Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014, Article ID 340752, 9 pages
http://dx.doi.org/10.1155/2014/340752
Research Article

Numerical Algorithm Based on Haar-Sinc Collocation Method for Solving the Hyperbolic PDEs

1Department of Computer Engineering, Islamic Azad University, Science and Research Branch, Tehran, Iran
2Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Shahed University, Tehran, Iran

Received 13 April 2014; Revised 20 October 2014; Accepted 27 October 2014; Published 16 November 2014

Academic Editor: Zacharias Anastassi

Copyright © 2014 A. Pirkhedri et al. 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.

Linked References

  1. M. Lakestani and B. N. Saray, “Numerical solution of telegraph equation using interpolating scaling functions,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 1964–1972, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. M. S. El-Azab and M. El-Gamel, “A numerical algorithm for the solution of telegraph equations,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 757–764, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus, London, UK, 1993.
  4. G. Roussy and J. A. Pearcy, Foundations and Industrial Applications of Microwaves and Radio Frequency Fields, John Wiley & Sons, New York, NY, USA, 1995.
  5. J. Biazar and M. Eslami, “Analytic solution for Telegraph equation by differential transform method,” Physics Letters A, vol. 374, no. 29, pp. 2904–2906, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. S. Momani, “Analytic and approximate solutions of the space- and time-fractional telegraph equations,” Applied Mathematics and Computation, vol. 170, no. 2, pp. 1126–1134, 2005. View at Publisher · View at Google Scholar · View at Scopus
  7. Siraj-ul-Islam, B. Sarler, and R. Vertnik, “Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations,” Applied Numerical Mathematics, vol. 67, pp. 136–151, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. J. Biazar, H. Ebrahimi, and Z. Ayati, “An approximation to the solution of telegraph equation by variational iteration method,” Numerical Methods for Partial Differential Equations, vol. 25, no. 4, pp. 797–801, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. A. Yıldırım, “He's homotopy perturbation method for solving the space- and time-fractional telegraph equations,” International Journal of Computer Mathematics, vol. 87, no. 13, pp. 2998–3006, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. S. Kumar, “A new analytical modelling for fractional telegraph equation via Laplace transform,” Applied Mathematical Modelling, vol. 38, no. 13, pp. 3154–3163, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. S. Kumar, “An analytical algorithm for nonlinear fractional Fornberg-Whitham equation arising in wave breaking based on a new iterative method,” Alexandria Engineering Journal, vol. 53, no. 1, pp. 225–231, 2014. View at Publisher · View at Google Scholar · View at Scopus
  12. S. Kumar, A. Yildirim, Y. Khan, and L. Wei, “A fractional model of the diffusion equation and its analytical solution using Laplace transform,” Scientia Iranica, vol. 19, no. 4, pp. 1117–1123, 2012. View at Publisher · View at Google Scholar · View at Scopus
  13. S. Kumar, H. Kocak, and A. Yildirim, “A fractional model of gas dynamics equations and its analytical approximate solution using laplace transform,” Zeitschrift fur Naturforschung, vol. 67, no. 6-7, pp. 389–396, 2012. View at Publisher · View at Google Scholar · View at Scopus
  14. S. Kumar, “Numerical computation of time-fractional equation arising in solid state physics and circuit theory,” Zeitschrift für Naturforschung, vol. 68a, pp. 1–8, 2013. View at Publisher · View at Google Scholar
  15. L. Wei, H. Dai, D. Zhang, and Z. Si, “Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation,” Calcolo, vol. 51, no. 1, pp. 175–192, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. J. Chen, F. Liu, and V. Anh, “Analytical solution for the time-fractional telegraph equation by the method of separating variables,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1364–1377, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. H. R. Karimi and B. Lohmann, “Haar wavelet-based robust optimal control for vibration reduction of vehicle engine-body system,” Electrical Engineering, vol. 89, no. 6, pp. 469–478, 2007. View at Publisher · View at Google Scholar · View at Scopus
  18. H. R. Karimi, “A computational method for optimal control problem of time-varying state-delayed systems by Haar wavelets,” International Journal of Computer Mathematics, vol. 83, no. 2, pp. 235–246, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. H. R. Karimi, P. Jabedar Maralani, B. Moshiri, and B. Lohmann, “Numerically efficient approximations to the optimal control of linear singularly perturbed systems based on Haar wavelets,” International Journal of Computer Mathematics, vol. 82, no. 4, pp. 495–507, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. H. R. Karimi, B. Moshiri, B. Lohmann, and P. J. Maralani, “Haar wavelet-based approach for optimal control of second-order linear systems in time domain,” Journal of Dynamical and Control Systems, vol. 11, no. 2, pp. 237–252, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. F. Stenger, “Integration formulae based on the trapezoidal formula,” Journal of the Institute of Mathematics and its Applications, vol. 12, pp. 103–114, 1973. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. K. Parand, M. Dehghan, and A. Pirkhedri, “The sinc-collocation method for solving the Thomas-Fermi equation,” Journal of Computational and Applied Mathematics, vol. 237, no. 1, pp. 244–252, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. K. Parand and A. Pirkhedri, “Sinc-collocation method for solving astrophysics equations,” New Astronomy, vol. 15, no. 6, pp. 533–537, 2010. View at Publisher · View at Google Scholar · View at Scopus
  24. K. Parand, M. Dehghan, and A. Pirkhedri, “Sinc-collocation method for solving the Blasius equation,” Physics Letters A, vol. 373, no. 44, pp. 4060–4065, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. C. F. Chen and C. H. Hsiao, “Haar wavelet method for solving lumped and distributed-parameter systems,” IEE Proceedings Control Theory and Applications, vol. 14, no. 1, pp. 87–94, 1997. View at Publisher · View at Google Scholar
  26. M. Razzaghi and Y. Ordokhani, “Solution of differential equations via rationalized Haar functions,” Mathematics and Computers in Simulation, vol. 56, no. 3, pp. 235–246, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. J. Lund and K. L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, Pa, USA, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  28. Siraj-ul-Islam, B. Šarler, I. Aziz, and Fazal-i-Haq, “Haar wavelet collocation method for the numerical solution of boundary layer fluid flow problems,” International Journal of Thermal Sciences, vol. 50, no. 5, pp. 686–697, 2011. View at Publisher · View at Google Scholar · View at Scopus
  29. Siraj-ul-Islam, I. Aziz, and B. Šarler, “The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1577–1590, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. L. de Su, Z. W. Jiang, and T. S. Jiang, “Numerical solution for a kind of nonlinear telegraph equations using radial basis functions,” Communications in Computer and Information Science, vol. 391, pp. 140–149, 2013. View at Publisher · View at Google Scholar · View at Scopus
  31. Y. D. Shang, “Explicit and exact solutions for a class of nonlinear wave equations,” Acta Mathematicae Applicatae Sinica, vol. 23, no. 1, pp. 21–30, 2000. View at Google Scholar · View at MathSciNet
  32. A. Saadatmandi, M. Dehghan, and M.-R. Azizi, “The sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4125–4136, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. Z. Mao, A. Xiao, Z. Yu, and L. Shi, “Sinc-chebyshev collocation method for a class of fractional diffusion-wave equations,” The Scientific World Journal, vol. 2014, Article ID 143983, 7 pages, 2014. View at Publisher · View at Google Scholar