Numerical solution of General Rosenau-RLW Equation using Quintic B-splines Collocation Method

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1 Available online at Volume 2012, Year 2012 Article ID cna-00129, 16 pages doi: /2012/cna Research Article Numerical solution of General Rosenau-RLW Equation using Quintic B-splines Collocation Method R.C. Mittal 1, R.K. Jain 2 (1) Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee , Uttarakhand, India (2) Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee , Uttarakhand, India Copyright 2012 c R.C. Mittal and R.K. Jain. 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. Abstract In this paper a numerical method is proposed to approximate the solution of the nonlinear general Rosenau-RLW Equation. The method is based on collocation of quintic B-splines over finite elements so that we have continuity of the dependent variable and its first four derivatives throughout the solution range. We apply quintic B-splines for spatial variable and derivatives which produce a system of first order ordinary differential equations. We solve this system by using SSP-RK54 scheme. This method needs less storage space that causes to less accumulation of numerical errors. The numerical approximate solutions to the nonlinear general Rosenau-RLW Equation have been computed without transforming the equation and without using the linearization. Illustrative example is included for different value of p = 2, 3 and 6, to demonstrate the validity and applicability of the technique. Easy and economical implementation is the strength of this method. Keywords: general Rosenau-RLW Equation; quintic B-splines basis functions; SSP-RK54 scheme; Thomas algorithm. Corresponding author. address: rkjain 2000@rediffmail.com Tel:

2 1 Introduction In this paper, we consider the following initial-boundary value problem of the general Rosenau-RLW equation: u t u xxt + u xxxxt + u x + (u p ) x = 0, a < x < b, 0 < t < T (1.1) with an initial condition and boundary conditions u(x, 0) = u 0 (x) (1.2) u(a, t) = u(b, t) = 0, u xx (a, t) = u xx (b, t) = 0, 0 < t < T. (1.3) where p 2 is an integer and u 0 (x) is a known smooth function. When p = 2, equation (1.1) is called as usual Rosenau-RLW equation. When p = 3, it is called as modified Rosenau-RLW (MRosenau-RLW) equation. The initial boundary value problem (1.1) - (1.3) possesses the following conservative quantities: I 2 = 1 2 I 1 = 1 2 b a b a udx (1.4) (u 2 + u 2 x + u 2 xx)dx (1.5) related to mass and energy. We know that the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation. The quantities I 1 and I 2 are applied to measure the conservation properties of the present method, calculated by trapezoidal rule for Rosenau-RLW equation. Some of the previous works on the GRLW equation are inclusive of an implicit secondorder accurate and stable energy preserving finite difference method based on the use of central difference equations for the time and space derivatives [23], the method of lines based on the discretization of the spatial derivatives by means of Fourier pseudo-spectral approximations [11], the Fourier spectral method for the initial value problem of the GRLW equation [12] and a linearized implicit pseudo-spectral method [4]. For p = 1, this equation reduces to the regularized long wave (RLW) equation as an important equation in physics media describing phenomena with weak nonlinearity and dispersion waves. Various numerical techniques such as the finite element methods based on least square principle [1, 2, 9], finite element methods based on Galerkin and collocation principles [5, 7, 20, 22, 3, 24, 18, 8], Petrov-Galerkin method [6] and Reduced Differential Transform Method [15] have been devised to find numerical solutions of the RLW equation. RBF collocation method has been developed for numerical simulation of GEW equation in [17]. The other special case of the GRLW equation is the modified regularized long wave (MRLW) equation for p = 2. MRLW equation was solved numerically by the collocation method with quintic B-splines [10, 19] and cubic B-splines [11] finite element method. 2 ISPACS GmbH

