A Comparison of Some Splines-Based Methods for the One-dimensional Heat Equation

Authors: Joan Goh, Ahmad Abd. Majid, Ahmad Izani Md. Ismail

Abstract:

In this paper, collocation based cubic B-spline and extended cubic uniform B-spline method are considered for solving one-dimensional heat equation with a nonlocal initial condition. Finite difference and θ-weighted scheme is used for time and space discretization respectively. The stability of the method is analyzed by the Von Neumann method. Accuracy of the methods is illustrated with an example. The numerical results are obtained and compared with the analytical solutions.

Keywords: Heat equation, Collocation based, Cubic Bspline, Extended cubic uniform B-spline.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1060545

References:

[1] H. Caglar, M. Ozer, and N. Caglar. The numerical solution of the one-dimensional heat equation by using third degree B-spline functions. Chaos, Solitons & Fractals, 38(4):1197-1201, 2008.
[2] I. Dag, D. Irk, and B. Saka. A numerical solution of the Burgers- equation using cubic B-splines. Applied Mathematics and Computation, 163(1):199-211, 2005.
[3] C. Deboor. A Practical Guide to Splines. Springer-Verlag, 1978.
[4] M. Dehghan. A finite difference method for a non-local boundary value problem for two-dimensional heat equation. Applied Mathematics and Computation, 112(1):133-142, 2000.
[5] A. Gorguis and W. K. Benny Chan. Heat equation and its comparative solutions. Computers & Mathematics with Applications, 55(12):2973- 2980, 2008.
[6] X. L. Han and S. J. Liu. An extension of the cubic uniform B-spline curves. Journal of Computer Aided Design and Computer Graphics, 15(5):576-578, 2003 (in chinese).
[7] M. Kumar and Y. Gupta. Methods for solving singular boundary value problems using splines: a review. Journal of Applied Mathematics and Computing, 32(1):265-278, 2010.
[8] A. Mohebbi and M. Dehghan. High-order compact solution of the onedimensional heat and advection-diffusion equations. Applied Mathematical Modelling, In Press, Corrected Proof, 2010.
[9] P. M. Prenter. Splines and Variational Methods. John Wiley & Sons, 1989.
[10] H. W. Sun and J. Zhang. A high-order compact boundary value method for solving one-dimensional heat equations. Numerical Methods for Partial Differential Equations, 19(6):846-857, 2003.
[11] M. Tatari and M. Dehghan. A method for solving partial differential equations via radial basis functions: Application to the heat equation. Engineering Analysis with Boundary Elements, 34(3):206-212, 2010.
[12] G. Xu and G. Z. Wang. Extended cubic uniform B-spline and ╬▒-Bspline. Acta Automat. Sinica, 34(8):980-983, 2008.