Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 325473, 17 pages
http://dx.doi.org/10.1155/2012/325473
Research Article

Formulation and Solution of th-Order Derivative Fuzzy Integrodifferential Equation Using New Iterative Method with a Reliable Algorithm

Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt

Received 11 July 2012; Revised 9 September 2012; Accepted 10 September 2012

Academic Editor: Fazlollah Soleymani

Copyright © 2012 A. A. Hemeda. 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. D. Dubois and H. Prade, “Operations on fuzzy numbers,” International Journal of Systems Science, vol. 9, no. 6, pp. 613–626, 1978. View at Google Scholar · View at Zentralblatt MATH
  2. B. Asady, S. Abbasbandy, and M. Alavi, “Fuzzy general linear systems,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 34–40, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. M. S. Hashemi, M. K. Mirnia, and S. Shahmorad, “Solving fuzzy linear systems by using the Schur complement when coefficient matrix is an M-matrix,” Iranian Journal of Fuzzy Systems, vol. 5, no. 3, pp. 15–29, 2008. View at Google Scholar
  4. S. Abbasbandy, T. A. Viranloo, Ó. López-Pouso, and J. J. Nieto, “Numerical methods for fuzzy differential inclusions,” Computers & Mathematics with Applications, vol. 48, no. 10-11, pp. 1633–1641, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. S. Abbasbandy, J. J. Nieto, and M. Alavi, “Tuning of reachable set in one dimensional fuzzy differential inclusions,” Chaos, Solitons and Fractals, vol. 26, no. 5, pp. 1337–1341, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. S. Abbasbandy, E. Babolian, and M. Alavi, “Numerical method for solving linear Fredholm fuzzy integral equations of the second kind,” Chaos, Solitons and Fractals, vol. 31, no. 1, pp. 138–146, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. E. Babolian, H. Sadeghi Goghary, and S. Abbasbandy, “Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method,” Applied Mathematics and Computation, vol. 161, no. 3, pp. 733–744, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. M. Friedman, M. Ma, and A. Kandel, “Numerical solutions of fuzzy differential and integral equations,” Fuzzy Sets and Systems, vol. 106, no. 1, pp. 35–48, 1999, Fuzzy modeling and dynamics. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. Y. C. Kwun, M. J. Kim, B. Y. Lee, and J. H. Park, “Existence of solu-tions for the semilinear fuzzy integro-differential equations using by successive iteration,” Journal of Korean Institute of Intelligent Systems, vol. 18, pp. 543–548, 2008. View at Google Scholar
  10. J. H. Park, J. S. Park, and Y. C. Kwun, Controllabillty for the Semilinear Fuzzy Integro-Differential Equations with Nonlocal conditions, Fuzzy Systems and Knowledge Discovery, vol. 4223 of Lecture Notes in Computer Science, Springer, Berlin, Germany.
  11. A. V. Plotnikov and A. V. Tumbrukaki, “Integrodifferential inclusions with Hukuhara's derivative,” Nelīnīĭnī Kolivannya, vol. 8, no. 1, pp. 80–88, 2005. View at Publisher · View at Google Scholar
  12. S. Abbasbandy and M. S. Hashemi, “Fuzzy integro-differential equations: formulation and solution using the variational iteration method,” Nonlinear Science Letters A, vol. 1, no. 4, pp. 413–418, 2010. View at Google Scholar
  13. P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, River Edge, NJ, USA, 1994.
  14. D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, vol. 144, Academic Press, New York, NY, USA, 1980.
  15. C. M. Tavassoli Kajani, M. Ghasemi, and E. Babolian, “Numerical solution of linear integro-differential equation by using sine-cosine wavelets,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 569–574, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. J. Saberi-Nadjafi and M. Tamamgar, “The variational iteration method: a highly promising method for solving the system of integro-differential equations,” Computers & Mathematics with Applications, vol. 56, no. 2, pp. 346–351, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. X. Shang and D. Han, “Application of the variational iteration method for solving nth-order integro-differential equations,” Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1442–1447, 2010. View at Publisher · View at Google Scholar · View at Scopus
