A 2nd-Order Numerical Scheme for Fractional Ordinary Differential Equation Systems

  • W. Li
  • S. WangEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)


We propose a new numerical method for fractional ordinary differential equation systems based on a judiciously chosen quadrature point. The proposed method is efficient and easy to implement. We show that the convergence order of the method is 2. Numerical results are presented to demonstrate that the computed rates of convergence confirm our theoretical findings.



This work is partially supported by US Air Force Office of Scientific Research Project FA2386-15-1-4095.


  1. 1.
    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Erturk, V.S., Momani, S.: Solving systems of fractional differential equations using differential transform method. J. Comput. Appl. Math. 215, 142–151 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gejji, V.D., Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301(2), 508–518 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Guo, B., Pu, X., Huang, F.: Fractional Partial Differential Equations and Their Numerical Solutions. World Scientific, Singapore (2015)CrossRefGoogle Scholar
  5. 5.
    Jafari, H., Gejji, V.D.: Solving a system of nonlinear fractional differential equation using adomain decomposition. Appl. Math. Comput. 196, 644–651 (2006)CrossRefGoogle Scholar
  6. 6.
    Kilbas, A.A., Marzan, S.A.: Cauchy problem for differential equation with caputo derivative. Fract. Calc. Appl. Anal. 7(3), 297–321 (2004)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Li, W., Wang, S., Rehbock, V.: A 2nd-order one-point numerical integration scheme for fractional ordinary differential Equation. Numer. Algebra Control Optim. 7(3), 273–287 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Li, W., Wang, S., Rehbock, V.: Numerical solution of fractional optimal control. J. Optim. Theory Appl. (2018).
  9. 9.
    Momani, S., Al-Khaled, K.: Numerical solutions for systems of fractional differential equations by the decomposition method. Appl. Math. Comput. 162(3), 1351–1365 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Momani, S., Odibat, Z.: Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 365(5–6), 345–350 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Momani, S., Odibat, Z.: Numberical approach to differential equations of fractional order. J. Comput. Appl. Math. 207, 96–110 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Odibat, Z., Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simulat. 1(7), 15–27 (2006)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Varga, R.S.: On Diagonal dominance arguments for bounding \(\Vert A^{-1}\Vert _{\infty }\). Linear Algebra Appl. 14(3), 211–217 (1976)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zurigat, M., Momani, S., Odibat, Z., Alawneh, A.: The homotopy analysis method for handling systems of fractional differential equations. Appl. Math. Model. 34, 24–35 (2010)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsCurtin UniversityPerthAustralia

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