osmangazi University, 2648 Eskişehir, Türkiye Received 31 December 24 Abstract The regularized long wave equation (RLW) is solved numerically by using the quintic B-spline Galerkin finite element method. The same method is applied to the time-split RLW equation. Comparison is made with both analytical solutions and some previous results. Propagation of solitary waves, interaction of two solitons are studied. 25 Elsevier B.V. All rights reserved. MSC: 65N3; 65D7; 76B25 Keywords: Finite elements; Galerkin; Splines; Solitary waves 1. Introduction and governing equation Nonlinear partial differential equations are useful in describing the various phenomena in disciplines. Analytical solutions of these equations are usually not available, especially when the nonlinear terms are involved. Since only limited classes of the equations are solved by analytical means, numerical solution of these nonlinear partial differential equations is of practical importance. The regularized long wave (RLW) equation is one of the model partial differential equation of the nonlinear dispersive waves which has many application in many areas, e.g. ion-acoustic waves in plasma, magnetohydrodynamics waves in plasma, longitudinal dispersive waves in elastic rods, pressure waves in liquid gas bubble Corresponding author. Fax: address: idag@ogu.edu.tr (İ. Dağ) /$ - see front matter 25 Elsevier B.V. All rights reserved. doi:1.116/j.cam

2 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) mixtures and rotating flow down a tube. Mathematical theory for the equation was developed in [1,2]. Various numerical techniques have been proposed to solve the equation. These include finite difference, Fourier, finite element, collocation, spline and variational methods. Spline functions, which are a class of piecewise polynomial having continuity properties of up to the degree lower than that of the spline functions, are employed to set up approximate functions. So both collocation and Galerkin methods are used together with types of splines known as B-splines to construct the approximation functions over the finite elements. Various types of B-spline finite element formulation are built up to get the solution of some partial differential equations [3 5,7 1,14 16]. As also known, prosperity of the numerical method depends on the choice of B-spline basis. Various forms of both B-spline collocation and B-spline Galerkin methods have been constructed in getting the numerical solution of the RLW equation. The quintic B- spline basis together with finite element methods are shown to provide very accurate solutions in solving some partial differential equations. For instance, quintic B-spline collocation finite element method for the numerical solution of the Korteweg de Vries equation is designed by Gardner and his coauthor [6]. A numerical method for the RLW equation was set up using a quintic B-spline Petrov Galerkin method over finite elements in the paper [8]. An algorithm based on the collocation method with quintic B-spline finite element is set up to simulate the solutions of the KdV, Burgers and KdVB equations [16]. Results of the calculations showed that accuracy of solution is improved if the Galerkin formulation together with quintic B-spline functions is used in getting the numerical solution of the partial differential equations, but computational cost of the B-spline Galerkin algorithm increases. The aim of the paper is to investigate the accuracy of Galerkin method when the quintic B-spline is used to express the approximate function in the finite element method. Solution of the RLW equation is also found by applying time-splitting up scheme. The split RLW equation is also solved by the quintic B-spline finite element method. Finally, comparison between analytical and numerical solution of the RLW equation for the proposed algorithms is made. We consider the RLW equation U t + U x + εuu x μu xxt =, (1) where ε and μ are parameters and the subscripts x and t denote differentiation. Boundary conditions will be selected from the homogeneous boundary conditions: U(a,t)= β 1, U(b, t) = β 2, U x (a, t) =, U x (b, t) =, t (,T] U xx (a, t) =, U xx (b, t) =, (2) and initial condition U(x,) = f(x), x [a,b]. 2. Quintic B-spline Galerkin Method I (QBGM1) We subdivide the interval [a,b], the space variable domain of Eq. (1), into subintervals by the set of the N + 1 distinct grid points x m,m=,...,n, such that a = x <x 1 < <x N = b.

3 534 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) The construction of the quintic B-spline interpolate U N to the analytical solution U for spaced knots a = x <x 1 < <x N = b can be performed with the help of the 1 additional knots such that x 5 <x 4 <x 3 <x 2 <x 1 and x N+1 <x N+2 <x N+3 <x N+4 <x N+5. Now quintic B-splines B m (x), m = 2,...,N + 2, are defined by (x x m 3 ) 5 [x m 3,x m 2 ], (x x m 3 ) 5 6(x x m 2 ) 5 [x m 2,x m 1 ], (x x m 3 ) 5 6(x x m 2 ) (x x m 1 ) 5 [x m 1,x m ], (x x m 3 ) 5 6(x x m 2 ) 5 B m (x) = 1 +15(x x m 1 ) 5 2(x x m ) 5 [x m,x m+1 ], h 5 (x x m 3 ) 5 6(x x m 2 ) (x x m 1 ) 5 2(x x m ) (x x m+1 ) 5. [x m+1,x m+2 ], (x x m 3 ) 5 6(x x m 2 ) 5 +15(x x m 1 ) 5 2(x x m ) 5 +15(x x m+1 ) 5 6(x x m+2 ) 5 [x m+2,x m+3 ],, x<x m 3,x m+3 <x. (3) The set of quintic B-splines B m (x), m = 2,...,N + 2, forms a basis over the region a x b [13]. A global quintic B-spline interpolate to Eq. (1) is given by U N (x, t) = N+2 m= 2 δ m (t)b m (x), (4) where δ m (t) are time-dependent nodal parameters to be determined from quintic Galerkin form of Eq. (1). The quintic B-spline function and its 4 derivatives are continuous, and thus have trial solution with continuity of up to the fourth order. The nodal values U and its derivatives of up to fourth order at the knots x m are given in terms of the parameters δ m from the use of the B-splines (3) and the trial solution (4) U m = U(x m ) = δ m δ m δ m + 26δ m+1 + δ m+2, U m = U (x m ) = 5 h (δ m+2 + 1δ m+1 1δ m 1 δ m 2 ), U m = U (x m ) = 2 h 2 (δ m+2 + 2δ m+1 6δ m + 2δ m 1 + δ m 2 ), U m = U (x m ) = 6 h 3 (δ m+2 2δ m+1 + 2δ m 1 δ m 2 ), U m = U (x m ) = 12 h 4 (δ m+2 4δ m+1 + 6δ m 4δ m 1 + δ m 2 ). (5) The local coordinate transformation ξ = x x m,< ξ <h, transforms an element along the x-axis into a standard interval [,h]. A quintic B-spline covers six intervals so that an element is covered by the six successive B-splines. Thus element shape functions over the interval [,h] are obtained from the six

4 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) successive quintic B-spline functions whose branch lie on the interval [,h]: Q m 2 = 1 5 ξ ( 2 ( ) 3 ( ) 4 ( ) 5 ξ ξ ξ ξ h h) , h h h Q m 1 = 26 5 ξ ( 2 ( ) 3 ( ) 4 ( ) 5 ξ ξ ξ ξ h h) , h h h ( ) 2 ( ) 4 ( ) 5 ξ ξ ξ Q m = , h h h Q m+1 = ξ ( 2 ( ) 3 ( ) 4 ( ) 5 ξ ξ ξ ξ h h) , h h h Q m+2 = ξ ( 2 ( ) 3 ( ) 4 ( ) 5 ξ ξ ξ ξ h h) , h h h ( ) 5 ξ Q m+3 =. (6) h Combination of the element shape functions Q i, i = m 2,...,m+ 3, together with element time parameters δ i, i = m 2,...,m+ 3, gives an approximation for the typical element [,h] m+3 UN e = U(ξ,t)= i=m 2 δ i (t)q i (ξ). (7) Galerkin method for the RLW equation over the particular element becomes xm+1 x m W(U t + U x + εuu x μu xxt ) dx =. (8) In the above Galerkin method formulation, weight functions W and exact solution are replaced with quintic B-splines shape functions (6) and approximation given by Eq. (7), respectively. Thus we obtain following element equation of the coupled ordinary nonlinear differential equations: m+3 j=m 2 + ε ( h m+3 k=m 2 δj ( h ) Q i Q j dξ) + Q i Q j dξ δ j ( h ) ( h ) δj Q i Q j Q k dξ δ k δ j μ Q i Q j dξ, (9) where i, j and k take only the values m 2, m 1, m, m + 1, m + 2, m + 3 and m =, 1,...,N 1 and denotes derivative with respect to time, which in the matrix norm is A e δ e +C e δ e + εδ T L e δ e μd e δ e, (1)

