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International Journal of Differential Equations
Volume 2011 (2011), Article ID 814132, 8 pages
http://dx.doi.org/10.1155/2011/814132
Research Article

New Method for Solving Linear Fractional Differential Equations

Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt

Received 4 May 2011; Revised 21 July 2011; Accepted 25 July 2011

Academic Editor: Shaher M. Momani

Copyright © 2011 S. Z. Rida and A. A. M. Arafa. 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. Y. I. Babenko, Heat and Mass Transfer, Chemia, Leningrad, Germany, 1986.
  2. M. Caputo and F. Mainardi, “Linear models of dissipation in anelastic solids,” La Rivista del Nuovo Cimento, vol. 1, no. 2, pp. 161–198, 1971. View at Publisher · View at Google Scholar · View at Scopus
  3. R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainar, Eds., pp. 223–276, Springer, New York, NY, USA, 1997.
  4. R. Gorenflo and R. Rutman, “On ultraslow and intermediate processes,” in Transform Methods and Special Functions, P. Rusev, I. Dimovski, and V. Kiryakova, Eds., pp. 61–81, Science Culture Technology Publishing, Singapore, 1995.
  5. F. Mainardi, “Fractional relaxation and fractional diffusion equations, mathematical aspects,” in Proceedings of the 12th IMACS World Congress, W. F. Ames, Ed., vol. 1, pp. 329–332, Georgia Tech Atlanta, 1994.
  6. F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., pp. 291–348, Springer, New York, NY, USA, 1997.
  7. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, New York, NY, USA, 1993.
  8. H. Beyer and S. Kempfle, “Definition of physically consistent damping laws with fractional derivatives,” Journal of Applied Mathematics and Mechanics, vol. 75, pp. 623–635, 1995.
  9. S. Kempfle and H. Beyer, “Global and causal solutions of fractional differential equations,” in Proceedings of the 2nd International Workshop on Transform Methods and Special Functions, pp. 210–216, Science Culture Technology Publishing, Varna, Bulgaria, 1996.
  10. 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.
  11. A. A. Kilbas and M. Saigo, “On mittag-leffler type function, fractional calculus operators and solutions of integral equations,” Integral Transforms and Special Functions, vol. 4, no. 4, pp. 355–370, 1996. View at Scopus
  12. Y. F. Luchko and H. M. Srivastava, “The exact solution of certain differential equations of fractional order by using operational calculus,” Computers and Mathematics with Applications, vol. 29, no. 8, pp. 73–85, 1995. View at Scopus
  13. M. W. Michalski, “On a certain differential equation of non-integer order,” Zeitschrift fur Analysis und ihre Anwendungen, vol. 10, pp. 205–210, 1991.
  14. M. W. Michalski, Derivatives of Non-integer Order and Their Applications, vol. 328 of Dissertationes Mathematicae, Polska Akademia Nauk, Institut Matematyczny, Warszawa, Poland, 1993.
  15. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  16. I. Podlubny, “Solution of linear fractional differential equations with constant coefficients,” in Transform Methods and Special Functions, P. Rusev, I. Dimovski, and V. Kiryakova, Eds., pp. 227–237, Science Culture Technology Publishing, Singapore, 1995.
  17. S. B. Hadid and Y. F. Luchko, “An operational method for solving fractional differential equations of an arbitrary real order,” Pan-American Mathematical Journal, vol. 6, pp. 57–73, 1996.
  18. J. Padovan, “Computational algorithms for FE formulations involving fractional operators,” Computational Mechanics, vol. 2, no. 4, pp. 271–287, 1987. View at Publisher · View at Google Scholar · View at Scopus
  19. S. Momani, “Non-perturbative analytical solutions of the space- and time-fractional Burgers equations,” Chaos, Solitons and Fractals, vol. 28, no. 4, pp. 930–937, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488–494, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. Z. M. Odibat and S. Momani, “Approximate solutions for boundary value problems of time-fractional wave equation,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 767–774, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. S. Momani, “An explicit and numerical solutions of the fractional KdV equation,” Mathematics and Computers in Simulation, vol. 70, no. 2, pp. 110–118, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. S. Momani and Z. Odibat, “Analytical approach to linear fractional partial differential equations arising in fluid mechanics,” Physics Letters, Section A, vol. 355, no. 4-5, pp. 271–279, 2006. View at Publisher · View at Google Scholar · View at Scopus
