Application of Quintic B-splines Collocation Method on Some Rosenau Type Nonlinear Higher Order Evolution Equations.

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.13(2012) No.2,pp Application of Quintic B-splines Collocation Metod on Some Rosenau Type Nonlinear Higer Order Evolution Equations R.C. Mittal, R.K. Jain Department of Matematics, Indian Institute of Tecnology Roorkee, Roorkee , Uttarakand,India (Received 27 September 2011, accepted 19 January 2012) Abstract: In tis work, we discuss a collocation metod for solving some Rosenau type non-linear iger order evolution equations wit Diriclet s boundary conditions. Te approac used, is based on collocation of a quintic B-splines over finite elements so tat we ave continuity of te dependent variable and its first four derivatives trougout te solution range. We apply quintic. B-splines for spatial variable and derivatives wic produce a system of first order ordinary differential equations. We solve tis system by using SSP- RK3 sceme. Tis metod needs less storage space tat causes to less accumulation of numerical errors. Te numerical approximate solutions to Rosenau type non-linear evolution equations ave been computed witout transforming te equations and witout using te linearization. Tis metod is tested on four test problems were one example is wit te variable coefficients. Easy and economical implementation is te strengt of tis metod. Keywords: Rosenau type nonlinear iger order evolution equation; quintic B-splines basis functions; SSP- RK3 sceme; Tomas algoritm 1 Introduction Non-linear iger order evolution equations are special classes of te category of partial differential equations, wic ave been studied intensively in past several decades. It is well known tat seeking approximate numerical solutions for non-linear iger order evolution equations, by using different numerous metods, as long been a major concern for matematicians, pysicists, and engineers. In particular, te travelling wave solutions play an important role in te study of te models arising from various natural penomena and scientific and engineering fields; for instance, te wave penomena observed in fluid dynamics, elastic media, optical fibres, nuclear pysics, ig-energy pysics, plasma pysics, gravitation and in statistical and condensed matter pysics, biology, solid-state pysics, cemical kinematics, cemical pysics and geocemistry, etc.[8, 12, 13, 15-17]. In tis paper, we consider te matematical model of Rosenau type non-linear iger order evolution equation of te form u t + u xxxxt = ϕ(u, u x, u xx ), (x, t) Ω (0, T ] (1.1) were ϕ is some non-linear expression in terms of u, u x, u xx wit boundary conditions and initial coundition u(x, t) = g 0 (t), u x (x, t) = g 1 (t), (x, t) Ω (0, T ] (1.2) u(x, 0) = u 0 (x), x Ω (1.3) were Ω = (a, b) and µ > 0. In literature, we ave found different Rosenau type of non-linear iger order evolution equations of te form (1.1), some of tem are as follows: 1. Rosenau equation u t + u xxxxt + u x + uu x = 0 (1.4) Corresponding autor. address: rcmmmfma@iitr.ernet.in (R.C. Mittal ), rkjain 2000@rediffmail.com (R.K.Jain ) Copyrigt c World Academic Press, World Academic Union IJNS /586

