- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 178378, 13 pages
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.
- 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.
- T. Allahviranloo and M. B. Ahmadi, “Fuzzy Laplace transforms,” Soft Computing, vol. 14, no. 3, pp. 235–243, 2010.
- 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.
- 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.
- 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.
- 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.
- 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.
- S. Salahshour and A. Allahviranloo, “Applications of fuzzy Laplace transforms,” Soft Computing, vol. 17, no. 1, pp. 145–178, 2013.
- 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.
- Y. I. Babenko, Heat and Mass Transfer, Chemia, Leningrad, Russia, 1986.
- 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.
- 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.
- 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.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- 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.
- V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific, Cambridge, UK, 2009.
- 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.
- 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.
- 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.
- 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.
- 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.
- I. Perfilieva, “Fuzzy transforms: theory and applications,” Fuzzy Sets and Systems, vol. 157, no. 8, pp. 993–1023, 2006.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- H.-J. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers, Boston, Mass, USA, 2nd edition, 1992.
- M. L. Puri and D. A. Ralescu, “Fuzzy random variables,” Journal of Mathematical Analysis and Applications, vol. 114, no. 2, pp. 409–422, 1986.
- H.-C. Wu, “The improper fuzzy Riemann integral and its numerical integration,” Information Sciences, vol. 111, no. 1–4, pp. 109–137, 1998.
- G. A. Anastassiou, Fuzzy Mathematics: Approximation Theory, vol. 251 of Studies in Fuzziness and Soft Computing, Springer, Berlin, Germany, 2010.
- G. A. Anastassiou, Intelligent Mathematics: Computational Analysis, vol. 5 of Intelligent Systems Reference Library, Springer, Berlin, Germany, 2011.
- 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.
- 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.
- 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.
- 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.