  24. S. Momani and Z. Odibat, “Numerical comparison of methods for solving linear differential equations of fractional order,” Chaos, Solitons and Fractals, vol. 31, no. 5, pp. 1248–1255, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998. View at Scopus
  26. Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential equations of fractional order,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 15–27, 2006. View at Scopus
  27. Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order,” Chaos, Solitons and Fractals, vol. 36, no. 1, pp. 167–174, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  28. S. Momani and Z. Odibat, “Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations,” Computers and Mathematics with Applications, vol. 54, no. 7-8, pp. 910–919, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  29. J. D. Munkhammar, “Fractional calculus and the Taylor–Riemann series,” Undergraduate Mathematics Journal, vol. 6, 2005.
  30. R. L. Magin, “Fractional calculus in bioengineering,” Critical Reviews in Biomedical Engineering, vol. 32, no. 1, pp. 1–104, 2004. View at Scopus
  31. R. L. Magin, “Fractional calculus in bioengineering, part 2,” Critical Reviews in Biomedical Engineering, vol. 32, no. 2, pp. 105–193, 2004. View at Publisher · View at Google Scholar · View at Scopus
  32. R. L. Magin, “Fractional calculus in bioengineering, part 3,” Critical Reviews in Biomedical Engineering, vol. 32, no. 3-4, pp. 195–377, 2004. View at Publisher · View at Google Scholar · View at Scopus
  33. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
  34. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach Science Publishers, Longhorne, Pa, USA, 1993.
  35. S. Z. Rida, A. M. A. El-Sayed, and A. A. M. Arafa, “Effect of bacterial memory dependent growth by using fractional derivatives reaction-diffusion chemotactic model,” Journal of Statistical Physics, vol. 140, no. 4, pp. 797–811, 2010. View at Publisher · View at Google Scholar · View at Scopus
  36. A. M. A. El-Sayed, S. Z. Rida, and A. A. M. Arafa, “On the Solutions of the generalized reaction-diffusion model for bacteria growth,” Acta Applicandae Mathematicae, vol. 110, pp. 1501–1511, 2010.
  37. S. Z. Rida, A. M. A. El-Sayed, and A. A. M. Arafa, “On the solutions of time-fractional reaction-diffusion equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3847–3854, 2010. View at Publisher · View at Google Scholar · View at Scopus
  38. S. Z. Rida, A. M. A. El-Sayed, and A. A. M. Arafa, “A Fractional Model for Bacterial Chemoattractant in a Liquid Medium,” Nonlinear Science Letters A, vol. 1, no. 4, pp. 415–420, 2010.
  39. A. M. A. El-Sayed, S. Z. Rida, and A. A. M. Arafa, “Exact solutions of fractional-order biological population model,” Communications in Theoretical Physics, vol. 52, no. 6, pp. 992–996, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  40. A. M. A. El-Sayed, S. Z. Rida, and A. A. M. Arafa, “On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber,” International Journal of Nonlinear Sciences, vol. 7, no. 4, p. 485, 2009.
  41. S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, “On the solution of the fractional nonlinear Schrödinger equation,” Physics Letters A, vol. 372, no. 5, pp. 553–558, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  42. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications to Methods of Their Solution and Some of Their Applications, Academic Press, New York, NY, USA, 1999.
  43. Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  44. A. A. Kilbas, M. Rivero, L. Rodríguez-Germá, and J. J. Trujillo, “α-analytic solutions of some linear fractional differential equations with variable coefficients,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 239–249, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus