Research Article Optimal Error Estimate of Chebyshev-Legendre Spectral Method for the Generalised Benjamin-Bona-Mahony-Burgers Equations

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1 Abstract and Applied Analysis Volume, Article ID 6343, pages doi:.55//6343 Research Article Optimal Error Estimate of Chebyshev-Legendre Spectral Method for the Generalised Benjamin-Bona-Mahony-Burgers Equations Tinggang Zhao,, Xiaoxian Zhang, Jinxia Huo, Wanghui Su, Yongli Liu, and Yujiang Wu School of Mathematics, Lanzhou City University, Lanzhou 737, China School of Mathematics and Statistics, Lanzhou University, Lanzhou 73, China Correspondence should be addressed to Tinggang Zhao, Received 9 December ; Revised 6 April ; Accepted 5 April Academic Editor: M. Victoria Otero-Espinar Copyright q Tinggang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Combining with the Crank-Nicolson/leapfrog scheme in time discretization, Chebyshev-Legendre spectral method is applied to space discretization for numerically solving the Benjamin-Bona- Mahony-Burgers gbbm-b equations. The proposed approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto CGL nodes are used in the computation. By using the proposed method, the computational complexity is reduced and both accuracy and efficiency are achieved. The stability and convergence are rigorously set up. Optimal error estimate of the Chebyshev-Legendre method is proved for the problem with Dirichlet boundary condition. The convergence rate shows spectral accuracy. Numerical experiments are presented to demonstrate the effectiveness of the method and to confirm the theoretical results.. Introduction We are interested in numerically solving initial boundary value problem of the generalised Benjamin-Bona-Mahony-Burgers gbbm-b equations in the following form: u t μu xxt αu xx βu x g u, x. where u u x, t represents the fluid velocity in horizontal direction x, parameters μ,, α>andβ are any given constants, and g u is a nonlinear function with certain smoothness. Notation u t denotes the first derivative of u with respect to temporal t and u x, u xx are the first and second derivatives with respect to space variable x.

2 Abstract and Applied Analysis When g u / u with α andβ,. is the alternative regularized longwave equation proposed by Peregrine and Benjamin et al.. Equation. features a balance between nonlinear and dispersive effects but takes no account of dissipation for the case α. In the physical sense,. with the dissipative term αu xx is proposed if the good predictive power is desired, such problem arises in the phenomena for both the bore propagation and the water waves. The well posedness and some asymptotic properties of solutions to this problem were discussed by Karch 3, Zhang 4, and so on. For more knowledge about BBM-B equations, please consult with 5, 6 and so forth. In 7, Al-Khaled et al. implemented the Adomian decomposition method for obtaining explicit and numerical solutions of the BBM-B equation.. By applying the classical Lie method of infinitesimals, Bruzón et al. 8 obtained, for a generalization of a family of BBM equations, many exact solutions expressed by various single and combined nondegenerative Jacobi elliptic functions. Tari and Ganji have applied two methods known as variational iteration and homotopy perturbation methods to derive approximate explicit solution for. with g u / u. El-Wakil et al. and Kazeminia et al. 3 used the expfunction method with the aid of symbolic computational system to obtain the solitary solutions and periodic solutions for. with g u / u. Variational iteration combining with the exp-function method to solve the generalized BBM equation with variable coefficients was conducted by Gómez and Salas 4. In 5, Fakhari et al. solved the BBM- B equation by the homotopy analysis method to evaluate the nonlinear equation. with g u / u, α, and β. A tanh method and sinc-galerkin procedure were used by Alqruan and Al-Khaled 6 to solve gbbm-b equations. Omrani and Ayadi 7 used Crank- Nicolson-type finite difference method for numerical solutions of the BBM-B equation in one space dimension. In 8, a local discontinuous Galerkin finite element method was used and an optimal error estimate was derived by the authors for numerical solution to BBM-B equation. To our knowledge, there is little work on spectral method for numerical solution of the gbbm-b equations. For completeness,. may be complemented with certain initial and boundary conditions. Here in this paper we will deal with the initial boundary value problem u t μu xxt αu xx βu x g u f, x Λ : x,, t>,. with Dirichlet-type boundary conditions u,t u,t, t,.3 and initial condition u x, u x, x Λ,.4 The appearance of the right-hand side term in. is just for convenience in order to test the numerical efficiency of the proposed method. For other kinds of boundary conditions, a penalty method may be used as in 9.