  18. A. A. Hemeda, “Variational iteration method for solving wave equation,” Computers & Mathematics with Applications, vol. 56, no. 8, pp. 1948–1953, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. A. A. Hemeda, “Variational iteration method for solving non-linear partial differential equations,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1297–1303, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. V. Daftardar-Gejji and H. Jafari, “An iterative method for solving nonlinear functional equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 753–763, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. S. Bhalekar and V. Daftardar-Gejji, “New iterative method: application to partial differential equations,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 778–783, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. G. Adomain, Solving Frontier Problems of Physics: the Decomposition Method, Kluwer, 1994.
  23. J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. A. F. Elsayed, “Comparison between variational iteration method and homotopy perturbation method for thermal diffusion and diffusion thermo effects of thixotropic fluid through biological tissues with laser radiation existence,” Applied Mathematical Modelling. In press. View at Publisher · View at Google Scholar
  25. S. Haji Ghasemi, “Fuzzy Fredholm-Volterra integral equations and existance and uniqueness of solution of them,” Australian Journal of Basic and Applied Sciences, vol. 5, no. 4, pp. 1–8, 2011. View at Google Scholar · View at Scopus
  26. T. Allahviranloo and M. Ghanbari, “Solving fuzzy linear systems by homotopy perturbation method,” International Journal of Computational Cognition, vol. 8, pp. 27–30, 2010. View at Google Scholar
  27. H.-K. Liu, “On the solution of fully fuzzy linear systems,” International Journal of Computational and Mathematical Sciences, vol. 4, no. 1, pp. 29–33, 2010. View at Google Scholar
  28. B. Bede and S. G. Gal, “Solutions of fuzzy differential equations based on generalized differentiability,” Communications in Mathematical Analysis, vol. 9, no. 2, pp. 22–41, 2010. View at Google Scholar · View at Zentralblatt MATH
  29. Y. Chalco-Cano and H. Román-Flores, “Comparation between some approaches to solve fuzzy differential equations,” Fuzzy Sets and Systems, vol. 160, no. 11, pp. 1517–1527, 2009. View at Publisher · View at Google Scholar · View at Scopus
  30. Y. Chalco-Cano, H. Román-Flores, and M. D. Jiménez-Gamero, “Generalized derivative and π-derivative for set-valued functions,” Information Sciences. An International Journal, vol. 181, no. 11, pp. 2177–2188, 2011. View at Publisher · View at Google Scholar
  31. L. Stefanini, “A generalization of Hukuhara difference and division for interval and fuzzy arithmetic,” Fuzzy Sets and Systems, vol. 161, no. 11, pp. 1564–1584, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. L. Stefanini and B. Bede, “Some notes on generalized Hukuhara differentiability of interval-valued functions and interval differential equations,” Working Paper, University of Urbino, 2012. View at Google Scholar
  33. B. Bede and L. Stefanini, “Generalized differentiability of fuzzy-valued functions,” Working Paper, University of Urbino, 2012. View at Google Scholar
  34. A. A. Hemeda, “New iterative method: application to nth-order integro-differential equations,” International Mathematical Forum, vol. 7, no. 47, pp. 2317–2332, 2012. View at Google Scholar
  35. V. Daftardar-Gejji and S. Bhalekar, “Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1801–1809, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. S. Bhalekar and V. Daftardar-Gejji, “New iterative method: application to partial differential equations,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 778–783, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. S. Bhalekar and V. Daftardar-Gejji, “Solving evolution equations using a new iterative method,” Numerical Methods for Partial Differential Equations, vol. 26, no. 4, pp. 906–916, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  38. H. Jafari, S. Seifi, A. Alipoor, and M. Zabihi, “An iterative method for solving linear and nonlinear fractional diffusion-wave equation,” International e-Journal of Numerical Analysis and Related Topics, vol. 3, pp. 20–32, 2009. View at Google Scholar
  39. S. Bhalekar and V. Daftardar-Gejji, “Convergence of the new iterative method,” International Journal of Differential Equations, vol. 2011, Article ID 989065, 10 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  40. O. S. Fard and M. Sanchooli, “Two successive schemes for numerical solution of linear fuzzy Fredholm integral equations of the second kind,” Australian Journal of Basic and Applied Sciences, vol. 4, no. 5, pp. 817–825, 2010. View at Google Scholar · View at Scopus
  41. V. Daftardar-Gejji and S. Bhalekar, “Solving fractional diffusion-wave equations using a new iterative method,” Fractional Calculus & Applied Analysis, vol. 11, no. 2, pp. 193–202, 2008. View at Google Scholar · View at Zentralblatt MATH