2 R.C. Mittal, R.K. Jain: Application of Quintic B-splines Collocation Metod on Some Rosenau Type Nonlinear equation of type (1.4) is considered in [9, 15] and type (1.5) in [16]. u t + µu xxxxt = f(u) x (1.5) 2. Generalized Rosenau equation u t + u xxxxt + u x + (u p ) x = 0 (1.6) equation of type (1.6) is considered in [7] and type (1.7) in [4, 5]. 3. Rosenau-Burger equation equation of type (1.8) is considered in [1, 2, 11, 13, 17]. 4. Generalized Rosenau-Burgers equation equation of type (1.9) is considered in [8, 10, 12]. u t + [a(x, t)u xxt ] xx = [ψ(u)] x (1.7) u t + u xxxxt u xx + u x + uu x = 0 (1.8) u t + u xxxxt αu xx + βu x + ( up+1 p + 1 ) x = 0 (1.9) Many algoritms ave been developed and simulations performed for (1.1) equation, for example, example finite difference sceme [9, 15] for equation (1.4), discontinuous Galerkin metod [16] for equation (1.5), energy conservative finite difference scemes [7] for equation (1.6), finite element Galerkin approximation [4, 5] for (1.7), CrankNicolson finite difference sceme [1], finite difference sceme [2], Fourier transforms metod wit energy estimates [11] and tree-level difference sceme [17] for equation (1.8), Crank-Nicolson difference sceme [10] and Average implicit linear difference sceme [8] for (1.9). Solution of equation (1.1) is not available analytically in general. So, one as to obtain its numerical solutions to develop an understanding of te non-linear penomena. Tere is te different type of equations, wic are found in literature of te form (1.1), namely (1.4)-(1.9), in wic eac equation represents several different pysical penomena. In tis paper, we design a collocation metod based on a quintic B-splines basis functions for solving some Rosenau type non-linear iger order evolution equations of te form (1.1) wit boundary conditions (1.2). We know tat B-splines ave some special features, wic are useful in numerical work. One feature is tat te continuity conditions are inerent, oter special features of B-splines is tat tey ave small local support, i.e. eac B-spline function is only non-zero over a few mes subintervals, so tat te resulting matrix for te discretization equation is tigtly banded. Due to teir smootness and capability to andle local penomena, B-splines offer distinct advantages. In combination wit collocation, tis significantly simplifies te solution procedure of differential equations. Tere is a great reduction of te numerical effort, because tere is no need to calculate te integrals (like in variational metods) in order to form te final set of algebraic equations, wic substitutes te given set of non-linear differential equations. Unlike some previous tecniques using various transformations to reduce te equation into more simple equation, te current metod does not require extra effort to deal wit te non-linear terms. Terefore, te equations are solved easily and elegantly using te present metod. Tis metod as also additional advantages over some rival tecniques, suc as ease in use and computational cost effectiveness in finding solutions of te given non-linear evolution equations. In te present metod, te combination of te quintic B- spline collocation metod in space wit te low-storage tird-order total variation diminising SSP-RK3 sceme in time provides an efficient explicit solution wit ig accuracy and minimal computational efforts for te problems represented by (1.1)-(1.3). Tis paper is organized as follows. In section 2, description of te quintic, B-splines collocation metod is explained. In section 3, procedure for implementation of present metod for equation (1.1) is described. In section 4, procedure to obtain an initial vector wic is required to start our metod is explained. We present four numerical examples to establis te adaptability of te proposed metod computationally in section 5. Conclusion is given in section 6 tat briefly summarizes te numerical outcomes. IJNS omepage: ttp://

