Khalifa, A. K.; Raslan, K. R.; Alzubaidi, H. M. A collocation method with cubic B-splines for solving the MRLW equation. (English) Zbl 1133.65085 J. Comput. Appl. Math. 212, No. 2, 406-418 (2008). The modified regularized long wave equation is solved by a collocation method based on cubic B-splines. The time differencing is done by a leapfrog method. A von Neumann stability analysis is given for a linearlized form of the method. Note that L. R. T. Gardner, G. A. Gardner, F. A. Ayoub, and N. K. Amein [Arab. J. Sci. Eng. 22, 183–193 (1997; Zbl 0893.35113)] previously solved this problem using quintic B-splines. Reviewer: Gerald W. Hedstrom (Pleasanton) Cited in 45 Documents MSC: 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35Q53 KdV equations (Korteweg-de Vries equations) 76B25 Solitary waves for incompressible inviscid fluids Keywords:regularized long wave equation; collocation; solitons; solitary waves; cubic B-splines; leapfrog method; stability Citations:Zbl 0893.35113 PDFBibTeX XMLCite \textit{A. K. Khalifa} et al., J. Comput. Appl. Math. 212, No. 2, 406--418 (2008; Zbl 1133.65085) Full Text: DOI References: [1] Alexander, M. E.; Morris, J. L., Galerkin methods applied to some model equations for nonlinear dispersive waves, J. Comput. Phys., 30, 428-451 (1979) · Zbl 0407.76014 [2] Dag, I., Least squares quadratic B-splines finite element method for the regularized long wave equation, Comput. Methods Appl. Mech. Eng., 182, 205-215 (2000) · Zbl 0964.76042 [3] Dag, I.; Saka, B.; Irk, D., Application of cubic B-splines for numerical solution of the RLW equation, Appl. Math. Comput., 195, 373-389 (2004) · Zbl 1060.65110 [4] Gardner, L. R.T.; Gardner, G. A., Solitary wave of the regularized long wave equation, J. Comput. Phys., 91, 441-459 (1990) · Zbl 0717.65072 [5] Gardner, L. R.T.; Gardner, G. A.; Dag, I., A B-spline finite element method for the regularized long wave equation, Comm. Numer. Meth. Eng., 11, 59-68 (1995) · Zbl 0819.65125 [6] Gardner, L. R.T.; Gardner, G. A.; Dogan, A., A least squares finite element scheme for the RLW equation, Comm. Numer. Meth. Eng., 12, 795-804 (1996) · Zbl 0867.76040 [7] Gardner, L. R.T.; Gardner, G. A.; Ayoub, F. A.; Amein, N. K., Approximations of solitary waves of the MRLW equation by B-spline finite element, Arab. J. Sci. Eng., 22, 183-193 (1997) · Zbl 0893.35113 [8] Kaya, D.; El-Sayed, S. M., An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons and Fractals, 17, 869-877 (2003) · Zbl 1030.35139 [9] A.K. Khalifa, Theory and applications of the collocation method with splines for ordinary and partial differential equations, Ph.D. Thesis, Heriot-Watt University, 1979.; A.K. Khalifa, Theory and applications of the collocation method with splines for ordinary and partial differential equations, Ph.D. Thesis, Heriot-Watt University, 1979. [10] Peregrine, D. H., Calculations of the development of an undular bore, J. Fluid Mech., 25, 2, 321-330 (1966) [11] Raslan, K. R., A computational method for the regularized long wave (RLW) equation, Appl. Math. Comput., 176, 1101-1118 (2005) · Zbl 1082.65582 [12] Smith, G. D., Numerical Solution of Partial Differential Equation: Finite Difference Method (1978), Learendom Press: Learendom Press Oxford · Zbl 0389.65040 [13] Soliman, A. A.; Hussien, M. H., Collocation solution for RLW equation with septic splines, Appl. Math. Comput., 161, 623-636 (2005) · Zbl 1061.65102 [14] Soliman, A. A.; Raslan, K. R., Collocation method using quadratic B-spline for the RLW equation, Internat. J. Comput. Math., 78, 399-412 (2001) · Zbl 0990.65116 [15] Zhang, L., A finite difference scheme for generalized long wave equation, Appl. Math. Comput., 168, 962-972 (2005) · Zbl 1080.65079 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.