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International Journal of Differential Equations
Volume 2015, Article ID 347864, 15 pages
http://dx.doi.org/10.1155/2015/347864
Research Article

Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm

1Department of Mathematics, USC Salkehatchie, Walterboro, SC 29488, USA
2Department of Mathematics and Statistics, Austin Peay State University, Clarksville, TN 37044, USA

Received 6 July 2015; Accepted 22 October 2015

Academic Editor: Salim Messaoudi

Copyright © 2015 F. F. Ngwane and S. N. Jator. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose a block hybrid trigonometrically fitted (BHT) method, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs), such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous two-step hybrid trigonometrically fitted method with two off-grid points. The BHT is implemented in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHT is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.