Cubic Trigonometric B-spline Approach to Numerical Solution of Wave Equation
The generalized wave equation models various
problems in sciences and engineering. In this paper, a new three-time
level implicit approach based on cubic trigonometric B-spline for the
approximate solution of wave equation is developed. The usual finite
difference approach is used to discretize the time derivative while
cubic trigonometric B-spline is applied as an interpolating function in
the space dimension. Von Neumann stability analysis is used to
analyze the proposed method. Two problems are discussed to exhibit
the feasibility and capability of the method. The absolute errors and
maximum error are computed to assess the performance of the
proposed method. The results were found to be in good agreement
with known solutions and with existing schemes in literature.
[1] F. Shakeri and M. Dehghan, "The method of lines for solution of the
one-dimensional wave equation subject to an integral conservation
condition,” Computer & Mathematics with Applications, vol. 56, no. 9,
pp. 2175-2188, 2008.
[2] M. Dehghan, "On the solution of an initial-boundary value problem that
combines Neumann and integral condition for the wave equation,”
Numerical Methods for Partial Differential Equations, vol. 21, no. 1, pp.
24-40, 2005.
[3] W. T. Ang, "A numerical method for the wave equation subject to a
non-local conservation condition,” Applied Numerical Mathematics, vol.
56, pp. 1054-1060, 2006.
[4] M. Dehghan and M. Lakestani, "The Use of Cubic B-Spline Scaling
Functions for Solving the One-dimensional Hyperbolic Equation with a
Nonlocal Conservation Condition,” Numerical Methods for Partial
Differential Equation, vol. 23, pp. 1277-1289, 2007.
[5] S. A. Khuri and A. Sayfy, "A spline collocation approach for a
generalized wave equation subject to non-local conservation condition,”
Applied Mathematics and Computation, vol. 217, no. 8, pp. 3993-4001,
2010.
[6] J. Goh, A. Abd. Majid and A. I. Md Ismail, "Numerical method using
cubic B-spline for the heat and wave equation,” Computer &
Mathematics with Application, vol. 62, no. 12, pp. 4492-4498, 2011.
[7] I. Dag, D. Irk and B. Saka, "A numerical solution of the Burgers’
equation using cubic B-splines,”Applied Mathematics and Computation,
vol. 163, no. 1, pp. 199-211, 2005.
[8] H. Caglar, N. Caglar and K. Elfauturi, "B-spline interpolation compared
with finite difference, finite element and finite volume methods which
applied to two-point boundary value problems,” Applied Mathematics
and Computation, vol. 175, no. 1, pp. 72-79, 2006.
[9] S. S. Siddiqi and S. Arshed, "Quintic B-spline for the numerical solution
of the good Boussinesq equation,” Journal of Egyption Mathematical
Society, to be published.
[10] M. Dehghan and A. Shokri, "A meshless method for numerical solution
of the one-dimensional wave equation with an integral condition using
radial basis functions,” Numerical Algorithms, vol. 52, no. 3, pp. 461-
477, 2009.
[1] F. Shakeri and M. Dehghan, "The method of lines for solution of the
one-dimensional wave equation subject to an integral conservation
condition,” Computer & Mathematics with Applications, vol. 56, no. 9,
pp. 2175-2188, 2008.
[2] M. Dehghan, "On the solution of an initial-boundary value problem that
combines Neumann and integral condition for the wave equation,”
Numerical Methods for Partial Differential Equations, vol. 21, no. 1, pp.
24-40, 2005.
[3] W. T. Ang, "A numerical method for the wave equation subject to a
non-local conservation condition,” Applied Numerical Mathematics, vol.
56, pp. 1054-1060, 2006.
[4] M. Dehghan and M. Lakestani, "The Use of Cubic B-Spline Scaling
Functions for Solving the One-dimensional Hyperbolic Equation with a
Nonlocal Conservation Condition,” Numerical Methods for Partial
Differential Equation, vol. 23, pp. 1277-1289, 2007.
[5] S. A. Khuri and A. Sayfy, "A spline collocation approach for a
generalized wave equation subject to non-local conservation condition,”
Applied Mathematics and Computation, vol. 217, no. 8, pp. 3993-4001,
2010.
[6] J. Goh, A. Abd. Majid and A. I. Md Ismail, "Numerical method using
cubic B-spline for the heat and wave equation,” Computer &
Mathematics with Application, vol. 62, no. 12, pp. 4492-4498, 2011.
[7] I. Dag, D. Irk and B. Saka, "A numerical solution of the Burgers’
equation using cubic B-splines,”Applied Mathematics and Computation,
vol. 163, no. 1, pp. 199-211, 2005.
[8] H. Caglar, N. Caglar and K. Elfauturi, "B-spline interpolation compared
with finite difference, finite element and finite volume methods which
applied to two-point boundary value problems,” Applied Mathematics
and Computation, vol. 175, no. 1, pp. 72-79, 2006.
[9] S. S. Siddiqi and S. Arshed, "Quintic B-spline for the numerical solution
of the good Boussinesq equation,” Journal of Egyption Mathematical
Society, to be published.
[10] M. Dehghan and A. Shokri, "A meshless method for numerical solution
of the one-dimensional wave equation with an integral condition using
radial basis functions,” Numerical Algorithms, vol. 52, no. 3, pp. 461-
477, 2009.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:68106", author = "Shazalina Mat Zin and Ahmad Abd. Majid and Ahmad Izani Md. Ismail and Muhammad Abbas", title = "Cubic Trigonometric B-spline Approach to Numerical Solution of Wave Equation", abstract = "The generalized wave equation models various
problems in sciences and engineering. In this paper, a new three-time
level implicit approach based on cubic trigonometric B-spline for the
approximate solution of wave equation is developed. The usual finite
difference approach is used to discretize the time derivative while
cubic trigonometric B-spline is applied as an interpolating function in
the space dimension. Von Neumann stability analysis is used to
analyze the proposed method. Two problems are discussed to exhibit
the feasibility and capability of the method. The absolute errors and
maximum error are computed to assess the performance of the
proposed method. The results were found to be in good agreement
with known solutions and with existing schemes in literature.
", keywords = "Collocation method, Cubic trigonometric B-spline,
Finite difference, Wave equation.", volume = "8", number = "10", pages = "1306-5", }