3 Abstract and Applied Analysis 3 Let L Λ be a square integrable function space with inner product and norm as follows: u, v Λ u x v x dx, u u, u..5 For any positive integer r, H r Λ denotes the Sobolev-type space defined as { } H r Λ u L Λ : k u x k L Λ, k r..6 We denote the norm and seminorm of H r Λ by r and r, respectively. For any real r, H r Λ is defined by space interpolation. H Λ is a subspace of H Λ in which the function vanishes at ±. We reformulate the problem..4 in weak form: find u t H Λ, such that, for all φ H Λ, ut,φ μ u xt,φ x α ux,φ x β u, φx g u,φx f, φ, t,t, u, φ u,φ, t..7 Due to the high accuracy, spectral/pseudospectral methods are increasingly popular during the last two decades 3 7. In the context of spectral methods, Legendre and Chebyshev orthogonal polynomials are extensively used for the nonperiodic cases. The Legendre method is natural in the theoretical analysis due to the unity weight function. However, it is well known that the applications of Legendre method are limited by the lack of fast transform between the physical space and the spectral space and by the lack of explicit evaluation formulation of the Legendre-Gauss-Lobatto LGL nodes. A Chebyshev-Legendre method that implements the Legendre method at Chebyshev nodes was proposed by Don and Gottlieb 9 for the parabolic and hyperbolic equations. The approach enjoys advantages of both the Legendre and Chebyshev methods. The Chebyshev-Legendre methods were also applied to elliptic problems 8, nonlinear conservative laws in 9, 3, Burgers-like equations in, KdV equations in 3, generalised Burgers-Fisher equation 3, nonclassical parabolic equation 33, and optimal control problems 34. In this paper, Chebyshev- Legendre spectral method will be applied to solving the initial boundary value problem..4. The proposed approach is based on Legendre-Galerkin formulation while the Chebyshev-Gauss-Lobatto CGL nodes are used in the computation. The computational complexity is reduced and both accuracy and efficiency are achieved by using the proposed method. Compared with the generalised Burgers equation, the gbbm-b equation has an additional term μu xxt, which serves as a stabilizer. Due to this reason, numerical analysis of the Chebyshev-Legendre schemes for gbbm-b equations is actually much easier than that for generalised Burgers equations in,. This paper is organized as follows. In Section, we set up Chebyshev-Legendre spectral method for gbbm-b equations and show how to implement the scheme. Section 3 is preliminary that involves some lemmas which will be used later. In Section 4, error analysis is performed for both semidiscretization and fully discretization schemes. We obtain an optimal

4 4 Abstract and Applied Analysis convergence rate in sense of H -norm. In Section 5, numerical experiments are presented to support the theoretical results. Conclusion is given in Section 6.. Chebyshev-Legendre Spectral Method In this section, we first set up semidiscretization and fully discretization Chebyshev-Legendre spectral method for problem. and then give how to implement the proposed scheme... Chebyshev-Legendre Spectral Method Let P N be the set of all algebraic polynomials of degree at most N. We introduce the operator of interpolation at the CGL nodes {η i cos iπ/n } i N, denoted by Π C N, which satisfies Π C N f P N and Π C N f η i f ηi, i N.. Denote V N P N Λ H Λ.. The weak form.7 leads to the following semidiscretization Legendre-Galerkin scheme: find u N t V N, such that, for all φ N V N, unt,φ N μ unxt,φ Nx α unx,φ Nx β u N,φ Nx g un,φ Nx f, φn, t,t,.3 un,φ N u,φ N, t. Now we give a Chebyshev-Legendre spectral scheme for the problem.. The semidiscretization Chebyshev-Legendre spectral method for. is to find u N t V N, such that, for all φ N V N, unt,φ N μ unxt,φ Nx α unx,φ Nx β u N,φ Nx Π C N g u N,φ Nx Π C N f, φ N, t,t, un,φ N Π C N u,φ N, t..4 Remark.. The difference between Legendre method.3 and Chebyshev-Legendre method.4 lies in the treatment of three terms: nonlinear term, source term and initial data. But such treatment leads to two advantages: one is improvement of convergence rate in theoretical analysis, the other is free from computing the LGL nodes. We only need Chebyshev transform and Chebyshev-Legendre transform in computing, and the former can perform through FFT.