3 Recently, Zuo et al. [24], developed a numerical scheme for solving (1.1) - (1.3) using conservative finite difference method. Existence of its difference solutions have proved by Brouwer fixed point theorem. They have shown that general Rosenau-RLW Equation possesses conservative quantities. They have also proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable, and second-order convergent. Pan et al. [16] developed a three level finite difference scheme for usual Rosenau-RLW equation. They have discussed the second order convergence of their scheme by discrete energy method. In this paper, we present a new method to the solution of Rosenau-RLW equation. The method is based on quintic B-splines basis functions for solving equation (1.1) - (1.3).We knows that B-splines have some special features which are useful in numerical work. One feature is that the continuity conditions are inherent, other special feature of B-splines is that they have small local support, i.e. each B-spline function is only non-zero over a few mesh subintervals, so that the resulting matrix for the discretization equation is tightly banded. Due to their smoothness and capability to handle local phenomena, B-splines offer distinct advantages. In combination with collocation, this significantly simplifies the solution procedure of differential equations. There is a great reduction of the numerical effort, because there is no need to calculate the integrals (like in variational methods) in order to form the final set of algebraic equations, which substitutes the given set of nonlinear differential equations. Unlike some previous techniques using various transformations to reduce the equation into more simple equation, the current method does not require extra effort to deal with the nonlinear terms. Therefore the equations are solved easily and elegantly using the present method. This method has also additional advantages over some rival techniques, ease in use and computational cost effectiveness in order to find solutions of the given nonlinear evolution equations. In the present method, the combination of the quintic B-spline collocation method in space with the low-storage fourth-order total variation diminishing SSP-RK54 scheme in time provides an efficient explicit solution with high accuracy and minimal computational effort for the problems represented by (1.1) - (1.3). This paper is organized as follows. In Section 2, description of quintic B-splines collocation method is explained. In Section 3, procedure for implementation of present method is described for equation (1.1) - (1.3). In Section 4, procedure to obtain initial vector which is required to start our method is explained. We present numerical example for different values of p to establish the adaptability of the present method computationally in Section 5. Conclusion is given in Section 6 that briefly summarizes the numerical outcomes. 2 Description of Method In quintic B-splines collocation method the approximate solution can be written as a linear combination of basis functions which constitute a basis for the approximation space under consideration. We consider a mesh a = x 0 < x 1,..., x N 1 < x N = b as a uniform partition of the solution domain a x b by the knots x j with h = x j x j 1, j = 1,... N. Our numerical treatment for solving equation (1.1) using the collocation method with quintic B-spline is to find an approximate solution U N (x, t) to the exact solution u(x, t) in the form: 3 ISPACS GmbH

4 U N (x, t) = N+2 j= 2 c j (t)b j (x) (2.6) where c j (t) are time dependent quantities to be determined from the boundary conditions and collocation from the differential equation. The quintic B-spline B j (x) at the knots is given by [13]. (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 ) h 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, otherwise (2.7) Where B 2, B 1, B 0, B 1,...,B N 1, B N, B N+1, B N+2 forms a basis over the region of solution domain a x b. Each quintic B-spline cover six elements so that each element is covered by six quintic B-splines. The values of B j (x) and its derivative may be tabulated as in Table- 1. TABLE-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 h B j (x) h 2 h 2 B j (x) 0 60 h 3 B iv j (x) h h 0 h h 2 h h 3 h h 4 h h h 0 h h 3 0 h 4 Using approximate function (2.6) and quintic B-spline functions (2.7), the approximate values of U N (x, t) and its four derivatives at the knots are determined in terms of the time parameters c j as follows: U j = c j c j c j + 26c j+1 + c j+2 hu j = 5(c j c j+1 10c j 1 c j 2 ) h 2 U j = 20(c j 2 + 2c j 1 6c j + 2c j+1 + c j+2 ) h 3 U j = 60(c j+2 2c j+1 + 2c j 1 c j 2 ) h 4 U iv j = 120(c j 2 4c j 1 + 6c j 4c j+1 + c j+2 ) (2.8) 3 Implementation of Method Our numerical treatment for solving equation (1.1) using the collocation method with quintic B-splines is to find an approximate solution U N (x, t) to the exact solution u(x, t) 4 ISPACS GmbH