3 144 International Journal of Nonlinear Science, Vol.13(2012), No.2, pp Description of metod In quintic B-splines collocation metod te approximate solution can be written as a linear combination of basis functions wic constitute a basis for te approximation space under consideration. We consider a mes a = x 0 < x 1,...,x N 1 < x N = b as a uniform partition of te solution domain a x b by te knots x j wit = x j x j 1, j = 1,..., N. Our numerical treatment for solving equation (1.1) using te collocation metod wit quintic B-spline is to find an approximate solution U N (x, t) to te exact solution u(x, t) in te form: U N (x, t) = c j (t)b j (x) (2.1) were c j (t) are time dependent quantities to be determined from te boundary conditions and collocation from te differential equation. Te quintic B-spline B j (x) at te knots is given by [14]. (x x j 3 ) 5, x [x j 3, x j 2 ) (x x j 3 ) 5 6(x x j 2 ) 5, x [x j 2, x j 1 ) B j (x) = 1 (x x j 3 ) 5 6(x x j 2 ) (x x j 1 ) 5, x [x j 1, x j ) 5 (x j+3 x) 5 6(x j+2 x) (x j+1 x) 5, x [x j, x j+1 ) (x j+3 x) 5 6(x j+2 x) 5, x [x j+1, x j+2 ) (x j+3 x) 5, x [x j+2, x j+3 ) 0, oterwise were B 2, B 1, B 0, B 1,..., B N 1, B N, B N+1, B forms a basis over te region of solution domain axb. Eac quintic B-spline cover six elements so tat eac element is covered by six quintic B-splines. Te values of B j (x) and its derivative may be tabulated as in Table TABLE- 2.1: Coefficient of quintic B-splines and derivatives at nodes x j x x j 3 x j 2 x j 1 x j x j+1 x j+2 x j+3 B j (x) B j (x) 0 5 B j (x) B j (x) B iv j (x) Using approximate function (2.1) and quintic B-spline functions (2.2), te approximate values of U N (x, t) and its four derivatives at te knots are determined in terms of te time parameters c j as follows: U j = c j c j c j + 26c j+1 + c j+2 U j = 5(c j c j+1 10c j 1 c j 2 ) 2 U j = 20(c j 2 + 2c j 1 6c j + 2c j+1 + c j+2 ) 3 U j = 60(c j+2 2c j+1 + 2c j 1 c j 2 ) 4 U iv j = 120(c j 2 4c j 1 + 6c j 4c j+1 + c j+2 ) (2.2) 3 Implementation of metod Our numerical treatment for solving equation (1.1) using te collocation metod wit quintic B-splines is to find an approximate solution U N (x, t) to te exact solution u(x, t) is given in (2.1), were c j (t) are times dependent quantities to be determined from te boundary conditions and collocation from te differential equation. From equation (1.1), we ave IJNS for contribution: editor@nonlinearscience.org.uk

4 R.C. Mittal, R.K. Jain: Application of Quintic B-splines Collocation Metod on Some Rosenau Type Nonlinear Now, using (2.1) in (3.1), we get c j B j (x) + µ c j B iv u t + µu xxxxt = ϕ(u, u x, u xx ), (3.1) j (x) = ϕ({ c j B j (x)}, { c j B j(x)}, { c j B j (x)}) (3.2) Using approximates function (2.1) and quintic B-splines functions (2.2), te approximate values of Ut N derivatives at te knots/nodes are determined in terms of te time parameters c j as follows: (x) and its four (U j ) t = ċ j ċ j ċ j + 26ċ j+1 + ċ j+2 (U j) t = 5(ċ j ċ j+1 10ċ j 1 ċ j 2 ) 2 (U j ) t = 20(ċ j 2 + 2ċ j 1 6ċ j + 2ċ j+1 + ċ j+2 ) 3 (U j ) t = 60(ċ j+2 2ċ j+1 + 2ċ j 1 ċ j 2 ) 4 (U iv j ) t = 120(ċ j 2 4ċ j 1 + 6ċ j 4ċ j+1 + ċ j+2 ) (3.3) Using (2.1),(2.2) and (3.3) in (3.2) we get a system of ordinary differential equations as follows: (ċ j ċ j ċ j + 26ċ j+1 + ċ j+2 ) + µ (ċ j 2 4ċ j 1 + 6ċ j 4ċ j+1 + ċ j+2 ) = ϕ(c j c j c j + 26c j+1 + c j+2, ( 5 )(c j c j+1 10c j 1 c j 2 ), (c j 2 4c j 1 + 6c j 4c j+1 + c j+2 )), 0 j N For simplicity, taking omogeneous boundary conditions in (1.2) and using (2.2) and (3.3) in it, we get ċ 2 = 41.25ċ ċ ċ 2, ċ 1 = 4.125ċ ċ ċ 2, ċ N+1 = 4.125ċ N 