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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 142175, 15 pages
http://dx.doi.org/10.1155/2010/142175
Research Article

Existence and Uniqueness of Solutions for the Cauchy-Type Problems of Fractional Differential Equations

1Department of Applied Mathematics, Donghua University, Shanghai 201620, China
2College of Information Sciences and Technology, Donghua University, Shanghai 201620, China

Received 30 October 2009; Revised 21 January 2010; Accepted 25 January 2010

Academic Editor: Guang Zhang

Copyright © 2010 Chunhai Kou 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. B. Bonilla, M. Rivero, L. Rodríguez-Germá, and J. J. Trujillo, “Fractional differential equations as alternative models to nonlinear differential equations,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 79–88, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. D. Fitt, A. R. H. Goodwin, K. A. Ronaldson, and W. A. Wakeham, “A fractional differential equation for a MEMS viscometer used in the oil industry,” Journal of Computational and Applied Mathematics, vol. 229, no. 2, pp. 373–381, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  3. E. Ahmed and A. S. Elgazzar, “On fractional order differential equations model for nonlocal epidemics,” Physica A: Statistical Mechanics and its Applications, vol. 379, no. 2, pp. 607–614, 2007. View at Publisher · View at Google Scholar
  4. Y. Ding and H. Ye, “A fractional-order differential equation model of HIV infection of CD4+ T-cells,” Mathematical and Computer Modelling, vol. 50, no. 3-4, pp. 386–392, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  5. H. Xu, “Analytical approximations for a population growth model with fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1978–1983, 2009. View at Publisher · View at Google Scholar
  6. K. M. Furati and N.-E. Tatar, “On chaos synchronization of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 441–454, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Yan and C. Li, “Long time behavior for a nonlinear fractional model,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 725–735, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. C. Kou, Y. Yan, and J. Liu, “Stability analysis for fractional differential equations and their applications in the models of HIV-1 infection,” Computer Modeling in Engineering & Sciences, vol. 39, no. 3, pp. 301–317, 2009. View at Google Scholar · View at MathSciNet
  9. F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), vol. 378 of CISM Courses and Lectures, pp. 291–348, Springer, Vienna, Austria, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. Metzler and T. F. Nonnenmacher, “Fractional diffusion: exact representations of spectral functions,” Journal of Physics A, vol. 30, no. 4, pp. 1089–1093, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Metzler and T. F. Nonnenmacher, “Fractional diffusion, waiting-time distributions, and Cattaneo-type equations,” Physical Review E, vol. 57, no. 6, pp. 6409–6414, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  12. W. M. Glöckle, R. Metzler, and T. F. Nonnenmacher, “Fractional model equation for anomalous diffusion,” Physica A, vol. 211, pp. 13–24, 1994. View at Publisher · View at Google Scholar
  13. B. J. West, P. Grigolini, R. Metzler, and T. F. Nonnenmacher, “Fractional diffusion and Levy stable processes,” Physical Review E, vol. 55, no. 1, part A, pp. 99–106, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  14. H. E. Roman and M. Giona, “Fractional diffusion equation on fractals: three-dimensional case and scattering function,” Journal of Physics A, vol. 25, no. 8, pp. 2107–2117, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. T. F. Nonnenmacher and D. J. F. Nonnenmacher, “Towards the formulation of a nonlinear fractional extended irreversible thermodynamics,” Acta Physica Hungarica, vol. 66, no. 1–4, pp. 145–154, 1989. View at Google Scholar · View at MathSciNet
  16. E. Pitcher and W. E. Sewell, “Existence theorems for solutions of differential equations of non-integral order,” Bulletin of the American Mathematical Society, vol. 44, no. 2, pp. 100–107, 1938. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. H. Barrett, “Differential equations of non-integer order,” Canadian Journal of Mathematics, vol. 6, no. 4, pp. 529–541, 1954. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. P. L. Butzer and A. A. Kilbas, “Mellin transform analysis and integration by parts for Hadamard-type fractional integrals,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 1–15, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, “Fractional calculus in the Mellin setting and Hadamard-type fractional integrals,” Journal of Mathematical Analysis and Applications, vol. 269, no. 1, pp. 1–27, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, “Compositions of Hadamard-type fractional integration operators and the semigroup property,” Journal of Mathematical Analysis and Applications, vol. 269, no. 2, pp. 387–400, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. A. A. Kilbas, “Hadamard-type fractional calculus,” Journal of the Korean Mathematical Society, vol. 38, no. 6, pp. 1191–1204, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. A. Kilbas, O. I. Marichev, and S. G. Samko, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 1993. View at MathSciNet
  23. A. A. Kilbas and J. J. Trujillo, Hadamard-Type Fractional Integrals and Derivatives, vol. 11, Trudy Instituta Matematiki, Minsk, Russia, 2002. View at Zentralblatt MATH
  24. H. A. H. Salem, “On the existence of continuous solutions for a singular system of non-linear fractional differential equations,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 445–452, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. A. Bashir and J. N. Juan, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838–1843, 2009. View at Publisher · View at Google Scholar
  26. G. Mehdi, “Solution of nonlinear fractional differential equations using homotopy analysis method,” Applied Mathematical Modelling, vol. 58, no. 9, pp. 1838–1843, 2009. View at Google Scholar
  27. O. Abdulaziz, I. Hashim, and S. Momani, “Solving systems of fractional differential equations by homotopy-perturbation method,” Physics Letters A, vol. 372, no. 4, pp. 451–459, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  28. A. A. Kilbas, H. M. Sprivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, Calif, USA, 2006.