Numerical Solution of the MRLW Equation Using Finite Difference Method. 1 Introduction

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1 ISSN print, online International Journal of Nonlinear Science Vol.1401 No.3,pp Nuerical Solution of the MRLW Equation Using Finite Difference Method Pınar Keskin, Dursun Irk Eskişehir Osangazi University, Departent of Matheatics and Coputer Science, 6480, Eskişehir, Türkiye Received 19 January 011, accepted 15 May 01 Abstract:Two finite difference approxiations for the space discretization and a ulti-tie step ethod for the tie discretization are proposed for the odified regularized long wave equations. The efficiency of the proposed two algoriths is exained by evaluating the L nor and the three invariants of the otion. The results arising fro the experients are copared with the theoretical solution. Keywords: Finite difference ethod; Soliton wave; Modified regularized long wave equation. 1 Introduction The odified regularized long-wave MRLW equation has the for [1] u t + u x + εu u x µu xxt = 0, 1 where u = ux, t is an appropriately differentiable function of the space and the tie variables, respectively. This equation has a ajor role in the propagation of nonlinear dispersive waves. The boundary conditions on the region a x b are ua, t = ub, t = 0, u x a, t = u x b, t = 0, u xx a, t = u xx b, t = 0, t 0, T ] and the initial condition is ux, 0 = fx. 3 The MRLW equation 1 was considered by Gardner et al. [1] using collocation ethod with quintic B-splines finite eleent. The equation was solved nuerically using the finite difference ethod and interaction of solitary waves and other properties of the MRLW equation were also studied in []. An algorith for the nuerical solution of the MRLW equation was proposed using quadratic B-spline finite eleent within collocation ethod and four test probles were worked out to exaine the perforance of the algorith in the paper [3]. The collocation ethod using cubic B-splines was applied to study the solitary waves of the MRLW equation [4]. Hassan and Alaery [5] proposed the collocation ethods using sextic B-splines for solving nuerically the MRLW equation. Raslan and Hassan [6] presented a coputational coparison study of quadratic, cubic, quartic and quintic splines for solving the MRLW equation. A nuerical ethod based on collocation ethod using quintic B-spline finite eleents within the collocation ethod leads to a syste of first order differential equations was solved by fourth order Runge Kutta ethod [7]. Collocation ethod based on quartic B-splines was applied to investigate propagation of nonlinear dispersive solitary waves of the MRLW equation [8]. A new 10-point ultisyplectic schee was proposed for the nuerical solution of the MRLW equation [9]. In this study, MRLW equation is solved nuerically using the finite difference ethod. The ai of the paper is to investigate the accuracy of the three steps ethod when it is used for the tie discretization of the equation. After the new tie discretization of the Eq. 1 is perfored, three and five-point stencils approxiating of the first and second derivatives for the space discretization are used to obtain a syste of algebraic equation. The nonlinear part of the resulting syste of the ethod is handled by using an inner iteration. In the nuerical experients section, the propagation of a Corresponding author. E-ail address: dirk@ogu.edu.tr Copyright c World Acadeic Press, World Acadeic Union IJNS /676