2.25ċ N ċ N 2, ċ = 41.25ċ N ċ N ċ N 2, c 2 = 41.25c c c 2, c 1 = 4.125c c c 2, c N+1 = 4.125c N 2.25c N c N 2, c = 41.25c N c N c N 2. (3.4) (3.5) We eliminate ċ 2, ċ 1, ċ N+1, ċ, c 2, c 1, c N+1 and c in (3.4) by using (3.5) ten te system of first order differential equations can be written in te compact form as were a 0 b 0 c 0 d 1 a 1 b 1 x A = x y z y x x y z y x x d N 1 a N 1 b N 1 e N d N a N AĊ = φ (3.6), Ċ = ċ 0 ċ 1 ċ 2 ċ N 2 ċ N 1 ċ N x = 1 + µ 120, y = 26 µ480, z = 66 + µ ,, φ = φ 0 φ 1 φ 2 φ N 2 φ N 1 φ N IJNS omepage: ttp://

5 146 International Journal of Nonlinear Science, Vol.13(2012), No.2, pp a 0 = 41.25x 4.125y + z, b 0 = 32.5x 1.125y, c 0 = 3.25x 0.125y, d 1 = 4.125x + y, a 1 = 2.25x + z, b 1 = 0.125x + y, d N 1 = 4.125x + y, a N 1 = 2.25x + z, b N 1 = 0.125x + y, a N = 41.25x 4.125y + z, d N = 32.5x 1.125y, e N = 3.25x 0.125y, φ 0 = φ(( 20 2 )(27c c 1 + 3c 2 )), forj = 0 φ 1 = φ((21.875c c c 2 + c 3 ), ( 5 )( 5.875c c c 2 + c 3 ), ( 20 2 )( 2.125c c c 2 + c 3 )), forj = 1 φ j = φ((c j c j c j + 26c j+1 + c j+2 ), ( 5 )(c j c j+1 10c j 1 c j 2 ), ( 20 2 )(c j 2 + 2c j 1 6c j + 2c j+1 + c j+2 )), forj = 2toN 2 φ N 1 = φ((21.875c N c N c N 2 + c N 3 ), ( 5 )( 5.875c N c N c N 2 + c N 3 ), ( 20 2 )( 2.125c N 8.25c N c N 2 + c N 3 )), forj = N 1 φ N = φ(( 20 2 )(27c N + 30c N 1 + 3c N 2 )), forj = 0 Here A is (N +1) (N +1) penta-diagonal matrix, Ċ and φ are (N +1) order vectors, wic depend on te boundary conditions. Now, we solve te first order ordinary differential equations system (3.6) by using SSP-RK3 sceme [3]. For computing te RHS of equation (3.6) we need an initial vector C 0 wic can be obtained to follow te procedure of section-4. 4 Te initial vector C 0 Te initial vector C 0 can be obtained from te initial condition and boundary values of te derivatives of te initial condition as te following expressions: U x (x j, 0) = U xx (x j, 0), forj = 0, 1 U(x j, 0) = u 0 (x j ), forj = 2, N 2 U x (x j, 0) = U xx (x j, 0), forj = N 1, N Tis yields a (N + 1) (N + 1) system of equations, of te form AC 0 = b (4.1) IJNS for contribution: editor@nonlinearscience.org.uk

6 R.C. Mittal, R.K. Jain: Application of Quintic B-splines Collocation Metod on Some Rosenau Type Nonlinear were A =, C 0 = c 0 0 c 0 1 c 0 2 c 0 N 2 c 0 N 1 c 0 N u 0 (x 0 ) u 0 (x 1 ) u 0 (x 2 ), b = u 0 (x N 2 ) u 0 (x N 1 ) u 0 (x N ) Te solution of (4.1) can be found by Tomas algoritm. 5 Numerical experiments and discussion In order to sow te utility and adaptability of te metod, it is tested on te following four test problems. To gain insigt into te performance of te present metod, te numerical approximation errors L and L 2 are obtained by following formulae: L = max u exact j N L 2 = ( 0 u exact j Te order of convergence for te numerical metod as been computed by te formula ] Example 5.1: orderof convergence = log [ u u i Lj u u i+1 Lj In equation (1.9), we take α = 1, β = 1 and p = 1, ten it becomes [17] U num j (5.1) U num j 2 ) (5.2) [ ], i = 1, 2andj = 2, (5.3) log i i+1 wit te boundary conditions u t + u xxxxt u xx + u x + uu x = 0, x [0, 1], t [0, T ] (5.1.1) and intial condition u(0, t) = u(1, t) = 0andu x (0, t) = u x (1, t) = 0, t [0, T ] (5.1.2) u(x, 0) = x 4 (1 x) 4, x [0, 1]. (5.1.3) Since te exact solution of (5.1.1) (5.1.3) is not known, we consider te solution Uref N wic is computed by te present metod on fine mes = 1 and k = 0.05 as a reference solution at T = 1. Error estimates and order of convergence for different step size at T = 1.0 are computed using (5.1) (5.3) and reported in Table Te grap depicted in Figure-1, sows te same caracteristics as sown by [17]. TABLE-5.1.1: Error Estimates and order of convergence at T = E E E E E E E E IJNS omepage: ttp://