5 is given in (2.6), where c j (t) are time dependent quantities to be determined from the boundary conditions and collocation from the differential equation. From equation (1.1), we have Now, using (2.6) in (3.9), we get u t u xxt + u xxxxt = u x pu p 1 u x (3.9) N+2 j= 2 N+2 j= 2 N+2 c j B j (x) j= 2 N+2 c j B j(x) p{ j= 2 c j B j (x) + N+2 j= 2 N+2 c j B j (x)} p 1 j= 2 c j B iv j (x) = c j B j(x)} (3.10) Using approximates function (2.6) and quintic B-splines functions (2.7), the approximate values of Ut N (x) and its four derivatives at the knots/nodes are determined in terms of the time parameters c j as follows: (U j ) t = ċ j ċ j ċ j + 26ċ j+1 + ċ j+2 h(u j) t = 5(ċ j ċ j+1 10ċ j 1 ċ j 2 ) h 2 (U j ) t = 20(ċ j 2 + 2ċ j 1 6ċ j + 2ċ j+1 + ċ j+2 ) h 3 (U j ) t = 60(ċ j+2 2ċ j+1 + 2ċ j 1 ċ j 2 ) h 4 (U iv j ) t = 120(ċ j 2 4ċ j 1 + 6ċ j 4ċ j+1 + ċ j+2 ) (3.11) Using (2.7),(2.8) and (3.11) in (3.10) we get a system of ordinary differential equations as follows: (ċ j ċ j ċ j + 26ċ j+1 + ċ j+2 ) 20 h 2 (ċ j 2 + 2ċ j 1 6ċ j + 2ċ j+1 + ċ j+2 ) h 4 (ċ j 2 4ċ j 1 + 6ċ j 4ċ j+1 + ċ j+2 ) = 5 h (c j c j+1 10c j 1 c j 2 ) p(c j c j c j + 26c j+1 + c j+2 ) p 1 5 h (c j c j+1 10c j 1 c j 2 ) On L.H.S. of (3.12) we substitute x = 1 20 h ; y = 26 h4 h ; z = 66 + h4 h h 4, and R.H.S. = ψ j, then equation (3.12) is written as, 0 j N (3.12) xċ j 2 + yċ j 1 + zċ j + yċ j+1 + xċ j+2 = ψ j (3.13) To eliminate ċ 2, ċ 1, ċ N+1, ċ N+2, c 2, c 1, c N+1 and c N+2, we consider boundary condition for collocation as follows: u(a, t) = 0, u(b, t) = 0, u xx (a, t) = 0, u xx (b, t) = 0 (3.14) 5 ISPACS GmbH

6 From (2.8), (3.11) and (3.14) we get ċ 2 = 12ċ 0 ċ 2, ċ 1 = 3ċ 0 ċ 1, ċ N+1 = 3ċ N ċ N 1, ċ N+2 = 12ċ N ċ N 2, c 2 = 12c 0 c 2, c 1 = 3c 0 c 2, c N+1 = 3c N c N 1, c N+2 = 12c N c N 2. (3.15) We eliminate ċ 2, ċ 1, ċ N+1, ċ N+2, c 2, c 1, c N+1 and c N+2 in (3.13) by using (3.7) then the system of first order differential equation can be written in compact form as where 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 = ψ (3.16), Ċ = a 0 = 12x 3y + z, b 0 = 0, c 0 = 0, ċ 0 ċ 1 ċ 2 ċ N 2 ċ N 1 ċ N, ψ = ψ 0 ψ 1 ψ 2 ψ N 2 ψ N 1 ψ N d 1 = 3x + y, a 1 = x + z, b 1 = y, d N 1 = 3x + y, a N 1 = x + z, b N 1 = y, ψ 0 = 5 h (18c c 1 + 2c 2 ); a N = 12x 3y + z, d N = 0, e N = 0, ψ 1 = 5 h ( 7c 0 +c 1 +10c 2 +c 3 ) p(23c 0 +65c 1 +26c 2 +c 3 ) p 1 ( 5 h )( 7c 0 +c 1 +10c 2 +c 3 ); ψ j = ( 5 h )(c j c j+1 10c j 1 c j 2 ) p(c j c j c j + 26c j+1 + c j+2 ) p 1 ( 5 h )(c j c j+1 10c j 1 c j 2 );for j = 2 to N 2 ψ N 1 = 5 h (7c N c N 1 10c N 2 c N 3 ) p(23c N +65c N 1 +26c N 2 +c N 3 ) p 1 ( 5 h )(7c N c N 1 10c N 2 c N 3 ); 6 ISPACS GmbH

