Total views : 59

Collocation Method using Quartic B-Splines for Solving the Modified RLW Equation

Affiliations

  • Department of Mathematics, College of Science & Arts, University of Bisha, Bisha – 61922, PO Box 344, Kingdom of Saudi Arabia

Abstract


The Modified Regularized Long Wave (MRLW) equation is solved numerically by giving a new algorithm based on collocation method using quartic B-splines at the mid knots points as element shape. Also, we use the Fourth Runge-Kutta method for solving the system of first order ordinary differential equations instead of finite difference method. Our test problems, including the migration and interaction of solitary waves, are used to validate the algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. The temporal evaluation of a Maxwellian initial pulse is then studied.

Keywords

Collocation Method, MRLW Equation, Quartic B-splines, Solitons.

Full Text:

 |  (PDF views: 50)

References


  • Peregrine DH. Calculations of the development of an undular bore. Journal of Fluid Mech. 1966; 25(2):321. Crossref.
  • Alexander ME, Morris JL. Galerkin methods applied to some model equations for nonlinear dispersive waves.Journal of Comput Phys. 1979; 30:428–51. Crossref.
  • Gardner LRT, Gardner GA. Solitary wave of the regularized long wave equation. Journal of Comput Phys. 1990; 91:441–59. Crossref.
  • Gardner LRT, Gardner GA, Dag I. A B-spline finite element method for the regularized long wave equation. Commun Numer Meth Eng. 1995; 11:59–68. Crossref.
  • Gardner LRT, Gardner GA, Dogan A. A least squares finite element scheme for the RLW equation. Commun Numer Meth. Eng. 1996; 12:795–804. Crossref, Crossref.
  • Dag I. Least squares quadratic B-splines finite element method for the regularized long wave equation. Comp Methods Appl Mech Eng. 2000; 182:205–15. Crossref.
  • Khalifa AK. Theory and applications of the collocation method with splines for ordinary and partial differential equations. [Ph.D. Thesis]. Heriot-Watt University; 1979.
  • Soliman AA, Raslan KR. Collocation method using quadratic B-spline for the RLW equation. International Journal of Comput Math. 2001; 78:399–412. Crossref.
  • Dag I, Saka B, Irk D. Application of cubic B-splines for numerical solution of the RLW equation. App Math Comp.2004; 195:373–89.
  • Raslan KR. A computational method for the Regularized Long Wave (RLW) equation. App Math Comp; 2005.
  • Soliman AA, Hussien MH. Collocation solution for RLW equation with septic splines. App Math Comp. 2005; 161:623–36. Crossref.
  • Zhang L. A finite difference scheme for generalized long wave equation. App Math Comp; 2004.
  • Kaya D, El-Sayed SM. An application of the decomposition method for the generalized KdV and RLW equations. Chaos Solitons and Fractals. 2003; 17:869–77. Crossref.
  • Gardner LRT, Gardner GA, Ayoub FA, Amein NK.Approximations of solitary waves of the MRLW equation by B-spline finite element. Arab Journal of Science Eng.1997; 22:183–93.
  • Rashidinia J, Jamalzadeh S, Mohebianfar E. B-spline Collocation Approach to the Solution of Options Pricing Model. Journal of Computer Science & Computational Mathematics. 2014; 4:5–9. Crossref.
  • Kimio T, Natarajan G, Hideki A, Taichi K, Nanao K.Higher involvement of subtelomere regions for chromosome rearrangements in leukemia and lymphoma and in irradiated leukemic cell line. Indian Journal of Science and Technology. 2012 Apr; 5 (1):1801–11.
  • Cunningham CH. A Laboratory Guide in Virology. 6th edition.Burgess Publication Company: Minnesota. 1973.
  • Kumar E, Rajan M. Microbiology of Indian Desert. In: D.N. Sen. [ed]. Ecology and vegetation of Indian desert. India: Agro Botanical Publication; 1990. p. 83–105.
  • Rajan M, Rao BS, Anjaria KB, Unny VKP, Thyagarajan S. Radiotoxicity of sulfur-35. Proceedings of 10th NSRP; India. 1993. p. 257–8.
  • Available from: http://www.indjst.org/index.php/vision.Date accessed: 01/01/2015.

Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.