Combining B-spline least-square schemes with different weight functions to solve the generalized regularized long wave equation

Document Type : Research Paper


Department Mathematics, Education College for pure Sciences, Basrah University, Basrah, Iraq


For solving differential equations, a variety of numerical methods are available, accuracy, performance, and application are all different. In this article, we proposed new numerical techniques for solving the generalized regularized long wave equation(GRLWE) that are based on types M and M-1 of B-splines-least-square method (BSLSM) and weight function of B-splines respectively, which were proposed previously for solving integro-differential equations [2] where $M\in {N}$.  We investigated linear stability using a Fourier method.


[1] H.O. Al-Humedi and A.K. Al-Wahed, Combine of B-spline galerkin schemes with change weight function, Int. J. Engin. Innov. Res. 2 (2013) 340-349.
[2] H.O. Al-Humedi and Z.A. Jameel, Cubic B-spline least-square method combine with a quadratic weight function for solving integro-differential equations, Earthline J. Math. Sci. 4 (2020) 99–113 .
[3] T.B. Benjamin, J.L. Bona and J.J. Mahony,Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A. 272 (1972) 47–78.
[4] S. Bhowmik and S. Karakoc, Numerical approximation of the generalizedregularized long wave equation using Petrov-Galerkin finite element method, Numer. Meth. Partial Diff. Eq. 15 (2019) 1–22.
[5] J.L. Bona and P.J. Bryant, A mathematical model for long waves generated by wave makers in nonlinear dispersive systems, Proc. Cambridge Philos. Soc. 73 (1973) 391–405.
[6] I. Dag and M.N. Ozer, Approximation of the RLW equation by the least square cubic B-spline finite element method, Appl. Math. Model. 25 (2001) 221–231.
[7] A.J. Davies, The Finite Element Method, a First Approach, Clarendon Press. Oxford, London, 1980.
[8] C. DeBoor, A Practical Guide to Splines, Springer-Verlag, New York, 1978.
[9] S. Dhawan and S. Kumar, A numerical solution of one dimensional heat equation using cubic B-spline basis functions, Int. J. Res. Rev Appl. Sci. 1 (2009) 71–77.
[10] S. Dhawan, S. Kumar and S. Kapoor, Approximation of Burger’s equation using B-spline finite element method, Int. J. Appl. Math. Mech. 7 (2011) 61–86.
[11] J.C. Eilbeck and G.R. McGuire, Numerical study of RLW equation numerical methods, J. Compt. Phys. 9 (1975) 43–57.
[12] L.R.T. Gardner, G.A. Gardner, F.A. Ayub and N.K. Amein, Approximations of solitary waves of the MRLW[13] K.H. Jwamer, Minimizing error bounds in (0,2,3) lacunary interpolation by sextic spline function, J. Math. Stat.
3 (2007) 249–256.
[14] S. Karakoc, T. Geyikli and A. Bashan, Petrov-Galerkin finite element method for solving the MRLW equation, TWMS J. App. Eng. Math. 3 (2013) 231–244
[15] S. Karakoc, Y. U¸car and N. Yaˇgmurlu, Numerical solutions of the MRLW equation by cubic B-spline Galerkin finite element method, Kuwait J. Sci. 24 (2015) 141–159.
[16] A. K. Khalifa, K. R. Raslan and H. M. Alzubaidi, A finite difference scheme for the MRLW and solitary wave interactions, Appl. Math. Comput, 189 (2007) 346-354.
[17] A.K. Khalifa, K.R. Raslan and H.M. Alzubaidi, Numerical study using ADM for the modified regularized long wave equation, Appl. Math. Model. 32 (2008) 2962–2972.
[18] 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 (2008) 406–418.
[19] A.R. Mitchell and D.F. Griffiths, The Finite Difference Equations in Partial Differential Equations, John Wiley and Sons., New York, 1980.
[20] J. Noye, Numerical Solutions of Partial Differential Equations, North Holland Publishimg company-Amsterdam. New York. Oxford, 1982.
[21] D.H. Pergrine, Calculations of the development of an Undular Bore, J. Fluid Mech. 25 (1996) 321–330.
[22] P. M. Prenter, Splines and Variational Method, John Wiley and Sons, New York , USA, 1975.
[23] G. D. Smith, Numerical Solution of Partial Differential Equation, Finite Difference Methods, Oxford University press, 1978.
[24] O.C. Zienkiewicz, The Finite Element Method, 3rd edition, McGraw Hill, London, 1979. equation by B-spline, finite element, Arab. J. Sci., Eng. 22 (1997) 183–193.
Volume 13, Issue 1
March 2022
Pages 159-177
  • Receive Date: 10 June 2021
  • Accept Date: 30 August 2021