More Efficient Solutions for Numerical Analysis of the Nonlinear Generalized Regularized Long Wave (Grlw) Using the Operator Splitting Method

Abstract

Through the use of two numerical techniques, the purpose of this study is to examine the approximate outcomes of the (GRLW) equation. The utilized methods are the collocation method with quintic B-spline, which is based on finite elements and yields good results for nonlinear evolution equations, and the strang splitting technique, which is simple to apply, practical, and quick. In order to provide approximate solutions for the main problem, the collocation method is combined with the Strang splitting method for this study. Three examples—the formation of the Maxwellian initial condition, the interaction of two solitary waves, and a single solitary wave—are taken into consideration in order to assess the accuracy of these algorithms. To demonstrate how closely the exact solutions close to numerical results and to contrast them with other solutions in the literature, error norms, and conservation quantities are computed. Tables and graphs are used to illustrate the solutions that have generated. Based on the results obtained and the practical, easy-to-use, and current features of the methodologies, this article stands out from the rest.

Keywords

• {Generalized regularized long wave
• B-splines
• Collocation method
• Strang splitting

References

1. [1] D.H. Peregrine, Long waves on a beach, J. Fluid Mech., 27(4) (1967), 815-827. $ \href{https://doi.org/10.1017/S0022112067002605}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84958426983}{\mbox{[Scopus]}} $
2. [2] T.B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. of the R. Soc., London, 227(1220) (1972), 47-78. $ \href{https://doi.org/10.1098/rsta.1972.0032}{\mbox{[CrossRef]}} $
3. [3] L. Zhang, A finite difference scheme for generalized regularized long-wave equation, Appl. Math. Comput., 168(2) (2005), 962-972. $\href{https://doi.org/10.1016/j.amc.2004.09.027}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/26044473354}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000232760000020}{\mbox{[Web of Science]}} $
4. [4] S.K. Bhowmik and S.B.G. Karakoç, Numerical approximation of the generalized regularized long wave equation using Petrov-Galerkin finite element method, Numer. Meth. Part Differ. Equ., 35(6) (2019), 2236-2257. $\href{https://doi.org/10.1002/num.22410}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85069680299}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000480355300001}{\mbox{[Web of Science]}} $
5. [5] T. A. Roshan, Petrov-galerkin method for solving the generalized regularized long wave (GRLW) equation, Comput. Math. Appl., 63(5) (2012) 943-956. $ \href{https://doi.org/10.1016/j.camwa.2011.11.059}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84865617142}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000302760400006}{\mbox{[Web of Science]}} $
6. [6] H. Zeybek and S.B.G. Karakoc¸, A numerical investigation of the GRLW equation using lumped Galerkin approach with cubic B-spline, Springer Plus., 5(1) (2016), 1-17. $ \href{https://doi.org/10.1186/s40064-016-1773-9}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84975761230&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28A+numerical+investigation+of+the+GRLW+equation+using+lumped+Galerkin+approach+with+cubic+B-spline%2C%29}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000371416800015}{\mbox{[Web of Science]}} $
7. [7] H. Zeybek and S.B.G. Karakoç, A collocation algorithm based on quintic B-splines for the solitary wave simulation of the GRLW equation, Sci. Iran., Trans. B Mechanical Engineering, 26(6) (2019), 3356-3368. $\href{https://doi.org/10.24200/sci.2018.20781}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85105336666}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000514808700011}{\mbox{[Web of Science]}} $
8. [8] S.B.G. Karakoç and H. Zeybek, Solitary-wave solutions of the GRLWequation using septic B-spline collocation method, Appl. Math. Comput. 289(1) (2016), 159-171. $ \href{https://doi.org/10.1016/j.amc.2016.05.021}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84971281822}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000380754700012}{\mbox{[Web of Science]}} $

9. [9] R. Akbari and R. Mokhtari, A new compact finite difference method for solving the generalized long wave equation, Numer. Funct. Anal. Optim., 35(2) (2014), 133-152. $\href{https://doi.org/10.1080/01630563.2013.830128}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84891607687}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000329139900001}{\mbox{[Web of Science]}} $
10. [10] R. Mokhtari and M .Mohammadi, Solution of GRLW equation using Sinc-collocation method, Comput. Phys. Commun., 181(7) (2010), 1266-1274. $ \href{https://doi.org/10.1016/j.cpc.2010.03.015}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/77953131119}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000278411200012}{\mbox{[Web of Science]}} $
11. [11] S.B.G. Karakoç, M.Liguan and K.A. Khalid, Two efficient methods for solving the generalized regularized long wave equation, Appl. Anal., 101(13) (2022), 4721-4742. $ \href{https://doi.org/10.1080/00036811.2020.1869942}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85099288998}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000605745900001}{\mbox{[Web of Science]}}$
12. [12] D. Bhardwaj and R.A. Shankar, Computational method for regularized long wave equation, Comput. Math. Appl., 40(12) (2000), 1397-1404. $ \href{https://doi.org/10.1016/S0898-1221(00)00248-0}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/0034500868}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000166073900006}{\mbox{[Web of Science]}} $
13. [13] A. Başhan and N.M. Yağmurlu, A mixed method approach to the solitary wave, undular bore and boundary-forced solutions of the Regularized Long Wave equation, Comput. Appl. Math., 41(4) (2022), 2-20. $ \href{https://doi.org/10.1007/s40314-022-01882-7}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85129699753}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000793167300002}{\mbox{[Web of Science]}} $
14. [14] A. Başhan, A novel outlook to the an alternative equation for modelling shallow water wave: Regularised Long Wave (RLW) equation, Indian J. Pure Appl. Math., 54 (2022), 133-145. $\href{https://doi.org/10.1007/s13226-022-00239-4}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85125144435}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000761265900002}{\mbox{[Web of Science]}} $
15. [15] A. Başhan, Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation, Gazi Univ. J. Sci., 35(4) (2022), 1597-1612. $ \href{https://doi.org/10.35378/gujs.892116}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85139526112}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000904870300025}{\mbox{[Web of Science]}}$
16. [16] İ. Dağ and M.N. Özer, Approximation of RLW equation by least square cubic B-spline finite element method, Appl. Math. Model., 25(3) (2021), 221-231. $ \href{https://doi.org/10.1016/S0307-904X(00)00030-5}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/0342420544}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000166355800004}{\mbox{[Web of Science]}} $
17. [17] İ. Dağ, B. Saka and D. Irk, Application of cubic B-splines for numerical solution of the RLW equation, Appl. Math. Comput., 159(2) (2004), 373-389. $ \href{https://doi.org/10.1016/j.amc.2003.10.020}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/5644289701}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000224774100007}{\mbox{[Web of Science]}} $
18. [18] A. Esen and S. Kutluay, Application of a lumped Galerkin method to the regularized long wave equation, Appl. Math. Comput., 174(2) (2006), 833-845. $ \href{https://doi.org/10.1016/j.amc.2005.05.032}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/33644585053}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000236407300004}{\mbox{[Web of Science]}} $
19. [19] L.R.T. Gardner, G.A. Gardner and İ. Dağ, A B-spline finite element method for the regularized long wave equation, Commun Numer Methods Eng., 11(1) (1995), 59-68. $ \href{https://doi.org/10.1002/cnm.1640110109}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/0028976282}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1995QE54000008}{\mbox{[Web of Science]}} $
20. [20] K.R. Raslan, A computational method for the regularized long wave equation, Appl. Math. Comput., 167(2) (2005), 1101-1118. $ \href{https://doi.org/10.1016/j.amc.2004.06.130}{\mbox{[CrossRef]}} \href{https://www.sciencedirect.com/science/article/pii/S0096300304005521}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000232501200023}{\mbox{[Web of Science]}} $
21. [21] S. Islam, S. Had and A. Ali, A meshfree method for the numerical solution of the RLW equation, J. Comput. Appl. Math., 223(2) (2009), 997-1012. $ \href{https://doi.org/10.1016/j.cam.2008.03.039}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/57049148019}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000261898900040}{\mbox{[Web of Science]}} $
22. [22] P.C. Jain , R. Shankara and Singh T.V., Numerical solution of regularized long-wave equation, Commun Numer Methods Eng., 9(7) (1993), 579-586. $ \href{https://doi.org/10.1002/cnm.1640090705}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/0027634117}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1993LP58700004}{\mbox{[Web of Science]}} $
23. [23] Ö. Oruç, A. Esen and F.A. Bulut, Strang splitting approach combined with Chebyshev wavelets to solve the regularized long-wave equation numerically, Mediterr. J. Math., 17(5) (2020), 1-18.$ \href{https://doi.org/10.1007/s00009-020-01572-w}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85089564543}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000567405300005}{\mbox{[Web of Science]}} $
24. [24] N.M. Yağmurlu and A.S. Karakaş, Numerical solutions of the equal width equation by trigonometric cubic B-spline collocation method based on Rubin-Graves type linearization, Numer. Meth. Part Differ. Equ., 36(5) 2020, 1170-1183. $ \href{https://doi.org/10.1002/num.22470}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85085066218}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000530146700001}{\mbox{[Web of Science]}} $
25. [25] N.M. Yağmurlu and A.S. Karakaş, A novel perspective for simulations of the MEW equation by trigonometric cubic B-spline collocation method based on Rubin-Graves type linearization, Comput. Methods Differ. Equ., 10(4) (2022) 1046-1058. $ \href{https://doi.org/10.22034/cmde.2021.47358.1981}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85149736170&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28A+novel+perspective+for+simulations+of+the+MEW+equation+by+trigonometric+cubic+B-spline+collocation+method+based+on+Rubin-Graves+type+linearization%29&sessionSearchId=852c791ba767584d3977cf978bce2cc6}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001009603700014}{\mbox{[Web of Science]}} $
26. [26] D.A. Hammada and M.S. El-Azab, Chebyshev-Chebyshev spectral collocation method for solving the generalized regularized long wave (GRLW) equation, Appl. Math. Comput., 285(20) (2016), 228-240. $ \href{https://doi.org/10.1016/j.amc.2016.03.033}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84963795586}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000375403400018}{\mbox{[Web of Science]}} $
27. [27] 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(2A) (1997), 183-193. $\href{https://www.scopus.com/pages/publications/0031508664}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1997XX02000006}{\mbox{[Web of Science]}} $
28. [28] A.K. Khalifa, K.R. Raslan and H.M. Alzubaidi, A collocation method with cubic B-splines for solving the MRLW equation, J. Comput. Appl. Math., 212 (2) (2008), 406-418. $ \href{https://doi.org/10.1016/j.cam.2006.12.029}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/37249015520}{\mbox{[Scopus]}} $
29. [29] D.A. Hamzah and M.C. Aprianto, On the numerical solution of Burgers-Fisher equation by the strang splitting method, J. Phys. Conf. Ser., 1764 (1) (2021), 012041, IOP Publishing. $\href{https://doi.org/10.1088/1742-6596/1764/1/012041}{\mbox{[Web]}} \href{https://www.scopus.com/pages/publications/85102361031}{\mbox{[Scopus]}} $
30. [30] J. Geiser, Decomposition Methods for Differential Equations: Theory and Applications, CRC Press, (2009). $ \href{https://www.scopus.com/pages/publications/85055809325}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000267516500011}{\mbox{[Web of Science]}} $
31. [31] R. Glowinski, S.J. Osher andW. Yin, Splitting Methods in Communication, Imaging, Science, and Engineering, Springer, (2017). $ \href{http://dx.doi.org/10.1007/978-3-319-41589-5}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000411308200024}{\mbox{[Web of Science]}}$
32. [32] S.B.G. Karakoç, N.M.Yağmurlu and Y. Uçar, Numerical approximation to a solution of the modified regularized long wave equation using quintic B-splines, Bound. Value Probl., 2013(27) (2013), 1-17. $ \href{https://doi.org/10.1186/1687-2770-2013-27}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84888065588}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000325254000001}{\mbox{[Web of Science]}}$
33. [33] P.M. Prenter, Splines and variational methods, J. Wiley, New York, (1975). $ \href{https://sampotter.github.io/courses/nyu-spring-2022-math-ua-252/prenter--splines-and-variational-methods.pdf}{\mbox{[Web]}} $