7 ψ N = 5 h ( 18c N 20c N 1 2c N 2 ); Here A is (N + 1) (N + 1) penta-diagonal matrix Ċ and ψ are (N + 1) order vectors which depend on the boundary conditions. Now, we solve the first order ordinary differential equation system (3.8) by using SSP-RK54 scheme [21]. Once the parameter C 0 has been determined at a specified time level, we can compute the solution at the required knots. In (3.8), first we solve this system for vector Ċ by using a variant of Thomas algorithm only once at each time level t > 0 then we get a first order system of ordinary differential equations which can be solved for vector C by using SSP-RK54 scheme and consequently the solution U N (x, t) is completely known. 4 The initial vector C 0 The initial vector C 0 can be obtained from the initial condition and boundary values of the derivatives of the initial condition as the following expressions: U x (a, 0) = 0; U x (b, 0) = 0, U(x j, 0) = u 0 (x j ), forj = 2, N 2 U xx (a, 0) = 0; U xx (b, 0) = 0. This yields a (N + 1) (N + 1) system of equations, of the form where A = AC 0 = b (4.17) The solution of (4.17) can be found by Thomas algorithm., C 0 = 5 Numerical Experiments and Discussion 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 ) In order to show the utility and adaptability of the method, it is tested on the following test problem. In this section the proposed method is apply for different values of p. The accuracy of the scheme is measured by using the following error norms: L 2 = N h( 0 u exact j U num j 2 ) (5.18) 7 ISPACS GmbH

8 L = max u exact j U num j, 1 j N (5.19) Where u and U represent the exact and approximate solutions respectively and h is the minimum distance between any two points of set of points for which the errors are evaluated. Example: We consider Rosenau-RLW equation (1.1) whose exact solution is given as (p+3)(3p+1)(p+1) ln 2(p u(x, t) = exp[ 2 +3)(p 2 +4p+7) (p 1) ]sech 4 where c = p4 +4p 3 +14p 2 +20p+25 p 4 +4p 3 +10p 2 +12p+21. (p 1) p 1 [ (4p 2 +8p+20) (x ct)] Initial condition is extract from exact solution and boundary conditions are taken as u( 30, t) = 0, u(120, t) = 0, u xx ( 30, t) = 0, u xx (120, t) = 0. CASE-1 In first numerical simulation, we take p = 2, h = t = 0.1. We compute L and L 2 errors for different time levels and results are reported in Table- 2. We compare our results with Pan et. al. It is clearly seen that our results are much better than [16]. We also compute CPU time (in seconds) for different time levels. TABLE-2 : The errors of numerical solutions and CPU time for t 10 (p = 2, h = t = 0.1) Present Method Pan et al. [16] t L L 2 CPU time (in seconds) L [16] E E E E E E E E E E E E E E E 04 CASE-2 In second numerical simulation, we compute L and L 2 errors for different values of h with p = 2, t = 0.1 at t = 10 and calculate order of convergence using these error norms. The results are reported in Table- 3. We observe that present method is nearly of second order of convergence with respect to these error norms. We compare L errors with Pan et al. [16] and found that our results are much better. TABLE-3 : The errors of numerical solutions and order of convergence at t = 10 (p = 2, t = 0.1) Present Method Pan et al. [16] h L Order of Conv. L 2 Order of Conv. L [16] E E E E E E E E E E E E E E 06 8 ISPACS GmbH

9 CASE-3 In third numerical simulation, we compute L and L 2 errors for different values of h with p = 4, t = 0.1 at t = 60 and calculate order of convergence using these error norms. The results are reported in Table- 4.It is clearly seen that present method is of second order of convergence with respect to L error norm during simulation. TABLE-4 : The errors of numerical solutions and order of convergence at t = 60 (p = 4, t = 0.1) Present Method h L Order of Conv. L 2 Order of Conv E E E E E E E E E E CASE-4 In fourth numerical simulation, we compute L and L 2 errors and invariants for different time levels with p = 2 and h= t = 0.1. The results are reported in Table- 5. It is clearly seen that the invariants I 1 and I 2 remains constant during simulation. We also depicted numerical approximate and exact solutions at t = 0, 30 and 60 in Figure- 1 and 2 respectively. We also show the CPU time (in seconds) for present method. It is clearly seen that numerical solutions are in good agreement with exact solutions. We compare our results and figures with Zuo et. al [24]. TABLE-5 : The errors of numerical solutions, invariants and CPU time for t 60 (p = 2 and h = t = 0.1) t L L 2 I 1 I 2 CPU time (in seconds) E E E E E E E E E E E E [24] t = CASE-5 In fifth numerical simulation, we compute L and L 2 errors and invariants for different time levels with p = 3 and h= t = 0.1. The results are reported in Table- 6. It is clearly seen that the invariants I 1 and I 2 remains constant during simulation. We also depicted numerical approximate and exact solutions at t = 0, 30 and 60 in Figure- 3 and 4 respectively. We also show the CPU time (in seconds) for present method. It is clearly seen that numerical solutions are in good agreement with exact solutions. We compare our results and figures with Zuo et. al [24]. 9 ISPACS GmbH

