International Journal of Differential Equations

New Method for Solving Linear Fractional Differential Equations

S. Z. Rida and A. A. M. Arafa

Full-text: Open access

Abstract

We develop a new application of the Mittag-Leffler Function method that will extend the application of the method to linear differential equations with fractional order. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided. The results reveal that the technique introduced here is very effective and convenient for solving linear differential equations of fractional order.

Article information

Source
Int. J. Differ. Equ., Volume 2011, Special Issue (2011), Article ID 814132, 8 pages.

Dates
Received: 4 May 2011
Revised: 21 July 2011
Accepted: 25 July 2011
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485399983

Digital Object Identifier
doi:10.1155/2011/814132

Mathematical Reviews number (MathSciNet)
MR2832512

Zentralblatt MATH identifier
1239.34007

Citation

Rida, S. Z.; Arafa, A. A. M. New Method for Solving Linear Fractional Differential Equations. Int. J. Differ. Equ. 2011, Special Issue (2011), Article ID 814132, 8 pages. doi:10.1155/2011/814132. https://projecteuclid.org/euclid.ijde/1485399983


Export citation

References

  • Y. I. Babenko, Heat and Mass Transfer, Chemia, Leningrad, Germany, 1986.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • M. W. Michalski, “On a certain differential equation of non-integer order,” Zeitschrift fur Analysis und ihre Anwendungen, vol. 10, pp. 205–210, 1991.
  • M. W. Michalski, Derivatives of Non-integer Order and Their Applications, vol. 328 of Dissertationes Mathematicae, Polska Akademia Nauk, Institut Matematyczny, Warszawa, Poland, 1993.
  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  • 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.
  • 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.
  • J. Padovan, “Computational algorithms for FE formulations involving fractional operators,” Computational Mechanics, vol. 2, no. 4, pp. 271–287, 1987.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • J. D. Munkhammar, “Fractional calculus and the Taylor–Riemann series,” Undergraduate Mathematics Journal, vol. 6, 2005.
  • R. L. Magin, “Fractional calculus in bioengineering,” Critical Reviews in Biomedical Engineering, vol. 32, no. 1, pp. 1–104, 2004.
  • R. L. Magin, “Fractional calculus in bioengineering, part 2,” Critical Reviews in Biomedical Engineering, vol. 32, no. 2, pp. 105–193, 2004.
  • R. L. Magin, “Fractional calculus in bioengineering, part 3,” Critical Reviews in Biomedical Engineering, vol. 32, no. 3-4, pp. 195–377, 2004.
  • K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007.
  • A. A. Kilbas, M. Rivero, L. Rodríguez-Germá, and J. J. Trujillo, “$\alpha $-analytic solutions of some linear fractional differential equations with variable coefficients,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 239–249, 2007.