An O(k^{2}+kh^{2}+h^{2}) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations ()

Ranjan Kumar Mohanty, Vijay Dahiya

.

**DOI: **10.4236/ajcm.2011.11002
PDF
HTML
6,443
Downloads
12,841
Views
Citations

.

In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0< x <1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.

Keywords

Quasi-Linear Parabolic Equation, Implicit Method, Cubic Spline Approximation, Diffusion-Convection Equation, Singular Equation, Burgers’ Equation, Reynolds Number

Share and Cite:

R. Mohanty and V. Dahiya, "An O(k^{2}+kh^{2}+h^{2}) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations," *American Journal of Computational Mathematics*, Vol. 1 No. 1, 2011, pp. 11-17. doi: 10.4236/ajcm.2011.11002.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | W. G. Bickley, “Piecewise Cubic Interpolation and Two Point Boundary Value Problems,” Computer Journal, Vol. 11, No. 2, 1968, pp. 206-208. |

[2] | D. J. Fyfe, “The Use of Cubic Splines in the Solution of Two Point Boundary Value Problems,” Computer Journal, Vol. 12, No. 2, 1969, pp. 188-192. doi:10.1093/comjnl/12.2.188 |

[3] | E. L. Albasiny and W. D. Hoskins, “Increased Accuracy Cubic Spline Solutions to Two Point Boundary Value Problems,” Journal of the Institute of Mathematics and its Applications, Vol. 9, No.1, 1972, pp. 47-55. doi:10.1093/imamat/9.1.47 |

[4] | S. G. Rubin and P. K. Khosla, “Higher Order Numerical Solutions Using Cubic Splines,” American Institute of Aeronautics and Astro-nautics Journal. Vol. 14, No.7, 1976, pp. 851-858. |

[5] | M. M. Chawla and R. Subramanian, “A New Spline Method for Singular Two Point Boundary Value Problems,” International Journal of Computer Mathematics, Vol. 24, No. 3-4, 1988, pp. 291-310. doi:10.1080/00207168808803650 |

[6] | M. M. Chawla, R. Subramanian and H. L. Sathi, “A Fourth Order Spline Method for Singular Two Point Boundary Value Prob-lems,” Journal of Computational and Applied Mathemat-ics, Vol. 21, No. 2, 1988, pp. 189-202. doi:10.1016/0377-0427(88)90267-1 |

[7] | 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, 1983, pp. 83-91. doi:10.1016/0045-7825(83)90075-0 |

[8] | 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, 2003, pp. 45-54. doi:10.1023/A:1024783323624 |

[9] | Manoj Kumar, “A Fourth Order Spline Finite Difference Method for Singular Two Point Boundary Value Problems,” International Journal of Computer Mathematics, Vol. 80, No.12, 2003, pp. 1499-1504. doi:10.1080/0020716031000148179 |

[10] | Manoj Kumar, “Higher Order Method for Singular Boundary Value Problems by Using Spline Function,” Applied Mathemat-ics and Computations, Vol. 192, No.1, 2007, pp. 175-179. doi:10.1016/j.amc.2007.02.156 |

[11] | Manoj Kumar and P. K. Srivastava, “Computational Techniques for Solving Differential Equations by Cubic, Quintic and Sextic Spline,” International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 10, No. 1, 2009, pp. 108-115. doi:10.1080/15502280802623297 |

[12] | J. Rashidinia, R. Mohammadi and M. Ghasemi, “Cubic Spline Solution of Singularly Perturbed Boundary Value Problems with Significant First Derivatives,” Applied Ma- thematics and Computations, Vol. 190, No. 2, 2007, pp. 1762-1766. |

[13] | J. Rashidinia, R. Mohammadi, R. Jalilian and M. Ghasemi, “Convergence of Cubic Spline Approach to the Solution of a System of Boundary Value Problems,” Applied Mathematics and Computations, Vol. 192, No. 2, 2007, pp. 319-331. doi:10.1016/j.amc.2007.03.008 |

[14] | N. Papamichael and J. R. Whiteman, “A Cubic Spline Technique for the One-dimensional Heat Equation,” IMA Journal of Applied Mathematics, Vol. 11, No. 1, 1973, pp. 111-113. doi:10.1093/imamat/11.1.111 |

[15] | J. A. Fleck Jr., “A Cubic Spline Method for Solving the Wave Equation of Non-linear Optics,” Journal of Computational Physics, Vol. 16, No. 4, 1974, pp. 324-341. doi:10.1016/0021-9991(74)90043-6 |

[16] | G. F. Raggett and P. D. Wilson, “A Fully Implicit Finite Difference Approximation to the One-dimensional Wave Equation Using a Cubic SplineTechnique,” Journal of the Institute of Mathematics and its Applications, Vol. 14, No. 1, 1974, pp. 75-77. doi:10.1093/imamat/14.1.75 |

[17] | D. Archer, “An O(h4) Cubic Spline Collocation Method for Qua-si-linear Parabolic Equation,” SIAM Journal of Numerical Analysis, Vol. 14, No. 4, 1977, pp. 620-637. doi:10.1137/0714042 |

[18] | P. C. Jain and B. L. Lohar, “Cubic Spline Technique for Coupled Non-linear Parabolic Equations,” Computers & Mathematics with Applications, Vol. 5, No. 3, 1979, pp. 179-195. doi:10.1016/0898-1221(79)90040-3 |

[19] | J. Rashidinia and R. Mohammadi, “Non-polynomial Cubic Spline Me-thods for the Solution of Parabolic Equations,” Interna-tional Journal of Computer Mathematics, Vol. 85, No.5, 2008, pp. 843-850. doi:10.1080/00207160701472436 |

[20] | M. K. Jain, R. K. Jain and R. K. Mohanty, “A Fourth Order Difference Method for the One-dimensional General Quasi-linear Parabolic Partial Differential Equation,” Numerical Me-thods for Partial Differential Equations, Vol. 6, No.4, 1990, pp. 311-319. doi:10.1002/num.1690060403 |

[21] | R. K. Mohanty, “An O(k2 + h4) Finite Difference Method for One Space Burg-ers’ Equation in Polar Coordinates,” Numerical Methods for Partial Differential Equations, Vol. 12, No. 5, 1996, pp. 579-583. doi:10.1002/(SICI)1098-2426(199609)12:5<579::AID-NUM3>3.0.CO;2-H |

[22] | R. K. Mohanty and M. K. Jain, “Single Cell Finite Difference Approximations of O(kh2 + h4) for ( ) for one Space Dimensional Non-linear Parabolic Equations,” Numerical Methods for Partial Differential Equations, Vol. 16, No.4, 2000, pp. 408-415. doi:10.1002/1098-2426(200007)16:4<408::AID-NUM5>3.0.CO;2-J |

[23] | C. T. Kelly, “Iterative Methods for Li-near and Non-linear Equations,” SIAM Publication, Philadelphia, 1995. |

[24] | L. A. Hageman and D. M. Young, “Applied Iterative Methods,” Dover Publication, New York, 2004. |

Journals Menu

Contact us

+1 323-425-8868 | |

customer@scirp.org | |

+86 18163351462(WhatsApp) | |

1655362766 | |

Paper Publishing WeChat |

Copyright © 2024 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.