5 Abstract and Applied Analysis 5 For the time advance, we adopt the second-order Crank-Nicolson/leapfrog. For a given time step τ, lets t {kτ : k,,...,m t,t M t τ}; the notations v t t and v t are used as v t v t τ v t τ t, v t τ v t τ v t τ..5 Let t k kτ and u k u t k. The fully discretization Chebyshev-Legendre spectral method for. is to find u k N k,,...,m T T/τ V N, such that, for all φ N V N, u k,φ N t N μ u k,φ Nx t Nx α û k Nx,φ Nx β û k N,φ Nx Π C N g u k N,φ Nx Π C f k N,φ N, k,...,m T, u N u N u Nx,φ N μ u Nx u Nx,φ Nx α u Nx,φ Nx τ τ β u N u N,φ Nx Π C N g u N,φ Nx Π C N f f,φ N,.6 u N ΠC N u... Implementation of the Chebyshev-Legendre Spectral Method Let L k be the kth degree Legendre polynomial that is mutually orthogonal in L Λ, thatis, Lk,L j L k x L j x dx k δ kj..7 We define in 35 Λ Φ k x L k x L k x..8 One useful property of the Legendre polynomials is n L n x L n x L n x,.9 which gives the following relation: Φ k x k 3 L k x.. It is easy to verify that, for N, V N span{φ, Φ,...,Φ N }..

6 6 Abstract and Applied Analysis Let us denote N u i N k û i k Φ k, û i û i, ûi,...,ûi N T, T, ĝ i k Π C N u g i N, Φ k, ĝ i ĝ i, ĝi N,...,ĝi û,k f i k Π C N f i, Φ k, f i f i T,, f i,..., f N i Π C N u, Φ k, û û,, û,,...,û,n T,. m ij Φ j, Φ i, M mij i,j,,...,n, s ij Φ j, Φ i, S sij i,j,,...,n, p ij Φ j, Φ i, P p ij i,j,,...,n. Then, scheme.6 leads to the following linear system series: For k,,...,m T [ M μ ατ P βτs ]ûk [ M μ ατ P βτs ] û k τĝ k τ f k f k,.3 starting with [ M μ ατ P βτ ] [ S û M μ ατ P βτ ] S û τĝ τ f f, Mû û..4 By using integration by parts and orthogonality of Legendre polynomials, one can easily determine that the matrix M is pentadiagonal and P diagonal. Moreover, the nonzero entries of the matrices M, P can be determined by using the orthogonal relation.7 and. as follows: k, j k, k 5 m jk m kj, j k ±, k 5, otherwise,

7 Abstract and Applied Analysis 7, j k, s jk, j k,, otherwise, 4k 6, j k; p jk p kj, otherwise..5 Hence, the matrices in the left-hand side of.3 are banded and the linear system.3 can be easily solved. For evaluation of the last three terms on the right-side hand in.3, it can be split into two steps: one step is the transform from its values at CGL nodes to the coefficients of its Chebyshev expansion, which can be done by using fast Chebyshev transform or fast Fourier transform. The other step involves the fast Legendre transform between the coefficients of the Chebyshev expansion and of the Legendre expansion, which has been addressed by Alpert and Rokhlin 36. It is worthy to note that Shen suggested an algorithm 8 that seems more attractive for moderate N. 3. Preliminaries In this section, we introduce a suitable comparison function and give some lemmas needed in error analysis. We denote by C a generic positive constant independent of N or any function. Now we introduce two projection operators and their approximation properties. One is P L N : L Λ P N such that P L N v, φ v, φ, φ P N. 3. The approximation property Theorem 6. and 6.8 in 4, page 58, 6 is the following. Lemma 3.. For any u H r Λ r >, there exists a positive constant C independent of N such that u P L N u s u P L N u CN r u r, CN u r, s, CN s / r u r, s The other projection operator is defined as follows: P N u x x P L N u y y dy. 3.4

