Table of Contents
ISRN Computational Mathematics
Volume 2013 (2013), Article ID 710529, 8 pages
http://dx.doi.org/10.1155/2013/710529
Research Article

Applying Cubic B-Spline Quasi-Interpolation to Solve 1D Wave Equations in Polar Coordinates

Department of Applied Mathematics, School of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht 41938, Iran

Received 27 August 2013; Accepted 14 October 2013

Academic Editors: Y. Peng and J. G. Zhou

Copyright © 2013 Hossein Aminikhah and Javad Alavi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. G. Bickley, “Piecewise cubic interpolation and two-point boundary problems,” Computer Journal, vol. 11, no. 2, pp. 206–208, 1968. View at Publisher · View at Google Scholar · View at Scopus
  2. G. F. Raggett and P. D. Wilson, “A fully implicit finite difference approximation to the one-dimensional wave equation using a cubic spline technique,” IMA Journal of Applied Mathematics, vol. 14, no. 1, pp. 75–78, 1974. View at Publisher · View at Google Scholar · View at Scopus
  3. T. C. Chawla, G. Leaf, W. L. Chen, and M. A. Grolmes, “The application of the collocation method using hermite cubic spline to nonlinear transient one-dimensional heat conduction problem,” Journal of Heat Transfer, vol. 97, no. 4, pp. 562–569, 1975. View at Google Scholar · View at Scopus
  4. S. G. Rubin and P. K. Khosla, “Higher-order numerical solution using cubic splines,” AIAA Journal, vol. 14, no. 7, pp. 851–858, 1976. View at Google Scholar · View at Scopus
  5. M. K. Jain and T. Aziz, “Cubic spline solution of two-point boundary value problems with significant first derivatives,” Computer Methods in Applied Mechanics and Engineering, vol. 39, no. 1, pp. 83–91, 1983. View at Google Scholar · View at Scopus
  6. H. M. El-Hawary and S. M. Mahmoud, “Spline collocation methods for solving delay-differential equations,” Applied Mathematics and Computation, vol. 146, no. 2-3, pp. 359–372, 2003. View at Publisher · View at Google Scholar · View at Scopus
  7. R. K. Mohanty, “An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation,” Applied Mathematics Letters, vol. 17, no. 1, pp. 101–105, 2004. View at Publisher · View at Google Scholar · View at Scopus
  8. A. Mohebbi and M. Dehghan, “High order compact solution of the one-space-dimensional linear hyperbolic equation,” Numerical Methods for Partial Differential Equations, vol. 24, no. 5, pp. 1222–1235, 2008. View at Publisher · View at Google Scholar · View at Scopus
  9. C.-G. Zhu and R.-H. Wang, “Numerical solution of Burgers' equation by cubic B-spline quasi-interpolation,” Applied Mathematics and Computation, vol. 208, no. 1, pp. 260–272, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. L. Ma, Z. Mo, and X. Xu, “Quasi-interpolation operators based on a cubic spline and applications in SAMR simulations,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 3853–3868, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. M. Dosti and A. Nazemi, “Solving one-dimensional hyperbolic telegraph equation using cubic B-spline quasi-interpolation,” International Journal of Mathematical & Computer Sciences, vol. 7, no. 2, p. 57, 2011. View at Google Scholar
  12. C.-C. Wang, J.-H. Huang, and D.-J. Yang, “Cubic spline difference method for heat conduction,” International Communications in Heat and Mass Transfer, vol. 39, no. 2, pp. 224–230, 2012. View at Publisher · View at Google Scholar · View at Scopus
  13. M. K. Kadalbajoo, L. P. Tripathi, and A. Kumar, “A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1483–1505, 2012. View at Publisher · View at Google Scholar · View at Scopus
  14. S. A. Khuri and A. Sayfy, “A spline collocation approach for a generalized parabolic problem subject to non-classical conditions,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9187–9196, 2012. View at Publisher · View at Google Scholar · View at Scopus
  15. R. K. Mohanty, R. Kumar, and V. Dahiya, “Cubic spline iterative method for Poisson’s equation in cylindrical polar coordinates,” International Scholarly Research Network ISRN Mathematical Physics, vol. 2012, Article ID 234516, 11 pages, 2012. View at Publisher · View at Google Scholar
  16. R. C. Mittal and R. K. Jain, “Numerical solutions of nonlinear Burgers' equation with modified cubic B-splines collocation method,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7839–7855, 2012. View at Publisher · View at Google Scholar · View at Scopus
  17. R. K. Mohanty, R. Kumar, and V. Dahiya, “Cubic spline method for 1D wave equation in polar coordinates,” International Scholarly Research Network ISRN Computational Mathematics, vol. 2012, Article ID 302923, 6 pages, 2012. View at Publisher · View at Google Scholar
  18. C. De Boor, A Practical Guide to Splines, Springer, New York, NY, USA, 1978.
  19. P. Sablonnière, “Univariate spline quasi-interpolants and applications to numerical analysis,” Rendiconti del Seminario Matematico, vol. 63, no. 3, pp. 211–222, 2005. View at Google Scholar · View at Scopus
  20. K. K. Sharma and P. Singh, “Hyperbolic partial differential-difference equation in the mathematical modeling of neuronal firing and its numerical solution,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 229–238, 2008. View at Publisher · View at Google Scholar · View at Scopus