**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**33041

##### Cubic Trigonometric B-spline Approach to Numerical Solution of Wave Equation

**Authors:**
Shazalina Mat Zin,
Ahmad Abd. Majid,
Ahmad Izani Md. Ismail,
Muhammad Abbas

**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.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1096411

**References:**

[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.