8 8 Abstract and Applied Analysis It is easy to know that P N : H Λ V N such that PN v x,φ x vx,φ x, φ VN. 3.5 Its approximation property Theorem 6. in 4, page 6 is the following. Lemma 3.. For any u H r Λ H Λ, there exists a positive constant C independent of N such that u P N u s CN s r u r, s r. 3.6 For the interpolation operator Π C N at the Chebyshev-Gauss-Lobatto points, we cite the following approximation result 9. Lemma 3.3. If u H Λ,then NΠ C Π N u u C N u C u, 3.7 moreover, if u H s Λ s, then Π C N u u l CN l s u s, l. 3.8 In general, the discretization inner product and norm are defined as follows: N u, v N u y j v yj ωj, j u N u, u N, 3.9 where y j and ω j j,,...,n are the Legendre-Gauss-Lobatto points and the corresponding quadrature weights. Associating with this quadrature rule, we denote by Π L N the Legendre interpolation operator. The following result is cited from 6. Lemma 3.4. If u H σ Λ σ, then Π L N u u l CN l σ u σ, l, u, v u, v N CN σ u σ v, v P N. 3. Further, if u P N,then u u N N u, 3. u L Λ N u. 3.

9 Abstract and Applied Analysis 9 4. Error Analysis of the Schemes In this section, we set up the stability and convergence results first for the semidiscretization Chebyshev-Legendre spectral method.4 and then for the fully discretization scheme Stability and Convergence of Scheme.4 Since the initial value and the right-hand side term cannot be exactly evaluated, we consider here how stable the numerical solution of.4 depending on the initial value and the righthand side term. As in, we suppose that u N and f are computed with errors ũ and f respectively. That is, ũ, ũt,φ N μ ũxt,φ Nx α ũx,φ Nx β φnx Π C N G, φ Nx Π C f,φ N N, φ N V N, 4. where G : g u N ũ g u N. Since ũ V N leads to ũ, ũ x, we come out to, by taking φ N ũ in 4., d [ ] ũ μ ũ α ũ dt Π C N G, ũ x Π C f,ũ N. 4. Now we need to estimate the nonlinear term Π C N G in the right-hand side 4.. LetC be a positive constant and C g z,z { } u M max s T u s L Λ u x s L Λ, max z z z g z z z max z z z g z. 4.3 For any given t,t,if ũ s L Λ C, s,t, 4.4 then G g u N θũ ũ Cg u M,C ũ, G g u N θũ u Nx ũ g u N ũ ũ x 4.5 C g u M,C ũ x s ũ s.

10 Abstract and Applied Analysis As an immediate consequence of the above estimates and inequality 3.8, itistruethat,for large N, Π C N G Π C N G G G CN G G CN C g u M,C ũ ũ C g u M,C ũ CC g u M,C N ũ CC g u M,C ũ N. 4.6 Hence, it is true that, for large N, Π C N G, ũ x C ũ α ũ. 4.7 Therefore, integrating 4. in time leads to s s ũ s μ ũ s α ũ t dt ũ μ ũ C ũ t dt s Π C f t N dt. 4.8 Denoting s E ũ, s ũ s μ ũ s α ũ t dt, s ρ ũ, s ũ μ ũ Π C f t N dt, 4.9 then, we have E ũ, t ρ ũ, t C t E ũ s,s ds. 4. We have the following stability result. Theorem 4.. If g C R and e CT ρ ũ, T C N, 4. then there holds the following inequality: E ũ, t ρ ũ, t e Ct, t,t. 4.

11 Abstract and Applied Analysis Proof. We need to verify that ũ s L Λ C, s,t. 4.3 Otherwise, there must exist t <Tsuch that max s t ũ s L Λ C, ũ t L Λ C, 4.4 while by 4., 4., and the Gronwall inequality, we have E ũ, t ρ ũ, f,t e Ct <ρ ũ, f,t e CT C N. 4.5 Thus, from Lemma 3.4, ũ t L Λ N ũ t N E ũ, t <C, 4.6 which is contradictory with 4.4.Thus, 4.35 holds, and we derive 4. from the Gronwall inequality. Remark 4.. Condition 4. in Theorem 4. means that errors occurrs during evaluation of the initial value and the right-hand side term should not be larger than C e CT / N, which becomes small when N goes large. This condition may be caused by the nonlinearity of the problem. Also, the condition could be satisfied because the orthogonal polynomial approximation goes faster than / N in general. As for the result 4. in Theorem 4., it is analogous to those in 9 33 just for different energy norm E u, t. Next we turn to convergence of the semidiscretization scheme.4.letu t and u N t be the solutions to.7 and.4, respectively. We denote u P N u, e t u N t u t. Then we have the following result. Theorem 4.3. If u C,T; H r Λ H Λ,f C,T; Hr Λ H Λ r >, and g z C R which satisfies the assumption of Theorem 4., then the following error estimate t e t μ e t α e s ds C μn N r, t T, 4.7 holds. Proof. From.7,.4, and 3.5, we know that [ et,φ μ e xt,φ x α ex,φ x ] β e, φx [Π C N g u N g u,φ x ] [ u u t,φ ] β [ u u,φ x ] [ f Π C N f, φ ]. 4.8

12 Abstract and Applied Analysis Taking φ e in 4.8, we have d [ ] e μ e α e dt I I I 3 I 4, I : Π C Π N g u N g u,e x C N g u N g u e Π C N α g u N g u α 4 e, I : u u t,e u t P N u t e CN r u t r e, 4.9 I 3 : β u u,e x u P N u e α CN r u r α 4 e, I 4 : f Π C f N f, e Π C N f e CN r f r e. 4. Next, we estimate Π C N g u N g u, Π C N g u N g u Π C N g u N g u N g un g u g u g u CN r g un r g u N θu e g u θu u u CN r u r C e. 4. Combining with the estimates above, inequality 4.9 becomes d [ ] e μ e dt α e CN r u r u t r f C r e. 4. Denoting t E t e t μ e t α e s ds, t ρ t e μ e CN r u s r u t s r f t, r ds 4.3 we have E t ρ t C t E s ds. 4.4 Thanks to the Gronwall inequality, we get E t ρ t e Ct, <t T. 4.5

13 Abstract and Applied Analysis 3 Because of e P N u Π C N u u P N u u Π C N u, we combine with the approximation properties to obtain e u P N u u Π C N u CN r u r, e u P N u u Π C N u CN r u r. 4.6 Then it is easy to show the desired result. Remark 4.4. The result in Theorem 4.3 is optimal in sense of H estimation. 4.. Stability and Convergence of the Fully Discretization Scheme In this section we consider the stability and convergence of the fully discretization scheme of.6. We assume that all functions below are valued at time s unless otherwise specified. Suppose that u N and f in.6 have the errors ũ and f, respectively. Then, we have ũ t,φ μ ũ x t,φ x α ũx,φ x β ũ, φx Π C N g,φ x Π C N f,φ, 4.7 where g g u k N ũ g uk N. Taking φ ũ in 4.7, weget ] [ ũ t μ ũ t α ũ Π C N g, ũ x Π C N f, ũ. 4.8 Now we need to estimate Π C N g, which would be easy to do, provided that ΠC N is replaced by Legendre interpolation operator or the norm is in the weighted Chebyshev one. But here we cannot deal with this directly. Just like in, by taking φ ũ t in 4.7, we again get [ ] ũ t μ ũ t α ũ ũ, β ũx t Π C t N g,ũ x t Π C N f,ũ t. 4.9 Combining 4.8 and 4.9 through the factor N, we arrive at [ ũ μ ũ N α ũ t N ũ t ] N μ ũ t ũ α Π C N g, ũ x Π C N f, ũ N β ũ, ũx t 4.3 N Π C N g,ũ x t N Π C N f,ũ t.

14 4 Abstract and Applied Analysis By using the Hölder inequality, we have Π C N g, ũ Π x C N ũ g 3 Π C N α g α ũ 3, Π C N f, ũ ΠC N f ũ 3 α ΠC N f α ũ 3, N β ũ, ũx t N β ũ ũ t N β ũ N ũ t, N Π C N x t g,ũ N Π C N g ũ t N Π C N g N ũ t, f,ũ t N Π C N N ΠC N f ũ t N μ Π C N f N μ ũ t. 4.3 Thus, if N β 6/α, we have [ ũ μ ũ N α ũ t N ũ t ] N μ ũ t ũ α 3 Π C N α g N Π C N g 3 α ΠC N f N μ Π C N f Π C C N g N Π C N g ΠC N f N Π C N f, 4.3 where C is a positive constant dependent on α, μ. Summing 4.3 for s S t τ gives E ũ, t ρ ũ, f,t Cτ Π C N g s N Π C N g, 4.33 s S t τ where E ũ, t ũ t μ ũ t N α ũ t τ N ũ t s N μ ũ t s ũ s α, s S t τ ρ ũ, f,t ũ μ ũ N α ũ ũ τ μ ũ τ N α ũ τ Cτ Π C N f s N Π C N f s s S t τ For any given t S T,if ũ s L Λ C, s S t τ, 4.35

15 Abstract and Applied Analysis 5 then, by 3.7 and 3.8, Π C N g N Π C N g g Π C N g g N Π C N g g CN g C g u M,C ũ N ũ, s S t τ Thus, we have shown that for any t S T,if 4.35 holds, then E ũ, t ρ ũ, f,t C τ E ũ, s, s S t τ 4.37 where C is a positive constant dependent on C g u M,C, α and μ. Theorem 4.5. Let τ be suitably small and N β 6/α. If g C R and ρ ũ, f,t C e C T / N,then E ũ, t ρ ũ, f,t e C t, t S T Proof. We verify the result by induction over t S T. It is easy to see that the result is true for t τ. Assume that it is true for all s S t τ that E ũ, s ρ ũ, f,s e C s Then, from the inverse inequality 3. we have ũ s L Λ N ũ s N ρ ũ, f,s e C s C, 4.4 which means that 4.35 holds. Therefore, we have, from 4.37 and 4.39, that E ũ, t ρ ũ, f,t C τ E ũ, s ρ ũ, f,t C τ ρ ũ, f,s e C s s S T τ s S T τ ρ ũ, f,t C τ e C s s S T τ ρ ũ, f,t e C t. 4.4 Thus, the proof is completed.

16 6 Abstract and Applied Analysis Next we consider the convergence of scheme.6.letu t and u k N be the solutions to.7 and.6, respectively. Setting v t P N u t,e k v k u k N, we have e k t,φ μ e k,φ x t x α êx,φ k x β ê k,φ x Π C G, N φ x Π C f k N f k,φ P N u k t ûk t,φ μ P N u kx t û xt,φ x β û k P N û k,φ x ĝ u k Π C N u g k N ek,φ x, 4.4 where G g u k N ek g u k N and ĝ uk g u t k τ g u t k τ /. It is easy to obtain the following estimate: Π C f k N f k CN r fr, P N u k t ûk t P N u u k t uk t k t ûk t CN r u r u k t u t t k u t t k û k t CN r u r C τ u ttt t k C τ u tt t k, P N u k x t ûk xt P N u k x t uk u k x t x t ûk xt CN r u r u k u x t xt t k u xt t k û k xt CN r u r C τ u xttt t k C τ u xtt t k, û k P N û k CN r u r, ĝ u k Π C N u g k N ek ĝ u k Π C Nĝ u k Π C N ĝ u k g u k N ek CN r u r C ĝ u k g u t k C g u k g u k N ek CN r u r C τ u C N r u r Combining with the stability results of Theorem 4.5, we can obtain the following convergence result. Theorem 4.6. Let u and u k N be the solutions of.7 and.6, respectively. If u x, t C 3,T; H r Λ H Λ r and g C R and τ is suitably small, then one has u t k u k N μ u t k u k N C τ N r Numerical Experiments In this section, we will present some numerical experiments to confirm the effectiveness and robustness of scheme.6 or.3.