10 TABLE-6 : The errors of numerical solutions, invariants and CPU time for t 60 (p = 3 and h = t = 0.1) t L L 2 I 1 I 2 CPU time (in seconds) E E E E E E E E E E E E Zuo et al. [24] t = CASE-6 In sixth numerical simulation, we compute L and L 2 errors and invariants for different time levels with p = 6 and h= t = 0.1. The results are reported in Table- 7. It is clearly seen that the invariants I 1 and I 2 remains constant during simulation. We also depicted numerical approximate and exact solutions at t = 0, 30 and 60 in Figure- 5 and 6 respectively. We also show the CPU time (in seconds) for present method. It is clearly seen that numerical solutions are in good agreement with exact solutions. We compare our results and figures with Zuo et. al [24]. TABLE-7 : The errors of numerical solutions, invariants and CPU time for t 60 (p = 6 and h = t = 0.1) t L L 2 I 1 I 2 CPU time (in seconds) E E E E E E E E E E E E [24] t = E Conclusion In this paper, we develop a collocation method for solving nonlinear general Rosenau-RLW equation with Dirichlet boundary conditions using quintic B-splines basis functions. In the present method we apply quintic B-splines for spatial variable and derivatives which produce a system of first order ordinary differential equations. The resulting systems of ordinary differential equations are solved by using SSP-RK54 scheme. The numerical approximate solutions to nonlinear general Rosenau-RLW equation have been computed without transforming the equation and without using the linearization. This method is tested for different values of p = 2, 3 and 6 and the numerical results obtained are quite satisfactory and comparable with the existing solutions found in literature. Easy and economical implementation is the strength of this method. The computed results justify the advantage of this method. 10 ISPACS GmbH

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14 Acknowledgment One of the authors R.K. Jain thankfully acknowledges the sponsorship under QIP, provided by Technical Education and Training Department, Bhopal (M.P.), India. The authors are very thankful to the reviewers for their valuable suggestions to improve the quality of the paper. References [1] I. Dag, Approximation of the RLW equation by the least square cubic B-spline finite element method, Appl. Math. Model. 25 (2001) [2] I. Dag, A. Dogan and B. Saka, B-spline collocation methods for numerical solutions of the RLW equation, Int. J. Comput. Math. 80 (2003) [3] I. Dag, B. Saka and D. Irk, Galerkin method for the numerical solution of the RLW equation using quintic B-splines, J. Comput. Appl. Math. 190 (2006) [4] K. Djidjeli, W.G. Price, E.H. Twizell and Q. Cao, A linearized implicit pseudospectral method for some model equations: the regularized long wave equations, Comm. Numer. Methods Engrg. 19 (2003) [5] A. Dogan, Numerical solution of RLW equation using linear finite elements within Galerkin method, Appl. Math. Model. 26 (2002) [6] A. Dogan, Numerical solution of regularized long wave equation using Petrov-Galerkin method, Comm. Numer. Methods Engrg. 17 (2001) [7] A. Esen and S. Kutluay, Application of a lumped Galerkin method to the regularized long wave equation, Appl. Math. Comput. 174 (2006) [8] L. R. T. Gardner, G. A. Gardner and I. Dag, A B-spline finite element method for the regularized long wave equation, Comm. Numer. Methods Engrg. 11 (1995) [9] L. R. T. Gardner, G. A. Gardner and A. Dogan, A least-square finite element method for the RLW equation, Comm. Numer. Methods Engrg. 12 (1996) ::AID- CNM CO;2-O [10] L. R. T. Gardner, G. A. Gardner, F. A. Ayoub and N. K. Amein, Approximations of solitary waves of the MRLW equation by B-spline finite element, Arab. J. Sci. Eng. 22 (1997) ISPACS GmbH

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16 [24] J.M. Zuo, Y.M. Zhang, Tian-De Zhang, and Feng Chang, A New Conservative Difference Scheme for the General Rosenau-RLW Equation, Boundary Value Problems, Hindawi Publishing Corporation, Volume 2010, Article ID ISPACS GmbH