Two Numerical Schemes for the Solution of the Generalized Rosenau Equation with the help of Operator Splitting Techniques

Year 2023, Volume: 11 Issue: 1, 14 - 28, 28.03.2023

Melike Karta

Abstract

In the present manuscript, numerical solution of generalized Rosenau equation are applied quintic B-spline collocation and cubic B-spline lumped-Galerkin finite element methods (FEMs) together with both Strang splitting technique and the Ext4 and Ext6 techniques based on Strang splitting and derived from extrapolation. In the first instance, the problem is divided into two sub-equations as linear $U_t=\hat{A}(U)$ and nonlinear $U_t=\hat{B}(U)$ in the time term. Later, these sub-equations is implemented collocation and lumped-Galerkin (FEMs) using quintic and cubic B-spline functions respectively, with Strang ($S\Delta t=\hat{A}-\hat{B}-\hat{A}$), Ext4 and Ext6 splitting techniques. The numerical solutions of the system of ordinary differential equations obtained in this way are solved with help fourth order Runge-Kutta method. The aim of this study is to obtain superior results. For this, a test problem is selected to show the accuracy and efficiency of the method and the error norm results produced by these techniques have been compared among themselves and with the current study in the literature. İt can be clearly stated that it is concluded that the approximate results obtained with the proposed method are better than the study in the literature. So that one can see that the study has achieved its purpose.

Keywords

Generalized Rosenau equation, quintik and cubic B-splines, collocation and Galerkin methods, Splitting techniques

