About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 823098, 12 pages
http://dx.doi.org/10.1155/2013/823098
Research Article

A Collocation Method Based on the Bernoulli Operational Matrix for Solving High-Order Linear Complex Differential Equations in a Rectangular Domain

1Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
2Center of Excellence on Modelling and Control Systems, Ferdowsi University of Mashhad, Mashhad, Iran
3Department of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa

Received 19 November 2012; Accepted 14 February 2013

Academic Editor: Douglas Anderson

Copyright © 2013 Faezeh Toutounian 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. L. Cveticanin, “Free vibration of a strong non-linear system described with complex functions,” Journal of Sound and Vibration, vol. 277, no. 4-5, pp. 815–824, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. L. Cveticanin, “Approximate solution of a strongly nonlinear complex differential equation,” Journal of Sound and Vibration, vol. 284, no. 1-2, pp. 503–512, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. G. A. Barsegian, Gamma Lines: On the Geometry of Real and Complex Functions, vol. 5 of Asian Mathematics Series, Taylor & Francis, New York, NY, USA, 2002. View at MathSciNet
  4. G. Barsegian and D. T. Lê, “On a topological description of solutions of complex differential equations,” Complex Variables, vol. 50, no. 5, pp. 307–318, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. K. Ishizaki and K. Tohge, “On the complex oscillation of some linear differential equations,” Journal of Mathematical Analysis and Applications, vol. 206, no. 2, pp. 503–517, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. Heittokangas, R. Korhonen, and J. Rättyä, “Growth estimates for solutions of linear complex differential equations,” Annales Academiæ Scientiarum Fennicæ, vol. 29, no. 1, pp. 233–246, 2004. View at Zentralblatt MATH · View at MathSciNet
  7. V. Andrievskii, “Polynomial approximation of analytic functions on a finite number of continua in the complex plane,” Journal of Approximation Theory, vol. 133, no. 2, pp. 238–244, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. V. A. Prokhorov, “On best rational approximation of analytic functions,” Journal of Approximation Theory, vol. 133, no. 2, pp. 284–296, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. Gülsu and M. Sezer, “Approximate solution to linear complex differential equation by a new approximate approach,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 636–645, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Gülsu, B. Gürbüz, Y. Öztürk, and M. Sezer, “Laguerre polynomial approach for solving linear delay difference equations,” Applied Mathematics and Computation, vol. 217, no. 15, pp. 6765–6776, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. Öztürk and M. Gülsu, “Approximate solution of linear generalized pantograph equations with variable coefficients on Chebyshev-Gauss grid,” Journal of Advanced Research in Scientific Computing, vol. 4, no. 1, pp. 36–51, 2012. View at MathSciNet
  12. M. Sezer and A. Akyuz-Das-cioglu, “A Taylor method for numerical solution of generalized pantograph equations with linear functional argument,” Journal of Computational and Applied Mathematics, vol. 200, no. 1, pp. 217–225, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. Sezer, M. Gülsu, and B. Tanay, “A Taylor collocation method for the numerical solution of complex differential equations with mixed conditions in elliptic domains,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 498–508, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Sezer and S. Yuzbasi, “A collocation method to solve higher order linear complex differential equations in rectangular domains,” Numerical Methods for Partial Differential Equations, vol. 26, no. 3, pp. 596–611, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. E. Tohidi, “Legendre approximation for solving linear HPDEs and comparison with Taylor and Bernoulli matrix methods,” Applied Mathematics, vol. 3, no. 5, pp. 410–416, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  16. S. Yuzbasi, M. Aynigul, and M. Sezer, “A collocation method using Hermite polynomials for approximate solution of pantograph equations,” Journal of the Franklin Institute, vol. 348, no. 6, pp. 1128–1139, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. Yuzbasi, “A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differential difference equations,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1691–1705, 2012. View at Publisher · View at Google Scholar
  18. S. Yuzbasi, N. Sahin, and M. Sezer, “A collocation approach for solving linear complex differential equations in rectangular domains,” Mathematical Methods in the Applied Sciences, vol. 35, no. 10, pp. 1126–1139, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. S. Yuzbasi, N. Sahin, and M. Gulso, “A collocation approach for solving a class of complex differential equations in elliptic domains,” Journal of Numerical Mathematics, vol. 19, no. 3, pp. 225–246, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  20. A. H. Bhrawy, E. Tohidi, and F. Soleymani, “A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals,” Applied Mathematics and Computation, vol. 219, no. 2, pp. 482–497, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  21. E. Tohidi, A. H. Bhrawy, and Kh. Erfani, “A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation,” Applied Mathematical Modelling, vol. 37, no. 6, pp. 4283–4294, 2013. View at Publisher · View at Google Scholar
  22. E. Tohidi, “Bernoulli matrix approach for solving two dimensional linear hyperbolic partial differential equations with constant coefficients,” Journal of Computational and Applied Mathematics, vol. 2, no. 4, pp. 136–139, 2012.
  23. O. R. N. Samadi and E. Tohidi, “The spectral method for solving systems of Volterra integral equations,” Journal of Applied Mathematics and Computing, vol. 40, no. 1-2, pp. 477–497, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  24. E. Tohidi and O. R. N. Samadi, “Optimal control of nonlinear Volterra integral equations via Legendre polynomials,” IMA Journal of Mathematical Control and Information, vol. 30, no. 1, pp. 67–83, 2013. View at Publisher · View at Google Scholar
  25. S. Mashayekhi, Y. Ordokhani, and M. Razzaghi, “Hybrid functions approach for nonlinear constrained optimal control problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1831–1843, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. M. Sezer and M. Gülsu, “Approximate solution of complex differential equations for a rectangular domain with Taylor collocation method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 844–851, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. V. I. Krylov, Approximate Calculation of Integrals, Dover, Mineola, NY, USA, 1962. View at MathSciNet
  28. F. A. Costabile and F. Dell'Accio, “Expansion over a rectangle of real functions in Bernoulli polynomials and applications,” BIT Numerical Mathematics, vol. 41, no. 3, pp. 451–464, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet