About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 694043, 11 pages
http://dx.doi.org/10.1155/2013/694043
Research Article

On Solution of Fredholm Integrodifferential Equations Using Composite Chebyshev Finite Difference Method

Department of Mathematics, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Received 25 February 2013; Accepted 26 May 2013

Academic Editor: Mustafa Bayram

Copyright © 2013 Z. Pashazadeh Atabakan 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. A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2nd edition, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. Dehghan and R. Salehi, “The numerical solution of the non-linear integro-differential equations based on the meshless method,” Journal of Computational and Applied Mathematics, vol. 236, no. 9, pp. 2367–2377, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. I. Berenguer, M. V. Fernández Muñoz, A. I. Garralda-Guillem, and M. Ruiz Galán, “A sequential approach for solving the Fredholm integro-differential equation,” Applied Numerical Mathematics, vol. 62, no. 4, pp. 297–304, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Khani and S. Shahmorad, “An operational approach with Pade approximant for the numerical solution of non-linear Fredholm integro-differential equations,” Sharif University of Technology Scientia Iranica, vol. 19, pp. 1691–1698, 2011.
  5. M. Dehghan and A. Saadatmandi, “Chebyshev finite difference method for Fredholm integro-differential equation,” International Journal of Computer Mathematics, vol. 85, no. 1, pp. 123–130, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. Yalçinbaş, M. Sezer, and H. H. Sorkun, “Legendre polynomial solutions of high-order linear Fredholm integro-differential equations,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 334–349, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. Z. P. Atabakan, A. Kılıçman, and A. K. Nasab, “On spectral homotopy analysis method for solving linear Volterra and Fredholm integro-differential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 960289, 16 pages, 2012. View at Publisher · View at Google Scholar
  8. Z. P. Atabakan, A. K. Nasab, A. Kılıçman, and K. Z. Eshkuvatov, “Numerical solution of nonlinear Fredholm integro-differential equations using Spectral Homotopy Analysis method,” Mathematical Problems in Engineering, vol. 2013, Article ID 674364, 9 pages, 2013. View at Publisher · View at Google Scholar
  9. M. Lakestani, M. Razzaghi, and M. Dehghan, “Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations,” Mathematical Problems in Engineering, Article ID 96184, 12 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. H. R. Marzban and S. M. Hoseini, “A composite Chebyshev finite difference method for nonlinear optimal control problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, pp. 1347–1361, 2012.
  11. H. R. Marzban and S. M. Hoseini, “Solution of linear optimal control problems with time delay using a composite Chebyshev finite difference method,” Optimal Control Applications and Methods, vol. 34, no. 3, pp. 253–274, 2013. View at Publisher · View at Google Scholar
  12. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, Germany, 1988. View at MathSciNet
  13. R. G. Voigt, D. Gottlieb, and M. Y. Hussaini, Spectral Methods for Partial Differential Equations, SIAM, Philadelphia, Pa, USA, 1984.
  14. L. Fox and I. B. Parker, Chebyshev Polynomials in Numerical Analysis, Clarendon Press, Oxford, UK, 1968. View at MathSciNet
  15. S. H. Behiry and H. Hashish, “Wavelet methods for the numerical solution of Fredholm integro-differential equations,” International Journal of Applied Mathematics, vol. 11, no. 1, pp. 27–35, 2002. View at Zentralblatt MATH · View at MathSciNet
  16. S. M. Hosseini and S. Shahmorad, “Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method,” Applied Mathematical Modelling, vol. 29, no. 11, pp. 1005–1021, 2005. View at Publisher · View at Google Scholar · View at Scopus
  17. H. Danfu and S. Xufeng, “Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration,” Applied Mathematics and Computation, vol. 194, no. 2, pp. 460–466, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. P. Darania and A. Ebadian, “A method for the numerical solution of the integro-differential equations,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 657–668, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. E. Yusufoǧlu (Agadjanov), “Improved homotopy perturbation method for solving Fredholm type integro-differential equations,” Chaos, Solitons and Fractals, vol. 41, no. 1, pp. 28–37, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. Z. H. Jiang and W. Schaufelberger, Block Pulse Functions and Their Applications in Control Systems, vol. 179, Springer, Berlin, Germany, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  21. K. G. Beauchamp, Applications of Walsh and Related Functions with an Introduction to Sequency Theory, Academic Press, London, UK, 1984. View at MathSciNet
  22. A. Deb, G. Sarkar, and S. K. Sen, “Block pulse functions, the most fundamental of all piecewise constant basis functions,” International Journal of Systems Science, vol. 25, no. 2, pp. 351–363, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. G. P. Rao, Piecewise Constant Orthogonal Functions and Their Application to Systems and Control, Springer, New York, NY, USA, 1983. View at MathSciNet
  24. E. Babolian and Z. Masouri, “Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions,” Journal of Computational and Applied Mathematics, vol. 220, no. 1-2, pp. 51–57, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. K. Maleknejad and Y. Mahmoudi, “Numerical solution of linear Fredholm integral equation by using hybrid Taylor and block-pulse functions,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 799–806, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. K. Maleknejad and K. Mahdiani, “Solving nonlinear mixed Volterra-Fredholm integral equations with two dimensional block-pulse functions using direct method,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 9, pp. 3512–3519, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. K. Maleknejad, S. Sohrabi, and B. Baranji, “Application of 2D-BPFs to nonlinear integral equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 3, pp. 527–535, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. K. Maleknejad, M. Shahrezaee, and H. Khatami, “Numerical solution of integral equations system of the second kind by block-pulse functions,” Applied Mathematics and Computation, vol. 166, no. 1, pp. 15–24, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. K. Maleknejad and M. T. Kajani, “Solving second kind integral equations by Galerkin methods with hybrid Legendre and block-pulse functions,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 623–629, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. A. Kılıçman and Z. A. A. Al Zhour, “Kronecker operational matrices for fractional calculus and some applications,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 250–265, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. C. W. Clenshaw and A. R. Curtis, “A method for numerical integration on an automatic computer,” Numerische Mathematik, vol. 2, pp. 197–205, 1960. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. E. M. E. Elbarbary and M. El-Kady, “Chebyshev finite difference approximation for the boundary value problems,” Applied Mathematics and Computation, vol. 139, no. 2-3, pp. 513–523, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. E. M. E. Elbarbary, “Chebyshev finite difference method for the solution of boundary-layer equations,” Applied Mathematics and Computation, vol. 160, no. 2, pp. 487–498, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. S. Yalçınbaş and M. Sezer, “The approximate solution of high-order linear Volterra-Fredholm integro-differential equations in terms of Taylor polynomials,” Applied Mathematics and Computation, vol. 112, no. 2-3, pp. 291–308, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. A. Akyüz-Daşcioğlu and M. Sezer, “A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations in the most general form,” International Journal of Computer Mathematics, vol. 84, no. 4, pp. 527–539, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet