Numerical Simulations of Boundary-Forced RLW Equation with Cubic B-Spline-based Differential Quadrature Methods

Arabian Journal for Science and Engineering

Volumn 38, Issue 5, 2013, Pages 1151-1160

  Korkmaz, Alper ^a   Daǧ, Idris ^b  

^a CANKIRI KARATEKIN UNIVERSITY   (Turkey)

^b ESKISEHIR OSMANGAZI UNIVERSITY   (Turkey)

Author keywords

B spline; Differential quadrature method; RLW equation; Solitary waves; Undular bore; Wave undulation

Indexed keywords

EID: 84876422849     PISSN: 2193567X     EISSN: 21914281     Source Type: Journal    
DOI: 10.1007/s13369-012-0353-8     Document Type: Article

References (47)

1. Peregrine D. H.: Calculations of the development of an undular bore. J. Fluid Mech. 25, 321-330 (1966).
2. Benjamin T. B., Bona J. L., Mahony J. J.: Model equations for long waves in non-linear dispersive systems. Philos. Trans. Roy. Soc. London A. 272, 47-78 (1972).
3. Bona J. L., Bryant P. J.: A mathematical model for long waves generated by wavemakers in nonlinear dispersive systems. Proc. Cambridge Philos. Soc. 73, 391-405 (1973).
4. Albert J. P., Bona J. L., Henry D. B.: Sufficient conditions for stability of solitary-wave solutions of model equations for long waves. Physica 24d, 343-366 (1987).
5. Eilbeck J. C., McGuire G. R.: Numerical study of the regularized long wave equation I: numerical methods. J. Comp. Phys. 19, 43-57 (1975).
6. Eilbeck J. C., McGuire G. R.: Numerical study of the regularized long wave equation II: interaction of solitary waves. J. Comp. Phys. 23, 63-73 (1977).
7. Abdulloev K. O., Bogolubsky I. L., Markhankov V. G.: One more example of inelastic soliton interaction. Phys. Lett. 56A, 427-428 (1976).
8. Gardner L. R. T., Gardner G. A., Doǧan A.: A least squares finite element scheme for the RLW equation. Commun. Numer. Meth. Eng. 12, 795-804 (1996).
9. Gardner L. R. T., Gardner G. A., Ayoub F. A., Amein N. K.: Modelling an undular bore with B-splines. Comput. Meth. Appl. Mech. Eng. 147, 147-152 (1997).
10. Daǧ İ.: Least squares quadratic B-spline finite element method for the regularized long wave equation. Comput. Meth. Appl. Mech. Eng. 182, 205-215 (2000).
11. Doǧan A.: Numerical solution of the RLW equation using linear finite elements within Galerkin's method. Appl. Math. Model. 26, 771-783 (2002).
12. Daǧ İ, Özer M. N.: Approximation of the RLW equation by the least square cubic B-spline finite element method. Appl. Math. Model. 25, 221-231 (2001).
13. Zaki S. I.: Solitary waves of the split RLW equation. Comput. Phys. Commun. 138, 80-91 (2001).
14. Saka B., Da ǧ., Doǧan A. A.: Galerkin method for the numerical solution of the RLW equation using quadratic B-splines. Int. J. Comput. Math. 81, 727-739 (2004).
15. Avilez-Valente P., Seabra-Santos F. J.: A Petrov-Galerkin finite element scheme for the regularized long wave equation. Comput. Mech. 34, 256-270 (2004).
16. Daǧ İ, Saka B., Irk D.: Galerkin method for the numerical solution of the RLW equation using quintic B-splines. J. Comput. Appl. Math. 190, 532-547 (2006).
17. Daǧ İ, Doǧan A., Saka B.: B-spline collocation methods for numerical solutions of RLW equation. Int. J. Comput. Math. 80, 743-757 (2003).
18. Saka B., Daǧ İ.: A collocation method for the numerical solution of the RLW equation using cubic B-spline basis. Arab. J. Sci. Eng. 30, 39-50 (2005).
19. Raslan K. R.: A computational method for the regularized long wave (RLW) equation. Appl. Math. Comput. 167(2), 1101-1118 (2005).
20. Soliman A. A., Hussien M. H.: Collocation solution for RLW equation with septic spline. Appl. Math. Comput. 161(2), 623-636 (2005).
21. Gardner L. R. T., Gardner G. A.: Solitary wave of the regularized long wave equation. J. Comput. Phys. 91, 441-459 (1990).
22. Bellman R., Kashef B. G., Casti J.: Differential quadrature: a technique for the rapid solution of nonlinear differential equations. J. Comput. Phys. 10, 40-52 (1972).
23. Bellman, R.; Kashef, B. G.; Lee, E. S.; Rasudevan, R.: Differential quadrature and splines. Comput. Math. Appl. 371-376 (1976).
24. Quan J. R., Chang C. T.: New sightings in involving distributed system equations by the quadrature methods-I. Comput. Chem. Eng. 13, 779-788 (1989).
25. Quan J. R., Chang C. T.: New sightings in involving distributed system equations by the quadrature methods-II. Comput. Chem. Eng. 13, 1017-1024 (1989).
26. Shu C., Richards B. E.: Application of generalized differential quadrature to solve two dimensional incompressible Navier Stokes equations. Int. J. Numer. Meth. Fluids 15, 791-798 (1992).
27. Bonzani I.: Solution of non-linear evolution problems by parallelized collocation-interpolation methods. Comput. Math. Appl. 34(12), 71-79 (1997).
28. Cheng J., Wang B., Du S.: A theoretical analysis of piezoelectric/composite laminate with larger-amplitude deflection effect, Part II: Hermite differential quadrature method and application. Int. J. Solids Struct. 42(24-25), 6181-6201 (2005).
29. Shu C.: Differential Quadrature and its Application in Engineering. Springer, London Limited (2000).
30. Shu, C.: Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation, Ph. D. Thesis. s. l.: University of Glasgow, UK (1991).
31. Zong Z., Lam K. Y.: A localized differential quadrature (LDQ) method and its applications to the 2D wave equation. Comput. Mech. 29, 382-391 (2002).
32. Daǧ İ, Korkmaz A.: A differential quadrature algorithm for simulations of nonlinear Schrödinger equation. Comput. Math. Appl. 56(9), 12222-2234 (2008).
33. Saka B., Daǧ İ, Dereli Y., Korkmaz A.: Three different methods for numerical solutions of the EW equation. Eng. Anal. Bound. Elem. 32, 556-566 (2008).
34. Korkmaz, A.; Daǧ, İ: A differential quadrature algorithm for nonlinear Schrödinger equation. Nonlinear Dynam. 56(1-2) (2009).
35. Korkmaz A., Daǧ İ.: Crank-Nicolson-differential quadrature algorithms for the Kawahara equation. Chaos Solitons Fractals 42(1), 65-73 (2009).
36. Korkmaz A., Daǧ İ.: Solitary wave simulations of complex modified Korteweg-de Vries equation using differential quadrature method. Comput. Phys. Commun. 180(9), 1516-1523 (2009).
37. Tomasiello, S.: Numerical solutions of the Burger-Huxley equation by the IDQ method. Int. J. Comput. Math. (2008). doi: 10. 1080/00207160801968672.
38. Malekzadeh P., Karami G.: Polynomial and harmonic differential quadrature methods for free vibration of variable thickness thick skew plates. Eng. Struct. 27, 1563-1574 (2005).
39. Li C., Wu X.: A new linearized method for Benjamin-Bona-Mahony equation. Numer. Meth. Partial Differ. Equ. 20, 948-961 (2004).
40. Korkmaz A., Daǧ İ, Saka B.: Cosine expansion based differential quadrature (CDQ) algorithms for numerical solution of the RLW equation. Numer. Meth. Partial Differ. Equ. 26(3), 544-560 (2010).
41. Guo Q., Zhong H.: Non-linear vibration analysis of beams by a spline-based differential quadrature method. J. Sound Vib. 269, 413-420 (2004).
42. Zhong H.: Spline-based differential quadrature for fourth order differential equations and its application to Kirchhoff plates. Appl. Math. Modell. 28, 353-366 (2004).
43. Zhong H., Lan M.: Solution of nonlinear initial-value problems by the spline-based differential quadrature method. J. Sound Vib. 296, 908-918 (2006).
44. Olver P. J.: Euler operators and conservation laws of the BBM equation. Math. Proc. Camb. Phil. Soc. 85, 143-160 (1979).
45. Daǧ İ., Saka B., Irk D.: Application of cubic B-splines for numerical solution of the RLW equation. Appl. Math. Comput. 159(2), 373-389 (2004).
46. Bhardwaj D., Shankar R.: A computational method for regularized long wave equation. Comput. Math. Appl. 40, 1397-1404 (2000).
47. Esen A., Kutluay S.: Application of a lumped Galerkin method to the regularized long wave equation. Appl. Math. Comput. 174, 833-845 (2006).