7 148 International Journal of Nonlinear Science, Vol.13(2012), No.2, pp Figure 1: Approximate solution at T = 1.0. Figure 2: Approximate and exact solution at T = 1.0( = 0.25, k = 0.1). Example 5.2: We consider te following generalized Rosenau equation given in [16] 2u t + u xxxxt + 3u x 60u 2 u x + 120u 4 u x = 0, x [ 10, 10], t [0, T ] (5.2.1) wit te boundary conditions u( 10, t) = sec( 10 t), u(10, t) = sec(10 t), u x ( 10, t) = sec( 10 t)tan( 10 t), u x (10, t) = sec(10 t)tan(10 t), t [0, t], (5.2.2) and intial condition u(x, 0) = sec(x), x [ 10, 10]. (5.2.3) Te exact solution for te Rosenau equation (5.2.1) is known to be u(x, t) = sec(x t). Te equation (5.2.1) can be rewritten as u t + 0.5u xxxxt = f(u) x (5.2.4) were f(u) = 10u 3 12u 5 1.5u. For numerical computation, we take = 1 4, 1 5, 1 6 and 1 7 wit k = Te error estimates and order of convergence are computed using (5.1) (5.3) at T = 0.2 and reported in Table Grap of exact and approximate solution depicted in Fig.- 2 at T = 1.0, wic sows te same caracteristics as sown by S.M. Coo et. al [17]. Example 5.3: TABLE : Error Estimates and order of convergence at T = E E E E E E E E We consider te following generalized Rosenau equation given in [5] u t + {a(x, t)u xxt } xx = {ψ(u)} x (5.3.1) Case 1: We take a(x, t) = 1.0 and ψ(u) = u, wit initial condition u 0 (x) = x 2 (1 x) 2 and boundary conditions (1.2), were Ω = (0, 1). IJNS for contribution: editor@nonlinearscience.org.uk

8 R.C. Mittal, R.K. Jain: Application of Quintic B-splines Collocation Metod on Some Rosenau Type Nonlinear Equation (5.3.1) takes form u t + u xxxxt = u x (5.3.2) Since we dont ave te exact solution to (5.3.2), we consider te solution on mes = 1 as te reference solution and obtain te error estimates by comparing te numerical solutions on mes = 0.1, = 0.05, = and = wit tose on mes = 1 wit t = 0.05 respectively. Te error estimates and order of convergence are computed from (5.1) (5.3) and reported in Table at T = 1.0. Te grap of approximate solution at T = 0.0 and 1.0 is depicted in Fig.- 3. TABLE : Error Estimates and order of convergence at T = E E E E E E E E Case 2: We take a(x, t) = 1.0 and ψ(u) = u 3, wit initial condition u 0 (x) = x 2 (1 x) 2 and boundary conditions (1.2), were Ω = (0, 1). Equation (5.3.1) takes form u t + u xxxxt = 3u 2 u x (5.3.3) Since we dont ave te exact solution to (5.3.3), we consider te solution on mes = 1 as te reference solution. We obtain te error estimates and order of convergence using (5.1) (5.3). We compare te numerical solutions on mes = 0.1, = 0.05, = and = wit tose on mes = 1 wit t = 0.05 respectively. Te error estimates and order of convergence are reported in Table at T = 1.0. TABLE : Error Estimates and order of convergence at T = E E E E E E E E Case 3: We take a(x, t) = (x 2 + 5)e t and ψ(u) = u, wit initial condition u 0 (x) = x 2 (1 x) 2 and boundary conditions (1.2), were Ω = (0, 1). Equation (5.3.1) takes form u t + 2e t u xxt + 4xe t u xxxt + (x 2 + 5)e t u xxxxt = u x (5.3.4) Since we dont ave te exact solution to (5.3.4), we consider te solution on mes = 1 as te reference solution. We obtain te error estimates and order of convergence using (5.1) (5.3). We compare te numerical solutions on mes = 0.1, = 0.05, = and = wit tose on mes = 1 wit t = 0.05 respectively. Te error estimates and order of convergence are reported in Table at T = 1.0. TABLE : Error Estimates and order of convergence at T = E E E E E E E E Case 4: We take a(x, t) = (x 2 + 5)e t and ψ(u) = u 3, wit initial condition u 0 (x) = x 2 (1 x) 2 and boundary conditions (1.2), were Ω = (0, 1). Equation (5.3.1) takes form u t + 2e t u xxt + 4xe t u xxxt + (x 2 + 5)e t u xxxxt = 3u 2 u x (5.3.5) IJNS omepage: ttp://