References

• [1] Rosenau, P.: Dynamics of dense discrete systems. Progress of Theoretical Physics. 79 (5), 1028-1042 (1988).
• [2] Chung, S. K.: Finite difference approximate solutions for the Rosenau equation. Applicable Analysis. 69 (1-2), 149-156 (1998).
• [3] Rosenau, P.: A quasi-continuous description of a non -linear transmissionline.Physica Scripta. 34, 827-829 (1986).
• [4] Park, M. A.: On the Rosenau equation. Math. Appl. Comput. 9, 145-152 (1990).
• [5] Chung,S. K., Ha, S. N.: Finite element Galerkin solutions for the Rosenau equation. Applicable Analysis.54, 30-56 (1994).
• [6] Manickam, S. A., Pani, A. K., Chung, S. K.: A second-order splitting combined with orthogonal cubik spline collocation method for the Rosenau equation. Numerical Methods for Partial Differential Equations: An International Journal. 14, 695-716 (1998).
• [7] Lee, H. Y., Ohm, M. R., Shin, J. Y.: The convergence of fully discrete Galerkin approximations of the Rosenau equation. The Korean Journal of Computational & Applied Mathematics. 6 (1), 1-13 (1999).
• [8] Sportisse, B.: An analysis of operator splitting techniques in the Stiff case. Journal of Computational Physics. 161, 140-168 (2000). https://doi:10.1006/jcph.2000.6495.
• [9] Chung, S. K., Pani, A. K.: Numerical methods for the Rosenau equation. Applicable Analysis. 77, 351-369 (2001).
• [10] Chung, S. K., Pani, A. K.: A second order splitting lumped mass finite element method for the Rosenau equation. Differ. Equ. Dyn. Syst. 15, 331-351 (2004).
• [11] Barreto, R. K., De Caldas, C. S . Q., Gamboa, P., Limaco, J.: Existence of solutions to the Rosenau and Benjamin-Banomahony equation in domains with moving boundary. Electronic Journal of Differential Equations. 2004, 35 (2004).
• [12] Choo, S. M., Chung, S. K., Kim, K. I.: A discontinuous Galerkin method for the Rosenau equation. Applied Numerical Mathematics. 58, 783-799 (2008) .
• [13] Omrani, K., Abidi, F., Achouri, T., Khiari, N.: A new conservative finite difference scheme for the Rosenau equation. Applied Mathematics and Computation. 201, 35-43 (2008).
• [14] Hu, J., Zheng, K.: Two conservative difference schemes for the generalized Rosenau equation. Boundary Value Problems. 2010, 543503 (2010). https://doi:10.1155/2010/543503.
• [15] Wang, M., Li, D., Cui, P.: A Conservative finite difference scheme for the generalized Rosenau equation. International Journal of Pure and Applied Mathematics. 71(4), 539-549 (2011).
• [16] Mittal,R. C., Jain,R.: Application of quintic B-splines collocation method on some Rosenau type nonlinear higher order evolution equations. International Journal of Nonlinear Science. 13(2), 142-152 (2012).
• [17] Atouani, N., Omrani, K.: A new conservative high-order accurate difference scheme for the Rosenau equation. Applicable Analysis. 94(12), 2435-2455 (2015). http://dx.doi.org/10.1080/00036811.2014.987134.
• [18] Abazari, R., Abazari, R.: Numerical solution of the Rosenau equation using quintic collocation B-spline method. Iranian Journal of Science and Technology. 39(3), 281-288 (2015).
• [19] Cai, W., Sun, Y., Wang, Y.: Variational discretizations for the generalized Rosenau-type equations. Applied Mathematics and Computation. 271, 860-873 (2015).
• [20] Ramos J. I., Garc´ıa-López, C. M.:Solitary Wave Formation from a Generalized Rosenau Equation. Hindawi Publishing Corporation Mathematical Problems in Engineering. 2016, 4618364 (2016). http://dx.doi.org/10.1155/2016/4618364.
• [21] Safdari-Vaighani, A., Larsson, E., Heryudono, A.:Radial basis function methods for the Rosenau equation and other higher order PDEs. Journal of Scientific Computing. 75, 1555-1580 (2018). https://doi.org/10.1007/s10915-017- 0598-1.
• [22] Kutluay, S., Karta, M., Ya˘gmurlu, N. M.:Operator time-splitting techniques combined with quintic B-spline collocation method for the generalized Rosenau–KdV equation. Numerical Methods for Partial Differential Equations. 35(6), 2221-2235 (2019).
• [23] Hundsdorfer,W.: Numerical solution of advection-diffusion-reaction equations. Lecture notes for PH.D. course, Thomas Stieltjes Institute, Amsterdam (2000).
• [24] McLachlan, R. I.:On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM Journal on Scientific Computing. 16(1), 151-168 (1995).
• [25] Smith, G. D.:Numerical solution of partial differential equations: Finite difference methods. Clarendon Press. Oxford (1985).
• [26] Özer, S., Ya˘gmurlu, N. M.:Numerical solutions of nonhomogeneous Rosenau type equations by quintic B-spline collocation method. Math. Meth. Appl. Sci. 45(9), 5545-5558 (2022). https://doi.org/10.1002/mma.8125.
• [27] Ya˘gmurlu, N. M., Karaka¸s, A. S.:Numerical solutions of the equal width equation by trigonometric cubic B-spline collocation method based on Rubin–Graves type linearization. Numerical Methods for Partial Differential Equations. 36(5), 1170-1183 (2020). https://doi.org/10.1002/num.22470.
• [28] Ya˘gmurlu, N. M., Karaagac, B., Kutluay, S.: Numerical solutions of Rosenau-RLW equation using Galerkin cubic B-spline finite element method. American Journal of Computational and Applied Mathematics. 7(1), 1-10 (2017). [29] Ya˘gmurlu, N. M., Uçar, Y., Çelikkaya, ˙I.: Operator splitting for numerical solutions of the RLW equation. J. Appl. Anal. Comput. 8(5), 1494-1510 (2018).
• [30] Ba¸shan, A.: Solitary wave, undular-bore and wave-maker solutions of the cubic, quartic and quintic nonlinear generalized equal width (GEW) wave equation. The European Physical Journal Plus. 138, 53 (2023). https://doi.org/10.1140/epjp/s13360-023-03648-4.
• [31] Ba¸shan, A., Esen, A.: Single soliton and double soliton solutions of thequadratic-nonlinear Korteweg-de Vries equation for small and long-times. Numer Methods Partial Differential Equation. 37, 1561–1582 (2021).
• [32] Ba¸shan, A., Uçar, Y., Ya˘gmurlu, N. M.: Numerical solution of the complex modified Korteweg-de Vries equation by DQM. Journal of Physics: Conference Series. 766, 012028 (2016).
• [33] Ba¸shan, A., Uçar, Y., Ya˘gmurlu, N. M., Esen, A.: Numerical approximation to the MEW equation for the single solitary wave and different types of interactions of the solitary waves. Journal of Difference Equations and applications. 28(9), 1193–1213 (2022). https://doi.org/10.1080/10236198.2022.2132154.
• [34] Ya˘gmurlu, N. M., Uçar, Y., Ba¸shan, A.: Numerical approximation of the combined KdV-mKdV equation via the quintic B-spline differential quadrature method. Adıyaman University Journal of Science. 9(2), 386-403 (2019). https://dergipark.org.tr/en/pub/adyujsci https://doi.org/10.37094/adyujsci.526264.
• [35] Ba¸shan, A.: Nonlinear dynamics of the Burgers’ equation and numerical experiments. Mathematical Sciences 16, 183–205 (2022). https://doi.org/10.1007/s40096-021-00410-8.
• [36] Atouani, N., Ouali, Y., Omrani, K.:Mixed finite element methods for the Rosenau equation. Journal of Applied Mathematics and Computing. 57, 393-420 (2018). https://doi.org/10.1007/s12190-017-1112-5.
• [37] Holden, H., Karlsen, K. H., Lie, K. A., Risebro, N. H.: In: Splitting methods for partial differential equations with Rough solutions. European Mathematical Society (2010).
• [38] Geiser, J.: Iterative splitting methods for differential equations. CRC Press. Boca Raton (2011).
• [39] Pazy, A.:Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences. 44, Springer, New York (1983).
• [40] Trotter, H. F.: On the product of semi-groups of operators. Proceedings of the American Mathematical Society. 10, 545-551 (1959).
• [41] Geiser, J.:Recent advances in splitting methods for multiphysics and multiscale: Theory and applications. Journal of Algorithms and Computational Technology. 9(1), 65-93 (2013).
• [42] Marchuk, G. I.:Some application of splitting-up methods to the solution of mathematical physics problems. Aplikace Matematiky. 13, 103-132 (1968).
• [43] Varadara jan, S.:Lie groups, Lie algebras, and their representations (First Edition). Springer-Verlag. New York (1985).
• [44] Strang, G.: On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis. 5, 506–517 (1968).
• [45] Seydaoglu, M., Erdogan, U., Özis, T.: Numerical solution of Burgers’ equation with high order splitting methods. Journal of Computational and Applied Mathematics. 291, 410–421 (2016). https://doi.org/10.1016/j.cam.2015.04.02.
• [46] Prenter, P. M.: Splines and variational methods. Wiley. New York (1975).