17 Abstract and Applied Analysis 7 Table : Convergence rates at t withμ, α, β, γ 3, n. N L -error Order L -error Order 4.875E 3.54E 8.79E 3 N E 4 N E 6 N E 7 N E 9 N E N 4. Table : Convergence rates at t withμ, α, β, γ 3, n. N L -error Order L -error Order 4.474E.96E E 4 N E 5 N E 7 N E 8 N E 9 N. 5.66E N. Table 3: Convergence rates at t withμ, α β, γ 3, and n. τ L -error Order L -error Order 8.4E 5.5E 5.6E 5 τ. 3.9E 6 τ E 7 τ..5e 7 τ. 3.63E 7 τ. 3.33E 8 τ E 9 τ..58e 9 τ. 4.76E 9 τ. 3.55E τ. Example 5.. We consider the initial boundary problem..4 with the exact solution as follows: u x, t t γ sin πx. 5. The data u x is zero function, and the source term f is determined by. and g u u n /n, that is, [ ] f x, t t γ sin πx γ μπ γ απ t βπt γ cos πx πt n sin n πx cos πx. 5. Because the exact solution is known, we can compute the error between the numerical solution and the exact solution. In Table, the convergent rate in L and L errors for spatial discretization is computed by fixing time step τ 5 and changing N from to 4 with the parameters μ, α, β, γ 3, and n att. In Table, the convergent rate in L and L errors for spatial discretization is computed by fixing time step τ 5 and changing N from to 4 with the parameters μ, α, β, γ 3, and n att. The results show clearly spectral accuracy.

18 8 Abstract and Applied Analysis Table 4: Convergence rates at t withμ., α, β., γ 3, and n. N τ L -error Order L -error Order 4.986E 4.7E 8.64E 3 N E 3 N 8.39 E E 6 N E 6 N E 9 N E 9 N E N E N E E E 7 τ..9e 7 τ E 8 τ E 9 τ E 9 τ..9e 9 τ E τ E τ. Table 5: Convergence rates at t withμ., α, β., γ 3, and n. N τ L -error Order L -error Order 4.7E.E 8.49E 5 N E 5 N.5 E 5 3.9E 9 N.3.845E 9 N E N E N E.57E 3.48E 7 3.5E E 8 τ E 9 τ E 9 τ. 3.5E τ E τ E τ. 5.8E τ E τ. For temporal discretization, the convergent rate in L and L errors is computed by fixing N 3 and changing time step τ from to 4 with the parameters μ, α β, γ 3, and n att. In Table 3, the results show clearly second-order convergence for time discretization. In Table 4, the results show clearly second-order convergence for time discretization and spectral accuracy for space discretization again. Example 5.. We consider the initial boundary problem..4 with the exact solution as follows: u x, t x e x t. 5.3 The data u x x e x and the source term f is determined by. and g u u n /n. In Table 5, the results show clearly second-order convergence for time discretization and spectral accuracy for space discretization again.

19 Abstract and Applied Analysis 9. μ = β = γ =. α =.. μ = β = γ =. α = t = t =.5 t = t =.5 t = t =.5 t = t =.5 a α. b α...8 μ = β = γ =. α = t = t =.5 t = t =.5 c α. Figure : The evolution of u x, t with different α. Example 5.3. We consider the gbbm-b equation u t μu xxt αu xx βu x γuu x, 5.4 with initial profile as u x e 4x. 5.5 In Figures and, the results show how the numerical diffusion does with the increasing diffusion α and reaction coefficient β.

20 Abstract and Applied Analysis. μ = α = γ =. β =.5. μ = α = γ =. β = t = t =.5 a β.5 t = t =.5 t = t =.5 b β. t = t =.5. μ = α = γ =. β = t = t =.5 t = t =.5 c β. Figure : The evolution of u x, t with different β. 6. Conclusion In this paper, we have developed the Chebyshev-Legendre spectral method to the generalised Benjamin-Bona-Mahony-Burgers gbbm-b equations. As is well known, Chebyshev method is popular for its explicit formulae and fast Chebyshev transform. But the Chebyshev weight makes much trouble in error analysis. Legendre method is natural in the theoretical analysis due to the unity weight function. However, it is well known that the applications of Legendre method are limited by the lack of fast transform between the physical space and the spectral space and by the lack of explicit evaluation formulation of the LGL nodes. Chebyshev- Legendre method adopts advantages of both Chebyshev method and Legendre method. Hence, it gives a better scheme. Also, error analysis shows that an optimal convergence