9 150 International Journal of Nonlinear Science, Vol.13(2012), No.2, pp Since we dont ave te exact solution to (5.3.4), we consider te solution on mes = 1 as te reference solution. We obtain te error estimates and order of convergence using (5.1) (5.3). We compare te numerical solutions on mes = 0.1, = 0.05, = and = wit tose on mes = 1 wit t = 0.05 respectively. Te error estimates and order of convergence are reported in Table at T = 1.0. Te grap of approximate solution at T = 0 and T = 1 are depicted in Fig.- 4. Figure 3: Approximate solution at T = 0.0 and 1.0( = 0.025, k = 0.05). Figure 4: Approximate solution at T = 0.0 and 1.0( = 0.025, k = 0.05). TABLE : Error Estimates and order of convergence at T = E E E E E E E E Example 5.4: (a) In equation (1.9), we take α = 0.1, β = 1 and p = 2, ten it becomes u t + u xxxxt 0.1u xx + u x + u 2 u x = 0, x [0, 1], t [0, T ] (5.4.1) wit te boundary conditions u(0, t) = u(1, t) = 0, u x (0, t) = u x (1, t) = 0, t [0, T ] (5.4.2) and initial condition u(x, 0) = x 4 (1 x) 4, x [0, 1] (5.4.3) Since we dont ave te exact solution to (5.4.1), we consider te solution on mes = 1 as te reference solution. We obtain te error estimates and order of convergence using (5.1) (5.3). We compare te numerical solutions on mes = 0.1, = 0.05, = and = wit tose on mes = 1 wit t = 0.1 respectively. Te error estimates and order of convergence are reported in Table at T = 1.0. IJNS for contribution: editor@nonlinearscience.org.uk

10 R.C. Mittal, R.K. Jain: Application of Quintic B-splines Collocation Metod on Some Rosenau Type Nonlinear TABLE : Error Estimates and order of convergence at T = E E E E E E E E (b) In equation (1.9), we take α = 0.5, β = 1 and p = 5, ten it becomes u t + u xxxxt 0.5u xx + u x + u 5 u x = 0, x [0, 1], t [0, T ] (5.4.4) wit te boundary conditions (5.5.2) and initial condition (5.4.3). Since we dont ave te exact solution to (5.4.4), we consider te solution on mes = 1 as te reference solution. We obtain te error estimates and order of convergence using (5.1) (5.3). We compare te numerical solutions on mes = 0.1, = 0.05, = and = wit tose on mes = 1 wit t = 0.1 respectively. Te error estimates and order of convergence are reported in Table at T = 1.0. TABLE : Error Estimates and order of convergence at T = E E E E E E E E (c) In equation (1.9), we take α = 1, β = 1 and p = 8, ten it becomes u t + u xxxxt u xx + u x + u 8 u x = 0, x [0, 1], t [0, T ] (5.4.5) wit te boundary conditions (5.5.2) and initial condition (5.4.3). Since we dont ave te exact solution to (5.4.4), we consider te solution on mes = 1 as te reference solution. We obtain te error estimates and order of convergence using (5.1) (5.3). We compare te numerical solutions on mes = 0.1, = 0.05, = and = wit tose on mes = 1 wit t = 0.1 respectively. Te error estimates and order of convergence are reported in Table at T = Conclusions TABLE : Error Estimates and order of convergence at T = E E E E E E E E In tis work, we ave developed a collocation metod for solving some Roseneu type non-linear iger order evolution equation wit Diriclet s boundary conditions using quintic B-splines basis functions. In te present metod, we apply quintic B-splines for spatial variable and derivatives, wic produce a system of first order ordinary