2 356 International Journal of Nonlinear Science, Vol.1401, No.3, pp solitary wave and the interaction of two solitary waves are investigated. The nuerical results obtained in this study deonstrate that the proposed two algoriths are a rearkably successful nuerical technique for solving the M RLW equation and also can be efficiently applied to the siilar physically iportant nonlinear evolution equations. Tie discretization For coputational work, the space-tie plane is discretized by grid with space step length h and the tie step t. The exact solution of unknown function at the grid point is denoted by ux, t n = u n, = 0, 1,..., N; n = 0, 1,,... and the notation U n is used to represent the nuerical value of u n. If we take v = u µu xx, the Eq. 1 can be rewritten as Consider the tie discretization of the for v t = u x + εu u x. 4 v n+1 + θ 1 v n + θ v n 1 = θ 3 v n+1 t + θ 4 v n t + θ 5 v n 1 t 5 where v n+i = vx, t n + i t, i = 1, 0, 1 and θ 1,..., θ 5 are real constants deterined in later to have higher accuracy of the proposed ethod with respect to tie discretization. Eq. 4 subject to Eq. 5 is written as U n+1 + θ ε U n+1 U n+1 x µu n+1 xx = θ 1 U n θ ε U n U n x + µθ 1 U n xx θ U n 1 θ ε U n 1 U n 1 x + µθ U n 1 xx It can be easily verified that the local truncation error error arising fro Eq. 6 is O t 3 when θ 1 = 1, θ = θ 5 = 0, θ 3 = θ 4 = t and it is O t 5 when θ 1 = 0, θ = 1, θ 4 = 4 t 3, θ 3 = θ 5 = t 3. 3 Finite difference ethod I FDM1 For the space discretization, we take three-point stencils approxiating first and second derivatives U x = U +1 U 1 h 6 + Oh 7 U xx = U 1 U + U +1 h + Oh 8 Then, substituting 7-8 into Eq. 6, the resulting algebraic syste of equations takes the for U 1 n+1 α 1 h µ h + U n µ h + U+1 n+1 α1 h µ h = where U n 1 α h + µ h θ 1 + U n θ 1 µ h θ 1 + U n +1 α h + µ h θ 1 + U 1 n 1 α3 h + µ h θ + U n 1 θ µ h θ + U+1 n 1 α 3 h + µ h θ α 1 = θ ε U n+1, α = θ ε U n, α 3 = θ ε U n 1. 9 IJNS eail for contribution: editor@nonlinearscience.org.uk

3 P. Keskin, D. Irk : Nuerical Solution of the MRLW Equation Using Finite Difference Method 357 Whereas the finite difference schee 9 with θ 1 = 1, θ = θ 5 = 0, θ 3 = θ 4 = t has a truncation error of O t 3 + th, the schee with θ 1 = 0, θ = 1, θ 4 = 4 t 3, θ 3 = θ 5 = t has a truncation error of 3 O t 3 + th. This set of equations is recurrence relationship for unknown paraeters vector d n+1 = U0 n+1,..., U n+1 N. This tridiagonal atrix syste is ade up N 1 equations including N + 1 unknown for = 1,..., N 1. Application of the boundary conditions ua, t = ub, t = 0 enables the eliination of the variables U0 n+1 and U n+1 N fro the syste 9. Then, the syste is reduced to N 1 N 1 atrix syste, which can be solved by using the Thoas algorith. After d 0 = U0 0,..., UN 0 unknown vector are found fro the initial condition, taking θ 1 = 1, θ = θ 5 = 0, θ 3 = θ 4 = t in the syste 9, we can find d 1 = U0 1,..., UN 1. Once the initial vectors d 0 and d 1 are coputed, d n+1, n = 1,, 3,... unknown vectors are found repeatedly by solving the recurrence relation 9 with θ 1 = 0, θ = 1, θ 4 = 4 t. Note that since the syste 9 is an iplicit syste, we have taken U n+1 5 ties to increase the accuracy of the syste. 4 Finite difference ethod II FDM 3, θ 3 = θ 5 = t 3 U n in the coefficient α 1 and done an inner iteration for For the space discretization, we take five-point stencils approxiating first and second derivatives U x = U 8U 1 + 8U +1 U + 1h + Oh 4, 10 U xx = U + 16U 1 30U + 16U +1 U + 1h + Oh Then, substituting into Eq. 6, we have U n+1 α1 1h + µ 1h + U 1 n+1 U+1 n+1 α1 where U n U n +1 U n 1 U n 1 +1 α 1 3h 4µ 3h 3h 4µ 3h + U+ n+1 α 1 1h + µ 1h = α 1h µ 1h θ 1 + U 1 n α 3h + 4µ 3h θ 1 α 3h + 4µ 3h θ 1 + U+ n α 1h µ 1h θ 1 α 3 1h µ 1h θ + U 1 n 1 α3 3h + 4µ 3h θ α 3 3h + 4µ 3h θ + U+ n 1 α3 1h µ 1h θ + U n µ h + + U n θ 1 5µ + α 1 = θ ε U n+1, α = θ ε U n, α 3 = θ ε U n 1. h θ U n 1 θ 5µ h θ + Whereas the finite difference schee 1 with θ 1 = 1, θ = θ 5 = 0, θ 3 = θ 4 = t has a truncation error of O t 3 + th 4, the schee with θ 1 = 0, θ = 1, θ 4 = 4 t 3, θ 3 = θ 5 = t has a truncation error of 3 O t 3 + th 4. 1 IJNS hoepage:

4 358 International Journal of Nonlinear Science, Vol.1401, No.3, pp This set of equations is recurrence relationship for unknown paraeters vector d n+1 = U n+1, U 1 n+1,..., U n+1 N+1, U n+1 This pentadiagonal atrix syste is ade up N + 1 equations including N + 5 unknown for = 0, 1,..., N 1, N. Application of the boundary conditions u x a, t = u x b, t = 0, u xx a, t = u xx b, t = 0, enables the eliination of the variables U n+1, U 1 n+1, U n+1 n+1 N+1 and UN+ fro the syste 1. Then, the syste is reduced to N + 1 N + 1 atrix syste, which can be solved by using the Thoas algorith. After d 0 and d 1 unknown vectors are coputed, d n+1 unknown vector can be found by solving the syste 1 with θ 1 = 0, θ = 1, θ 4 = 4 t 3, θ 3 = θ 5 = t as the previous section. 3 5 Nuerical experients In this section, to illustrate the effectiveness of the presented nuerical schee, two test probles are studied for the MRLW equation. Since the analytical solution of the MRLW equation is known for the first test proble, the L error nor is used to easure the error between the analytical and nuerical results at the node points: L = ax u U. Since an accurate nuerical schee ust keep the conservation properties of evolution equations, we will onitor the three invariants of nuerical solution for the MRLW equation corresponding to conservation of ass, oentu and energy given by the following integrals [1]: I 1 = udx, I = u + µu x dx, I 3 = 5.1 Motion of single solitary wave for MRLW equation N+. 6 u 4 µ u x dx 13 ε It is known [1] that Eq. 1 has a solitary wave solution of the for 6c ux, t = ε sechk[x x 0 c + 1t] 14 c 6c where k =. This solution corresponds to a solitary wave of aplitude A =, initially centered on the µc + 1 ε peak position x 0 propagating towards the right without change of shape at a steady velocity c + 1. For this proble the analytical values of the invariants by using the solitary wave solution 14 when t = 0 can be found as I 1 = πa k, I = A k + µka, I 3 = 4A A ε 3µk kε Nuerical experients of this otion are perfored for the aplitudes of A = 0.3, 1 and paraeters ε = 6, µ = 1. In the first case for the first test proble, we have used various space and tie steps, peak position x 0 = 40 and aplitude 1 over the region 0 x 100. After the coputation is done until tie t = 10 to find error nor L and nuerical invariants I 1, I, I 3, results of the proposed two algoriths together with the analytical values of the invariants are docuented in Table 1. It is clearly seen that the result obtained by the FDM are ore accurate then obtained by the FDM1. We can also say that when we use saller tie and space steps, nuerical invariants for the FDM are alost the sae as the their exact values. In the second case after the progra is rerun up to tie t = 10 with various space and tie steps and ε = 6, µ = 1, c = 0.3, x 0 = 40, the coputed invariants and error nor for the two proposed algoriths are given in Table. According to the Table, FDM gives ore accurate results then FDM1. It can also be seen fro the table that the coputed values of invariants for the FDM are in good agreeent with the analytical values of the invariants when we use sall tie and tie steps. Absolute error analytical-nuerical distributions for the two algoriths with aplitude 1 and x 0 = 40, h = t = 0.1 is drawn at tie t = 10 in Fig1 Fig.It can be easily observed that axiu error is taken place around the iddle of the space interval. IJNS eail for contribution: editor@nonlinearscience.org.uk

5 P. Keskin, D. Irk : Nuerical Solution of the MRLW Equation Using Finite Difference Method 359 Table 1: Invariants and error nors for a single solitary wave at tie t = 10,ε = 6, µ = 1, c = 1, x 0 = 40, [a, b] = [0, 100]. FDM1 FDM h = t L I 1 I I 3 L I 1 I I Analytical values of invariants Table : Invariants and error nors for a single solitary wave at tie t = 10, ε = 6, µ = 1, c = 0.3, x 0 = 40, [a, b] = [0, 100]. FDM1 FDM h = t L I 1 I I 3 L I 1 I I Analytical values of invariants Figure 1: Absolute error distribution at t = 10 for FDM1 Figure : Absolute error distribution at t = 10 for FDM IJNS hoepage:

6 360 International Journal of Nonlinear Science, Vol.1401, No.3, pp Figure 3: Interaction of two solitary waves 5. Interaction of two solitary waves for MRLW equation In the second test proble, interaction of two positive solitary waves for MRLW equation is studied by using the initial condition given by the linear su of two separate solitary waves of various aplitudes ux, 0 = A 1 sechk 1 [x x 1 ] + A sechk [x x ] 16 6ci where A i = ε, k ci i = µc i + 1, i = 1,, and x i, c i are arbitrary constants. Calculations are carried about with ε = 6, µ = 1, c 1 = 1, c = 1/4, x 1 = 30, x = 60, h = 0.1, t = 0.01 over the space interval [0, 00] fro t = 0 to t = 60. These paraeters provide two solitary waves of aplitude 1 and 0.5 and peak positions of the are located at x = 30 and 60. The analytical invariants can be found as I 1 = π k A 1 + k 1 A , k 1 k I = k A 1 + k 1 A µ + k k 1 k 3k 1 k 1 k A 1 + k 1 ka , I 3 = 4 3k 1 k ε εk1 A 4 3µk 1 ka + εk A 4 1 3µk1k A Siulations of interaction of two solitary waves at ties t = 0, t = 35 and t = 60 are plotted in Figure for the FDM. Since solitary waves don t obey superposition, when a taller wave overtakes a shorter wave, they don t cobine and add together. As seen in Fig3, two solitary waves at the tie t = 0 are propagated to the right with velocities dependent upon their agnitudes and reached a stage that the larger solitary wave has passed through the saller solitary wave and eerged in their original positions. The absolute value of the difference between exact and nuerical values of the invariants for the both algoriths are plotted in Fig4 and Fig5. We can easily seen that the first invariant reain alost constant while the second and third invariants are affected ore fro the interaction of the solitary waves during run of the two algoriths. We can also say that FDM gives ore accurate results than FDM1 too. 6 Conclusion The MRLW equation is solved nuerically using finite difference ethod with ulti-tie step discretization. The perforance of the algoriths has been exained by studying the propagation of the single solitary wave and interaction of two solitary waves. Nuerical results show that since the proposed ethod is an accurate and efficient nuerical technique and also application of the ethod is easier than any other nuerical techniques, proposed schee especially FDM can be efficiently applied to the MRLW and this type of nonlinear probles. IJNS eail for contribution: editor@nonlinearscience.org.uk

7 P. Keskin, D. Irk : Nuerical Solution of the MRLW Equation Using Finite Difference Method 361 Figure 4: Absolute errors for the invariants FDM1 Figure 5: Absolute errors for the invariants FDM Acknowledgeents Dursun Irk gratefully acknowledges the financial support given by the Higher Educational Council of Turkey during his stay at the Departent of Matheatics at Sion Fraser University, Burnaby. References [1] L.R.T. Gardner, G.A. Gardner, F.A. Ayoub and N.K. Aein. Approxiations of solitary waves of the MRLW equation by B-spline finite eleent, Arabian Journal for Science and Engineering, 1997: [] A.K. Khalifa, K.R. Raslan and H.M. Alzubaidi. A finite difference schee for the MRLW and solitary wave interactions, Applied Matheatics and Coputation,189007: [3] K.R. Raslan. Nuerical study of the Modified Regularized Long Wave MRLW equation, Chaos, Solitons and Fractals,4009: [4] A.K. Khalifa, K.R. Raslan and H.M. Alzubaidi. A collocation ethod with cubic B-splines for solving the MRLW equation, Journal of Coputational and Applied Matheatics,1008: [5] Saleh M. Hassan and D.G. Alaery. B-splines collocation algoriths for solving nuerically the MRLW equation, International Journal of Nonlinear Science,8009: [6] K.R. Raslan and Saleh M. Hassan. Solitary waves for the MRLW equation, Applied Matheatics Letters,009: [7] K.R. Raslan and Talaat S. EL-Danaf. Solitary waves solutions of the MRLW equation using quintic B-splines, Journal of King Saud University Science,010: [8] Fazal-i-Haq, Siraj-ul-Isla and Ikra A. Tirizi. A nuerical technique for solution of the MRLW equation using quartic B-splines, Applied Matheatical Modelling,34010: [9] J. Cai. A ultisyplectic explicit schee for the odified regularized long-wave equation, Journal of Coputational and Applied Matheatics,34010: IJNS hoepage: