A Comparison of Some Splines-Based Methods for the One-dimensional Heat Equation
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:54261", author = "Joan Goh and Ahmad Abd. Majid and Ahmad Izani Md. Ismail", title = "A Comparison of Some Splines-Based Methods for the One-dimensional Heat Equation", 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.", volume = "4", number = "10", pages = "1364-4", }