differential equations. Te resulting systems of ordinary differential equations are solved by using SSP-RK3 sceme. Te numerical approximate solutions to Roseneu type non-linear equations ave been computed witout transforming te equation and witout using te linearization. Tis metod is tested on four problems, and te numerical results obtained are quite satisfactory and comparable wit te existing solutions found in literature. Easy and economical implementation is te strengt of tis metod. Te metod is capable of solving problems wit variable coefficients. Te computed results justify te advantage of tis metod. Acknowledgments One of te autors R.K. Jain tankfully acknowledges te sponsorsip under QIP, provided by Tecnical Education and Training Department, Bopal (M.P.), India. IJNS omepage: ttp://

11 152 International Journal of Nonlinear Science, Vol.13(2012), No.2, pp References [1] Bing Hu, Youcai Xu and Jinsong Hu. CrankNicolson finite difference sceme for te RosenauBurgers equation. Applied Matematics and Computation, 204 (2008): [2] Zuangzuang Wang, Yongbing Wu and Yao Lu. A finite difference simulation for Rosenau-Burgers Equation /09/ IEEE. [3] Graslan G. and Sari M. Numerical solutions of linear and nonlinear diffusion equations by a differential quadrature metod (DQM). Int. J. Numer. Met. in Biomed. Engng, 27 (2011): [4] H. Y. Lee, M. R. Om and J. Y. Sin. Finite Element Galerkin Approximations of te Rosenau Equation. Proceedings of Nonlinear Functional Analysis and Applications, 2 (1997): [5] H. Y. Lee, M. R. Om and J. Y. Sin. Convergence of Fully Discrete Galerkin Approximations of te Rosenau Equation. Korean J. Comput. and Appl. Mat.,6(1)(1999): [6] Huabing Jia and Wei Xu. Solitons solutions for some nonlinear evolution equations. Applied Matematics and Computation, 217 (2010): [7] Jinsong Hu and Kelong Zeng. Two Conservative Difference Scemes for te Generalized Rosenau E- quation. Hindawi Publising Corporation, Boundary Value Problems Volume 2010, Article ID , doi: /2010/ [8] Jinsong Hu, Bing Hu and Youcai Xu. Average implicit linear difference sceme for generalized RosenauBurgers equation. Applied Matematics and Computation, 217 (2011): [9] Kaled Omrani, Faycal Abidi, Tala Acouri and Noomen Kiari: A new conservative finite difference sceme for te Rosenau equation, Applied Matematics and Computation 201 (2008): [10] Kelong Zeng and Jinsong Hu. Crank-Nicolson difference sceme for te generalized Rosenau-Burgers equation. International Journal of Matematical and Computer Sciences, 5:4 (2009). [11] Liping Liu and Ming Mei. A better asymptotic profile of RosenauBurgers equation. Applied Matematics and Computation 131 (2002): [12] MI AI PARK. Point wise Decay Estimates of Solutions of te Generalized Rosenau Equation. J. Korean Mat. Soc., 29 (1992)(2): [13] Ming Mei. Long-Time Beavior of Solution for Rosenau-Burgers Equation (I). Applicable Analysis, 63(3) 1996: [14] R.C. Mittal and Geeta Arora. Quintic B-Spline Collocation Metod for Numerical Solution of te Extended Fiser- Kolmogrov Equation. Int. J. of Appl. Mat and Mec., (1): [15] S. K. Cung. Finite Difference Approximate Solutions for te Rosenau Equation. Applicable Analysis, 69(1-2): [16] S.M. Coo, S.K. Cung, and K.I. Kim. A discontinuous Galerkin metod for te Rosenau equation. Applied Numerical Matematics, 58 (2008): [17] Weiyuan Ma, Aili Yang and Yang Wang. A Second-Order Accurate Linearized Difference Sceme for te Rosenau- Burgers Equation. Journal of Information and Computational Science, 7(8) (2010): IJNS for contribution: editor@nonlinearscience.org.uk