21 Abstract and Applied Analysis rate can be obtained by using the Chebyshev-Legendre method. The numerical results show that this scheme is an efficient one. For the problems of high dimension and with complex geometry boundary, Chebyshev-Legendre method could perform well. This will be left for our further research. Acknowledgments The authors thank the anonymous referees for their valuable suggestions and helpful comments. The work of the first author was partly supported by the National Natural Science Foundation of China under Grant no. 66. The work of Y. Wu was partially supported by the National Basic Research Program of China, 973 Program no. CB7693. References D. H. Peregrine, Calculations of the development of an undular bore, The Journal of Fluid Mechanics, vol. 5, no., pp. 3 33, 996. T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philosophical Transactions of the Royal Society of London, vol. 7, no., pp , G. Karch, Asymptotic behaviour of solutions to some pseudoparabolic equations, Mathematical Methods in the Applied Sciences, vol., no. 3, pp. 7 89, L. Zhang, Decay of solution of generalized Benjamin-Bona-Mahony-Burgers equations in n-space dimensions, Nonlinear Analysis, vol. 5, no., pp , S. Kinami, M. Mei, and S. Omata, Convergence to diffusion waves of the solutions for Benjamin- Bona-Mahony-Burgers equations, Applicable Analysis, vol. 75, no. 3-4, pp ,. 6 N. A. Larkin and M. P. Vishnevskii, Dissipative initial boundary value problem for the BBMequation, Electronic Journal of Differential Equations, vol. 49, pp., 8. 7 K. Al-Khaled, S. Momani, and A. Alawneh, Approximate wave solutions for generalized Benjamin- Bona-Mahony-Burgers equations, Applied Mathematics and Computation, vol. 7, no., pp. 8 9, 5. 8 M. S. Bruzón, M. L. Gandarias, and J.-C. Camacho, Symmetry for a family of BBM equations, Journal of Nonlinear Mathematical Physics, vol. 5, Supplement, pp. 8 9, 8. 9 M. S. Bruzón and M. L. Gandarias, Travelling wave solutions for a generalized Benjamin-Bona- Mahony-Burgers equation, International Journal of Mathematical Models and Methods in Applied Sciences, vol., no., pp. 3 8, 8. M. S. Bruzón, M. L. Gandarias, and J. C. Camacho, Symmetry analysis and solutions for a generalization of a family of BBM equations, Journal of Nonlinear Mathematical Physics, vol. 5, supplement 3, pp. 8 9, 8. H. Tari and D. D. Ganji, Approximate explicit solutions of nonlinear BBMB equations by Hes methods and comparison with the exact solution, Physics Letters A, vol. 367, pp. 95, 7. S. A. El-Wakil, M. A. Abdou, and A. Hendi, New periodic wave solutions via Exp-function method, Physics Letters A, vol. 37, no. 6, pp , 8. 3 M. Kazeminia, P. Tolou, J. Mahmoudi, I. Khatami, and N. Tolou, Solitary and periodic solutions of BBMB equation via Exp-function method, Advanced Studies in Theoretical Physics, vol. 3, no., pp , 9. 4 C. A. Gómez and A. H. Salas, Exact solutions for the generalized BBM equation with variable coefficients, Mathematical Problems in Engineering, vol., Article ID 49849, pages,. 5 A. Fakhari, G. Domairry, and Ebrahimpour, Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution, Physics Letters A, vol. 368, no. -, pp , 7. 6 M. Alqruan and K. Al-Khaled, Sinc and solitary wave solutions to the generalized Benjamin-Bona- Mahony-Burgers equations, Physica Scripta, vol. 83, no. 6,. 7 K. Omrani and M. Ayadi, Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation, Numerical Methods for Partial Differential Equations, vol. 4, no., pp , 8.

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