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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 178378, 13 pages
http://dx.doi.org/10.1155/2013/178378
Research Article

On Solutions of Linear Fractional Differential Equations with Uncertainty

1Department of Electronic and Communications, Faculty of Engineering, Izmir University, Izmir, Turkey
2Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
3Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran
4Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
5Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey
6Institute of Space Sciences, Magurele-Bucharest, Bucharest, RO 7690, Romania

Received 8 June 2013; Revised 29 August 2013; Accepted 30 August 2013

Academic Editor: Ali H. Bhrawy

Copyright © 2013 T. Allahviranloo 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. Ahmadian, M. Suleiman, S. Salahshour, and D. Baleanu, “A Jacobi operational matrix for solving a fuzzy linear fractional differential equation,” Advances in Difference Equations, vol. 2013, article 104, pp. 1–29, 2013. View at Publisher · View at Google Scholar
  2. T. Allahviranloo and M. B. Ahmadi, “Fuzzy Laplace transforms,” Soft Computing, vol. 14, no. 3, pp. 235–243, 2010. View at Publisher · View at Google Scholar · View at Scopus
  3. T. Allahviranloo, S. Salahshour, and S. Abbasbandy, “Explicit solutions of fractional differential equations with uncertainty,” Soft Computing, vol. 16, no. 2, pp. 297–302, 2012. View at Publisher · View at Google Scholar · View at Scopus
  4. T. Allahviranloo, S. Abbasbandy, S. Salahshour, and A. Hakimzadeh, “A new method for solving fuzzy linear differential equations,” Computing, vol. 92, no. 2, pp. 181–197, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  5. T. Allahviranloo and S. Salahshour, “A new approach for solving first order fuzzy differential equations,” Communications in Computer and Information Science, vol. 81, part 5, part 6, pp. 522–531, 2010.
  6. J. Li, A. Zhao, and J. Yan, “The Cauchy problem of fuzzy differential equations under generalized differentiability,” Fuzzy Sets and Systems, vol. 200, pp. 1–24, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  7. S. Salahshour and T. Allahviranloo, “Application of fuzzy differential transform method for solving fuzzy Volterra integral equations,” Applied Mathematical Modelling, vol. 37, no. 3, pp. 1016–1027, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  8. S. Salahshour and A. Allahviranloo, “Applications of fuzzy Laplace transforms,” Soft Computing, vol. 17, no. 1, pp. 145–178, 2013. View at Publisher · View at Google Scholar
  9. A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an unbounded domain,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 580–586, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  10. Y. I. Babenko, Heat and Mass Transfer, Chemia, Leningrad, Russia, 1986.
  11. R. L. Bagley, “On the fractional order initial value problem and its engineering applications,” in Fractional Calculus and Its Applications, K. Nishimoto, Ed., pp. 12–20, College of Engineering, Nihon University, Tokyo, Japan, 1990.
  12. H. Beyer and S. Kempfle, “Definition of physically consistent damping laws with fractional derivatives,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 75, no. 8, pp. 623–635, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  13. K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  14. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  15. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherland, 2006. View at MathSciNet
  16. V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific, Cambridge, UK, 2009.
  17. V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, vol. 6 of Series in Mathematical Analysis and Applications, Taylor & Francis, London, UK, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  18. V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  19. R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 6, pp. 2859–2862, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  20. B. Bede and S. G. Gal, “Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 581–599, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  21. B. Bede, I. J. Rudas, and A. L. Bencsik, “First order linear fuzzy differential equations under generalized differentiability,” Information Sciences, vol. 177, no. 7, pp. 1648–1662, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  22. I. Perfilieva, “Fuzzy transforms: theory and applications,” Fuzzy Sets and Systems, vol. 157, no. 8, pp. 993–1023, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  23. I. Perfilieva, H. De Meyer, B. De Baets, and D. Plšková, “Cauchy problem with fuzzy initial condition and its approximate solution with the help of fuzzy transform,” in Proceedings of IEEE International Conference on Fuzzy Systems (FUZZ '08), pp. 2285–2290, June 2008. View at Publisher · View at Google Scholar · View at Scopus
  24. S. Salahshour, T. Allahviranloo, S. Abbasbandy, and D. Baleanu, “Existence and uniqueness results for fractional differential equations with uncertainty,” Advances in Difference Equations, vol. 2012, article 112, pp. 1–12, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  25. M. Mazandarani and A. V. Kamyad, “Modified fractional Euler method for solving fuzzy fractional initial value problem,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 1, pp. 12–21, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  26. R. P. Agarwal, S. Arshad, D. O'Regan, and V. Lupulescu, “Fuzzy fractional integral equations under compactness type condition,” Fractional Calculus and Applied Analysis, vol. 15, no. 4, pp. 572–590, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  27. J. Xu, Z. Liao, and Z. Hu, “A class of linear differential dynamical systems with fuzzy initial condition,” Fuzzy Sets and Systems, vol. 158, no. 21, pp. 2339–2358, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  28. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  29. M. Ma, M. Friedman, and A. Kandel, “Numerical solutions of fuzzy differential equations,” Fuzzy Sets and Systems, vol. 105, no. 1, pp. 133–138, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  30. H.-J. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers, Boston, Mass, USA, 2nd edition, 1992. View at MathSciNet
  31. M. L. Puri and D. A. Ralescu, “Fuzzy random variables,” Journal of Mathematical Analysis and Applications, vol. 114, no. 2, pp. 409–422, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  32. H.-C. Wu, “The improper fuzzy Riemann integral and its numerical integration,” Information Sciences, vol. 111, no. 1–4, pp. 109–137, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  33. G. A. Anastassiou, Fuzzy Mathematics: Approximation Theory, vol. 251 of Studies in Fuzziness and Soft Computing, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  34. G. A. Anastassiou, Intelligent Mathematics: Computational Analysis, vol. 5 of Intelligent Systems Reference Library, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  35. S. Salahshour, T. Allahviranloo, and S. Abbasbandy, “Solving fuzzy fractional differential equations by fuzzy Laplace transforms,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1372–1381, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  36. Y. Chalco-Cano and H. Román-Flores, “On new solutions of fuzzy differential equations,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 112–119, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  37. A. Khastan, J. J. Nieto, and R. Rodríguez-López, “Variation of constant formula for first order fuzzy differential equations,” Fuzzy Sets and Systems, vol. 177, pp. 20–33, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  38. A. Khastan, J. J. Nieto, and R. Rodríguez-López, “Periodic boundary value problems for first-order linear differential equations with uncertainty under generalized differentiability,” Information Sciences, vol. 222, pp. 544–558, 2013. View at Publisher · View at Google Scholar · View at MathSciNet