Quartic trigonometric B-spline algorithm for numerical solution of the regularized long wave equation

Quartic trigonometric B-spline algorithm for numerical solution of the regularized long wave equation

In this paper, an application of the quartic trigonometric B-spline finite element method is used to solve theregularized long wave equation numerically. This approach involves a Galerkin method based on the quartic trigonometricB-spline function in space discretization together with second and fourth order schemes in time discretization. Theaccuracy of the proposed methods are demonstrated by test problems and numerical results are compared with the exactsolution and some previous results.

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  • [1] Abbas M, Majid AA, Ismail AIM, Rashid A. The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems. Appl Math Comput 2014; 239: 74-88.
  • [2] Ay B, Dag I, Gorgulu MZ. Trigonometric quadratic B-spline subdomain Galerkin algorithm for the Burgers’ equation. Open Phys 2015; 13: 400-406.
  • [3] Chang Q, Wang G, Guo B. Conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion. J Comput Phys 1991; 93: 360-375.
  • [4] Chegini NG, Salaripanah A, Mokhtari R, Isvand D. Numerical solution of the regularized long wave equation using nonpolynomial splines. Nonlinear Dynam 2012; 69: 459-471.
  • [5] Dag I, Ersoy O, Kacmaz O. The trigonometric cubic B-spline algorithm for Burgers’ equation. arXiv:1407.5434, 2014.
  • [6] Dag I, Saka B, Irk D. Galerkin methods for the numerical solution of the RLW equation using quintic B-splines. J Comput Appl Math 2006; 190: 532-547.
  • [7] Dogan A. Numerical solution of RLW equation using linear finite elements within Galerkin’s method. Appl Math Model 2002; 26: 771-783.
  • [8] Esen A, Kutluay S. Application of a lumped Galerkin method to the regularized long wave equation. Appl Math Comput 2006; 174: 833-845.
  • [9] Gardner LRT, Dag I. The boundary-forced regularised long-wave equation. Il Nuovo Cimento B (1971-1996) 1995; 110: 1487-1496.
  • [10] Hamid NNA, Majid AA, Ismail AIM. Cubic trigonometric B-spline applied to linear two-point boundary value problems of order two. International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 2010; 4: 1377-1382.
  • [11] Hosseini MM, Ghaneai H, Mohyud-Din ST, Usman M. Tri-prong scheme for regularized long wave equation. Journal of the Association of Arab Universities for Basic and Applied Sciences 2016; 20: 68-77.
  • [12] Irk D. Solitary wave solutions for the regularized long-wave equation. Phys Wave Phenom 2012; 20: 174-183.
  • [13] Irk D, Keskin P. Cubic trigonometric B-spline Galerkin methods for the regularized long wave equation. Journal of Physics Conference Series 2016; 766: 012032.
  • [14] Irk D, Keskin P. Quadratic trigonometric B-spline Galerkin methods for the regularized long wave equation. J Appl Anal Comput 2017; 7: 617-631.
  • [15] Koch PE. Multivariate trigonometric B-splines. J Approx Theory 1988; 54: 162-168.
  • [16] Lin B. Parametric spline solution of the regularized long wave equation. Appl Math Comput 2014; 243: 358-367.
  • [17] Lyche T, Winther R. A stable recurrence relation for trigonometric B-splines. J Approx Theory 1979; 25: 266-279.
  • [18] Mei L, Chen Y. Numerical solution of RLW equation using Galerkin method with extrapolation techniques. Comput Phys Commun 2012; 183: 1609-1616.
  • [19] Nikolis A. Numerical solutions of ordinary differential equations with quadratic trigonometric Splines. Applied Mathematics E-Notes 2004; 4: 142-149.
  • [20] Olver PJ. Euler operators and conservation laws of the BBM equation. Math Proc Cambridge 1979; 85: 143-159.
  • [21] Oruc O, Bulut F, Esen A. Numerical solutions of regularized long wave equation by Haar wavelet method. Mediterr J Math 2016; 13: 3235-3253.
  • [22] Peregrine DH. Calculations of the development of an undular bore. J Fluid Mech 1966; 25: 321-330.
  • [23] Saka B, Dag I. A numerical solution of the RLW equation by Galerkin method using quartic B-splines. Commun Numer Meth En 2008; 24: 1339-1361.
  • [24] Saka B, Dag I, Dogan A. Galerkin method for the numerical solution of the RLW equation using quadratic B-splines. Int J Comput Math 2004; 81: 727-739.
  • [25] Schoenberg IJ. On trigonometric spline interpolation. J Math Mech 1964; 13: 795-825.
  • [26] Walz G. Identities for trigonometric B-splines with an application to curve design. BIT 1997; 37: 189-201.
  • [27] Zaki SI. Solitary waves of the splitted RLW equation. Comput Phys Commun 2001; 138: 80-91.
  • [28] Zheng K, Hu J. High-order conservative Crank-Nicolson scheme for regularized long wave equation. Adv Differ Equ-Ny 2013; 2013: 287.
  • [29] Zorsahin Gorgulu M, Dag I, Irk D. Simulations of solitary waves of RLW equation by exponential B-spline Galerkin method. Chinese Phys B 2017; 26: 080202.