5 536 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) where the dimension of the element matrices A e, C e and D e are 6 6, matrix L e is and δ e = (δ m 2, δ m 1, δ m, δ m+1, δ m+2, δ m+3 ) and h h A e i,j = Q i Q j dξ, L e i,j,k = Q i Q j Q k dξ, h h Ci,j e = Q i Q j dξ, De i,j = The matrix L e is organized to be in the dimension 6 6 as matrix B e m+3 Bi,j e = k=m 2 L e ij k δe k Q i Q j dξ. (11) so the matrix B e is expressed depending on the element parameter δ e. Assembling contributions from all elements produce the first-order matrix differential equation system (A μd) δ +(C + εb)δ =, (13) where global element parameter δ = (δ 2, δ 1, δ,...,δ N+1, δ N+2 ) (14) and A, B, C, D are derived from the corresponding element matrices A e, B e, C e, D e, respectively. Replacing the time derivative of the parameter δ by usual finite difference approximation δ =(δ n+1 δ n )/Δt and parameter δ by the Crank Nicholson formulation δ=(δ n +δ n+1 )/2 gives nonlinear recurrence relationship for time parameters between consecutive times n and n + 1as (2A 2μD + Δt C + Δt εb)δ n+1 = (2A 2μD Δt C Δt εb)δ n. (15) This relationship between two successive time steps gives a matrix system of bandwidth 11 to be solved, having N +5 linear equations in N +5 unknown. We impose boundary conditions U(a,t)=β 1, U(b,t)= β 2, U xx (a, t) = U x (b, t) = to have equations U = U(x ) = δ δ δ + 26δ 1 + δ 2 = β 1, U = U (x ) = 2 h 2 (δ 2 + 2δ 1 6δ + 2δ 1 + δ 2 ) =, U N = U(x N ) = δ N δ N + 66δ N δ N+2 + δ N+3 = β 2, U N = U (x N ) = 5 h (δ N+3 + 1δ N+2 1δ N 1 δ N ) =. (16) The first and the last two equations are not used in system (15), so that unknown parameters δ 2, δ 1, δ N+1, δ N+2 are eliminated from resulting system (15) using Eqs. (16). Thus remaining 11-banded (N + 1) (N + 1) matrix system is solved by the Gauss elimination procedure at every time step. Before moving to the next step to calculating the unknown parameters, the following iteration procedure should be carried out at least two or three times (δ ) n+1 = δ n (δn+1 δ n ) to increase the accuracy of the system. (12) (17)

6 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) Initial vector parameter δ must be determined from the boundary and initial conditions. Hence we require to satisfy the boundary conditions at the knots. (U N ) x (a, ) = (U N ) x (b, ) =, (U N ) xx (a, ) = (U N ) xx (b, ) =, U N (x, ) = U(x m, ), m =,...,N. (18) Conditions with their corresponding quintic B-spline representation give matrix equation of the system δ 2 δ 1 δ δ 1. δ N 1 δ N δ N+1 δ N+2 = U(x ) U(x 1 ). U(x N 1 ) U(x N ) This matrix system is solved to get the initial condition parameters. On determining the initial parameters from the system above, calculation of the solutions are iterated using system (15) at successive times. By using the obtained parameters from system (15), nodal values and its derivatives of order 4 can be worked out from Eqs. (5).. 3. Quintic B-spline Galerkin Method II (QBGM2) We split the RLW equation into two equations as follows: (U μu xx ) t + 2εUU x =, (U μu xx ) t + 2U x =. (19) Applying the Galerkin method to Eqs. (19) produces the following form: b a b a W((U μu xx ) t + 2εUU x ) dx =, W((U μu xx ) t + 2U x ) dx =. (2) In this Galerkin formulation for the typical element [x i,x i+1 ], quintic B-spline shape functions are used in place of weight function W and the unknown function U is approximated by the series of the form (4)

7 538 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) so that we have element equations m+3 ( h δj ( h ) m+3 δj ( h ) Q i Q j dξ) μ Q i Q j dξ + 2ε Q i Q j Q k dξ δ k δ j j=m 2 k=m 2, m+3 {( h δj ( h ) δj ( h ) } Q i Q j dξ) μ Q i Q j dξ + 2 Q i Q j dξ δ j, (21) j=m 2 where i, j and k take only the values m 2, m 1, m, m + 1, m + 2, m + 3 and m =, 1,...,N 1 and denotes derivative with respect to time. This result has the matrix form A e δ e μd e δ e +2εδ T L e δ e, A e δ e μd e δ e +2C e δ e, (22) where the dimensions of the element matrices A e, C e, and D e are 6 6, matrix L e is and δ e = (δ m 2, δ m 1, δ m, δ m+1, δ m+2, δ m+3 ). h A e i,j = Q i Q j dξ, h L e ij k = Q i Q j Q k dξ, h Ce i,j = Q i Q j dξ, We use the associated 6 6 matrix B e instead of L e in our algorithm De i,j = h Q i Q j dξ. m+3 Bi,j e = k=m 2 L e ij k δe k. (23) Collecting contributions from all elements yield the following set of nonlinear equations: (A μd) δ +2εBδ =, (24) (A μd) δ +2Cδ =, (25) where δ = (δ 2, δ 1, δ,...,δ N+1, δ N+2 ) is global element parameter vector and A, B, C, D are derived from the corresponding element matrices A e, B e, C e, D e, respectively. If we apply the Crank Nicholson procedure for parameters δ m and the usual finite difference approximation for the time parameters δ m, relating the times n and n + 1/2: δ m = δn m + δn+1/2 m, δ m = δn+1/2 m δ n m, (26) 4 Δt then we will have the following iterative relationship: (2A 2μD + ε Δt B)δ n+1/2 = (2A 2μD ε Δt B)δ n, (27) Similarly semi-discrete Eq. (25) is also discretized fully by using the Crank Nicholson formulation for the time parameters vector δ and difference approximation for time derivatives vector δ between times

8 n + 1/2 and n + 1 as follows: δ m = δn+1 m İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) δn+1/2 m, 4 δ m = δn+1 m Thus we have the iterative relationship δn+1/2 m. (28) Δt (2A 2μD + Δt C)δ n+1 = (2A 2μD Δt C)δ n+1/2. (29) The above two sets of Eqs. (27) (29) has a matrix equation of bandwidth 11 consisting of N + 5 equations in N + 5 parameters δ n = (δ n 2, δn 1,...,δn N+2 ). Application of the boundary conditions at both ends of the region U(a,t) = β 1, U xx (a, t) = and U x (b, t) = U xx (b, t) = yields to eliminate parameters δ n 2, δn 1, δn N+1, δn N+2 after reducing the number of equations by removing the first and the last two equations from systems (27) (29). So the resulting set of equations becomes 11 banded (N + 1) (N + 1) matrix equation which is solved by way of Gauss elimination procedure. After finding the parameter vector δ n+1/2 from system (27), the next time unknown parameter vector δ n+1 is determined from system (29). Thus time evolution of time parameters and nodal values together with its derivatives up to order 4 from Eqs. (5) are determined by the above-mentioned iteration procedures after finding initial parameters δ as in the previous section. 4. The test problems RLW equation possesses only three conservation constants, which correspond to mass, momentum and energy, respectively [12]: C 1 = U dx, C 2 = (U 2 + μ(u x ) 2 ) dx, C 3 = (U 3 + 3U 2 ) dx. (3) Since the conservation constants are expected to remain constant during the run of the algorithm to have the efficient numerical scheme, conservation constants will be monitored. Composite rectangle rule will be used to calculate the integrals (3) at the discrete points numerically. To measure the accuracy of numerical solutions, difference between analytical and numerical solutions at some specified times is computed by using the discrete root mean square error norm [ ] 1/2 N L 2 = U U N 2 = h U i (U N ) i 2 i=1 and maximum error norm L = max i U i (U N ) i,i= 1,...,N 1. We solve two test problems to illustrate the efficiency of the proposed algorithm by studying the motion of a single solitary and two solitary wave interaction Motion of single solitary wave We adopt the single solitary wave solution of the RLW equation with the initial condition at t = U(x,) = 3c sec h 2 (k[x x ]),

9 54 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) Table 1 Invariants and error norms for single solitary wave: amplitude =.3, Δt =.1, h =.125, 4 x 6 (QBGM1) Time L L 1 3 C 1 C 2 C Table 2 Invariants and error norms for single solitary wave: amplitude =.3, Δt =.1, h =.125, 4 x 6 (QBGM2) Time L L 1 3 C 1 C 2 C where k = 1 2 (εc/μ(1+εc))1/2 and boundary conditions U( 4,t)=U(6,t)= together with derivative boundary conditions. The exact solution is given by U(x,t)= 3c sec h 2 (k[x x vt]) (31) which describes a single bell-shape solitary wave of amplitude 3c, travelling with velocity v = 1 + εc in the positive x-direction. The RLW equation is solved with the parameters μ = ε = 1. The parameters Δt =.1, h =.125, c =.1 and x =, used in some previous studies, are adopted to make comparison. The run of the algorithms is carried up to time t = 2 over the problem domain 4 x 6. The maximum, root mean square errors and conversation invariants are presented in Tables 1 and 2 for both schemes. From this tables, QBGM1 gives better approximation results than the QBGM2. So that time-splitting of the RLW equation causes the error to increase a little. Single solitary wave solution is drawn at time t = 2 in Figs. 1 and 2. Absolute error versus x-position are drawn for both schemes. Error deviation is in the range of <x< for QBGM1 and an <x< for QBGM2. The values of the C 1,C 2,C 3 throughout the simulation are shown in Table 1. The percentage of the relative error of the conserved quantities C 1,C 2,C 3 is calculated with respect to the conserved quantities at t =. Percentage of relative changes of C 1,C 2,C 3 for QBGM1 are found by.1%,.2%,.2%, respectively. The relative percentages of C 1,C 2,C 3 change by.1%,.1%,.1%, respectively, for the QBGM2.

