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Efficient Mittag-Leffler Collocation Method for Solving Linear and Nonlinear Fractional Differential Equations

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Abstract

In this paper, a new approximation method for fractional differential equations based on Mittag-Leffler function is developed. Finite Mittag-Leffler function and its fractional-order derivatives are investigated. An efficient technique for solving linear and nonlinear fractional order differential equations is developed. The proposed method combines Mittag-Leffler collocation method and optimization technique. Error estimation of the approximation is stated and proved. We present numerical results and comparisons of previous treatments to demonstrate the efficiency and applicability of the proposed method. Making use of small number of unknowns, the resulting solution converges to the exact one in the linear case and it has a very small error in the nonlinear case.

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References

  1. Baleanu, D., Gven, Z.B., Tenreiro Machado, J.A.: New Trends in Nanotechnology and Fractional Calculus Applications. Springer Science+Business Media B.V, New York (2010)

    Book  Google Scholar 

  2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier B.V, Dordrecht (2006)

    MATH  Google Scholar 

  3. Nategh, M.: A novel approach to an impulsive feedback control with and without memory involvement. J. Differ. Equ. 263(5), 2661–2671 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Deng, J., Ma, L.: Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. Appl. Math. Lett. 23(6), 676–680 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Yang, X., Wei, Z., Dong, W.: Existence of positive solutions for the boundary value problem of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(1), 85–92 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zhou, Y., Ahmad, B., Alsaedi, A.: Existence of nonoscillatory solutions for fractional neutral differential equations. Appl. Math. Lett. 72, 70–74 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Pedas, A., Tamme, E.: Numerical solution of nonlinear fractional differential equations by spline collocation methods. J. Comput. Appl. Math. 255, 216–230 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Yan, Y., Pal, K., Ford, N.J.: Higher order numerical methods for solving fractional differential equations. BIT Numer. Math. 54(2), 555–584 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: MittagLeffler Functions, Related Topics and Applications. Springer, Berlin (2014)

    MATH  Google Scholar 

  10. Rida, S.Z., Arafa, A.A., Mohammadein, M.A.A., Ali, H.M.: New method for solving linear fractional differential equations. Int. J. Differ. Equ. 814132, 1–8 (2011)

    MathSciNet  Google Scholar 

  11. Arafa, A.A.M., Rida, S.Z., Mohammadein, A.A., Ali, H.M.: Solving nonlinear fractional differential equation by generalized Mittag-Leffler function method. Commun. Theor. Phys. 59(6), 661663 (2013)

    Article  MathSciNet  Google Scholar 

  12. Arafa, A.A.M., Rida, S.Z., Ali, H.M.: Generalized Mittag-Leffler function method for solving Lorenz system. Int. J. Innov. Appl. Stud. 3(1), 105–111 (2013)

    Google Scholar 

  13. Yasmin, G., Khan, S., Ahmad, N.: Operational methods and truncated exponential-based Mittag-Leffler polynomials. Mediterr. J. Math. 13(4), 1555–1569 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Arafa, A.A.M., Rida, S.Z.: Numerical solutions for some generalized coupled nonlinear evolution equations. Math. Comput. Model. 56(11), 268–277 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Model. 37(7), 5498–5510 (2012)

    Article  MathSciNet  Google Scholar 

  16. Mathai, A.M., Haubold, H.J.: Special Functions for Applied Scientists. Springer Science+Business Media, New York (2008)

    Book  MATH  Google Scholar 

  17. El-Khateb, M.A., Hussien, H.S.: An optimization method for solving some differential algebraic equations. Commun. Nonlinear Sci. Numer. Simul. 14(5), 1970–1977 (2009)

    Article  Google Scholar 

  18. Li, Z., Yan, Y., Ford, N.J.: Error estimates of a high order numerical method for solving linear fractional differential equations. Appl. Numer. Math. 114, 201–220 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sakar, M.G., Akgl, A., Baleanu, D.: On solutions of fractional Riccati differential equations. Adv. Differ. Equ. 2017(1), 1–39 (2017)

    Article  MathSciNet  Google Scholar 

  20. Jia, Y., Xu, M., Lin, Y.: A new algorithm for nonlinear fractional BVPs. Appl. Math. Lett. 57, 121–125 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jafari, H., Yousefi, S.A., Firoozjaee, M.A., Momani, S., Khalique, C.M.: Application of Legendre wavelets for solving fractional differential equations. Comput. Math. Appl. 62(3), 1038–1045 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Keshavarz, E., Ordokhani, Y., Razzaghi, M.: Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl. Math. Model. 38(24), 6038–6051 (2014)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Saad Zagloul Rida & Hussien Shafei Hussien

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  1. Saad Zagloul Rida
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  2. Hussien Shafei Hussien
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Correspondence to Hussien Shafei Hussien.

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Rida, S.Z., Hussien, H.S. Efficient Mittag-Leffler Collocation Method for Solving Linear and Nonlinear Fractional Differential Equations. Mediterr. J. Math. 15, 130 (2018). https://doi.org/10.1007/s00009-018-1174-0

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  • DOI: https://doi.org/10.1007/s00009-018-1174-0

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