ISRN Computational Mathematics

Volume 2012 (2012), Article ID 302923, 6 pages

http://dx.doi.org/10.5402/2012/302923

Research Article

## Cubic Spline Method for 1D Wave Equation in Polar Coordinates

^{1}Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110 007, India^{2}Department of Mathematics, Deenbandhu Chhotu Ram University of Science & Technology, Murthal 131039, India

Received 10 September 2011; Accepted 28 September 2011

Academic Editors: P. B. Vasconcelos and J. G. Zhou

Copyright © 2012 R. K. Mohanty et al. 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

- W. G. Bickley, “Piecewise cubic interpolation and two-point boundary value problems,”
*The Computer Journal*, vol. 11, pp. 206–208, 1968. View at Google Scholar - D. J. Fyfe, “The use of cubic splines in the solution of two-point boundary value problems,”
*The Computer Journal*, vol. 12, pp. 188–192, 1969. View at Publisher · View at Google Scholar - 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 Publisher · View at Google Scholar - A. Khan and T. Aziz, “Parametric cubic spline approach to the solution of a system of second-order boundary-value problems,”
*Journal of Optimization Theory and Applications*, vol. 118, no. 1, pp. 45–54, 2003. View at Publisher · View at Google Scholar - G. F. Raggett and P. D. Wilson, “A fully implicit finite difference approximation to the one-dimensional wave equation using a cubic spline technique,”
*Journal of the Institute of Mathematics and its Applications*, vol. 14, pp. 75–77, 1974. View at Publisher · View at Google Scholar - J. A. Fleck Jr., “A cubic spline method for solving the wave equation of nonlinear optics,”
*Journal of Computational Physics*, vol. 16, pp. 324–341, 1974. View at Publisher · View at Google Scholar - 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 - F. Gao and C. Chi, “Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation,”
*Applied Mathematics and Computation*, vol. 187, no. 2, pp. 1272–1276, 2007. View at Publisher · View at Google Scholar - J. Rashidinia, R. Jalilian, and V. Kazemi, “Spline methods for the solutions of hyperbolic equations,”
*Applied Mathematics and Computation*, vol. 190, no. 1, pp. 882–886, 2007. View at Publisher · View at Google Scholar - J. Rashidinia, R. Mohammadi, and R. Jalilian, “Spline methods for the solution of hyperbolic equation with variable coefficients,”
*Numerical Methods for Partial Differential Equations*, vol. 23, no. 6, pp. 1411–1419, 2007. View at Publisher · View at Google Scholar - 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 - H. Ding and Y. Zhang, “Parameters spline methods for the solution of hyperbolic equations,”
*Applied Mathematics and Computation*, vol. 204, no. 2, pp. 938–941, 2008. View at Publisher · View at Google Scholar - H. Ding and Y. Zhang, “A new unconditionally stable compact difference scheme of $O({\tau}^{2}+{h}^{4})$ for the 1D linear hyperbolic equation,”
*Applied Mathematics and Computation*, vol. 207, no. 1, pp. 236–241, 2009. View at Publisher · View at Google Scholar - H.-W. Liu and L.-B. Liu, “An unconditionally stable spline difference scheme of $O({k}^{2}+{h}^{4})$ for solving the second-order 1D linear hyperbolic equation,”
*Mathematical and Computer Modelling*, vol. 49, no. 9-10, pp. 1985–1993, 2009. View at Publisher · View at Google Scholar - R. K. Mohanty, “New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations,”
*International Journal of Computer Mathematics*, vol. 86, no. 12, pp. 2061–2071, 2009. View at Publisher · View at Google Scholar - R. K. Mohanty, M. K. Jain, and K. George, “On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients,”
*Journal of Computational and Applied Mathematics*, vol. 72, no. 2, pp. 421–431, 1996. View at Publisher · View at Google Scholar - R. K. Mohanty and U. Arora, “A new discretization method of order four for the numerical solution of one-space dimensional second-order quasi-linear hyperbolic equation,”
*International Journal of Mathematical Education in Science and Technology*, vol. 33, no. 6, pp. 829–838, 2002. View at Publisher · View at Google Scholar - R. K. Mohanty and V. Dahiya, “An $O({k}^{2}+k{h}^{2}+{h}^{4})$ accurate two-level implicit cubic spline method for one space dimensional Quasi-linear parabolic equations,”
*American Journal of Computational Mathematics*, vol. 1, pp. 11–17, 2011. View at Google Scholar - R. K. Mohanty, “Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms,”
*Applied Mathematics and Computation*, vol. 190, no. 2, pp. 1683–1690, 2007. View at Publisher · View at Google Scholar