10 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) t= t= U.15 ERROR (a) X (b) X Fig. 1. Solitary wave solution. Parameters: h =.1, Δt =.1, c =.1, x = t= t= U.15 ERROR (a) X (b) X Fig. 2. Error distribution 1 3 at t = 2. Parameters: h =.1, Δt =.1, c =.1, x =. We compute both time and space pointwise rate of convergence for the numerical methods using the following formulas: order = log 1 ( U U h j / U U hj+1 ) log 1 (h j /h j+1 ), order = log 1 ( U U Δt j / U U Δtj+1 ), log 1 (Δt j /Δt j+1 )

11 542 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) Table 3 The order of convergence at t = 2, Δt =.1, c =.1 h j U U hj L2 QBGM1 Order QBGM1 U U hj L2 QBGM2 Order QBGM Table 4 The order of convergence at t = 2, Δt =.1, c =.1 h j U U hj L QBGM1 Order QBGM1 U U hj L QBGM2 Order QBGM where U is exact solution and U hj and U Δtj are the numerical solutions with step size h j and time steps Δt j, respectively. First, algorithm has run for various space steps with fixed the time step Δt =.1 and order of convergence for U in L 2 and L norms is documented in Tables 3 and 4 for both the schemes. The rate of order of the convergence diminishes with the smaller space step used for the proposed schemes. But if we choose much smaller space steps, convergence rate started to increase for the numerical methods. The time rate of the convergence for the numerical method is also computed with various time steps and fixed space step and recorded in Tables 5 and 6. When h =.2 is fixed, time variable pointwise convergence is not as good as the space ones. But the convergence order still is within the acceptable limits. In Tables 7 and 8, some of the results of both previous studies and quintic B-spline finite element methods are given to make comparison. Therefore the presented numerical methods provide the comparable errors. But splitting the equations cause a little loss of the accuracy in the numerical calculations. The proposed algorithms provide almost the same accuracy as [9,15] when h =.125. When the smaller amplitude of soliton with c =.3 is used, the results are the same as that of methods in [7,9,14,15] Interaction of two solitary waves In this section, we study the behavior of the interaction of two solitary waves having different amplitudes and travelling in the same direction. Initial condition of the two well-separated solitary waves of different

12 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) Table 5 The order of convergence at t = 2, h =.2, c =.1 Δt j U U hj L2 QBGM1 Order QBGM1 U U hj L2 QBGM2 Order QBGM Table 6 The order of convergence at t = 2, h =.2, c =.1 Δt j U U hj L QBGM1 Order QBGM1 U U hj L QBGM2 Order QBGM amplitudes has the following form: U(x,) = U 1 + U 2, U j = 3A j sec h 2 (k j (x x j )), A j = 4k2 j 1 4kj 2, j = 1, 2. (32) In order to run the algorithm in the finite space range, boundary conditions U(,t)= U x (12,t)= U xx (,t)= U xx (12,t)= are used. Earlier used parameters k 1 =.4, x 1 = 15, k 2 =.3, x 2 = 35 are selected to have solitary waves of magnitudes about and and peak positions of them are located at x = 15 and The differential equation is integrated from the time t = to 3 with time step Δt =.1 and the region x 12 is subdivided into 4 subintervals. A graph of the 2-soliton collision, plotted for time steps t =, 15, 3, is shown in Figs. 3a and 4a for both proposed algorithms. Two solitary waves are seen in these graphs at time t = 15 about the collision and after the separation of solitary waves. To observe the interaction event explicitly, amplitudes of the solitary wave versus time are graphed in Figs. 3b and 4b. Developing solution in time indicates that solitary wave started nonlinear collision about the time t = 1 and interaction region is finalized at about time t = 2 and then reappeared with almost original amplitude. Conserved quantities are given in Table 9. The percentage of the relative

13 544 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) Table 7 Invariants and error norms for single solitary wave at t = 2, x =, Δt =.1, amp. =.3, 4 x 6 Method h C 1 C 2 C 3 L L 1 3 QBGM QBGM [15] [3] [1] [11] [5] [4] [14] [17] QBGM QBGM [15] [7] [11] [9] [17] [5] [4] [14] Table 8 Invariants and error norms for single solitary wave at t = 2, x =, Δt =.1, amp. =.9, 4 x 6 Method h C 1 C 2 C 3 L L 1 3 QBGM QBGM [15] [3] [1] [11] [5] [14] [4] [17] QBGM QBGM [15] [7] [11] [9] [17] [14] [4] [5]

14 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) t= U 3. Amplitude t= t=3 1. (a) X 12.. (b) t 3. Fig. 3. (a) Interaction of two solitary waves (QBGM1); (b) Time-amplitude graph (QBGM1) t=15 5. U Amplitude t= t=3 1. (a) X (b) t Fig. 4. (a) Interaction of two solitary waves (QBGM2); (b) Time-amplitude graph (QBGM2). error for the QBCM1 is changed by.3% for C 1, 1.5% for C 2, and 2 for C 3 during computer run. The percentage of the relative error for the QBCM2 is changed by.5% for C 1,2%forC 2, and 2.7% for C 3. QBCM1 provides a little less conserved quantities than the QBCM2. The numerical integration of the RLW equation is much easy with low order B-spline functions. Hence, use of the low-order B-spline functions causes low-order system in the Galerkin finite element formulation. Although discretization of the RLW equation with quintic B-spline results in higher order

15 546 İ. Dağ et al. / Journal of Computational and Applied Mathematics 19 (26) Table 9 QBCM1 QBCM2 t C 1 C 2 C 3 C 1 C 2 C matrix equation to be solved, quintic B-spline method is applied to the RLW equation in getting the numerical solution to make comparison with the existing results found by using some other splines in finite element methods. The proposed method produced the same results with best of the other methods documented in Tables 7 and 8. Propagation of the single solitary wave and two soliton integration are simulated well with the proposed algorithms and conservation invariants do not change much during the computer run. Thus quintic B-spline functions can be used to construct approximate numerical methods over finite elements. In conclusion, higher-order differential equations can be integrated by using the quintic B-splines to have discretization of the PDE. References [1] T.B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves in non-linear dispersive systems, Philos. Trans. Roy. Soc. London A 272 (1972) [2] J.L. Bona, P.J. Bryant, A mathematical model for long waves generated by wavemakers in nonlinear dispersive systems, Proc. Cambridge Philos. Soc. 73 (1973) [3] İ. Daǧ, Least squares quadratic B-spline finite element method for the regularised long wave equation, Comput. Methods Appl. Mech. Engrg. 182 (2) [4] İ. Dağ, A. Doğan, B. Saka, B-spline collocation methods for numerical solutions of RLW equation, Internat. J. Comput. Math. 8 (23) [5] İ. Daǧ, M.N. Özer, Approximation of the RLW equation by the least square cubic B-spline finite element method, Appl. Math. Modelling 25 (21) [6] G.A. Gardner, L.R.T. Gardner, Modelling solitons of the Korteweg de Vries equation with quintic splines, University of Wales, Bangor, UK, Maths Preprint series, No: 9.3, 199. [7] L.R.T. Gardner, İ. Daǧ, The boundary-forced regularised long-wave equation, IL Nuova Cimento 11B (12) (1995) [8] L.R.T. Gardner, G.A. Gardner, F.A. Ayoub, N.K. Amein, Modelling an undular bore with B-splines, Comput. Methods Appl. Mech. Engrg. 147 (1997) [9] L.R.T. Gardner, G.A. Gardner, İ. Daǧ, A B-spline finite element method for the regularised long wave equation, Comm. Numer. Methods Engrg. 11 (1995) [1] L.R.T. Gardner, G.A. Gardner, A. Doǧan, A least squares finite element scheme for the RLW equation, Comm. Numer. Methods Engrg. 12 (1996) [11] P.C. Jain, L. İskandar, Numerical solutions of the regularised long wave equation, Comput. Methods Appl. Mech. Engrg. 2 (1979)

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