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Original Articles

Numerical algorithms for Caputo fractional-order differential equations

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Pages 1201-1211 | Received 28 Jun 2015, Accepted 22 Feb 2016, Published online: 23 Mar 2016

References

  • Agrawal, O.P., & Baleanu, D. (2007). A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. Journal of Vibration and Control, 13(9–10), 1269–1281.
  • Ali, I., Kiryakov, V., & Kalla, S.L. (2002). Solution of fractional multi-order integral and differential equations using a Poisson-type transform. Journal of Mathematical Analysis and Applications, 269(1), 172–199.
  • Baleanu, D., & Agrawal, O.P. (2006). Fractional Hamilton formalism within Caputo’s derivative. Czechoslovak Journal of Physics, 56(10–11), 1087–1092.
  • Baleanu, D., Defterli, O., & Agrawal, O.P. (2009). A central difference numerical scheme for fractional optimal control problems. Journal of Vibration and Control, 15(4), 583–597.
  • Daftardar-Gejji, V., & Bhalekara, S. (2008). Boundary value problems for multi-term fractional differential equations. Journal of Mathematical Analysis and Applications, 345(2), 754–765.
  • Daftardar-Gejji, V., & Jafar, H. (2007). Solving a multi-order fractional differential equation using Adomian decomposition. Applied Mathematics and Computation, 189(1), 541–548.
  • Diethelm, K. (2003). Efficient solution of multi-term fractional differential equations using P(EC)mE methods. Computing, 71(4), 305–319.
  • Diethelm, K. (2008). An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives. Numerical Algorithms, 47(4), 361–390.
  • Diethelm, K. (2010). The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Lecture Notes in Mathematics, 265(2), 229–248.
  • Diethelm, K. (2011). An efficient parallel algorithm for the numerical solution of fractional differential equations. Fractional Calculus and Applied Analysis, 14(3), 475–490.
  • Diethelm, K., & Ford, N.J. (2004). Multi-order fractional differential equations and their numerical solution. Applied Mathematics and Computation, 154(3), 621–640.
  • Diethelm, K., Ford, N.J., & Freed, A.D. (2002). A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29(1–4), 3–22.
  • Diethelm, K., Ford, N.J., & Freed, A.D. (2004). Detailed error analysis for a fractional Adams method. Numerical Algorithms, 36(1), 31–52.
  • Edwards, J.T., Ford, N.J., & Simpson, A.C. (2002). The numerical solution of linear multi-term fractional differential equations: Systems of equations. Journal of Computational and Applied Mathematics, 148(2), 401–418.
  • Lubich, C. (1983). On the stability of linear multistep methods for Volterra convolution equations. IMA Journal of Numerical Analysis, 3(4), 439–465.
  • Lubich, C. (1986). Discretized fractional calculus. SIAM Journal on Mathematical Analysis, 17(3), 704–719.
  • Lubich, C. (1987). Fractional linear multistep methods for Abel–Volterra integral equations of the first kind. IMA Journal of Numerical Analysis, 7(1), 97–106.
  • Matignon, D. (1996). Stability results on fractional differential equations with applications to control processing. In Proceedings of Multi-Conference on Computational Engineering in Systems and Application, IEEE-SMC (pp. 963–968). Lille, France.
  • Matignon, D. (1998). Stability properties for generalized fractional differential systems. Proceedings of Fractional Differential Systems: Models, Methods and Applications, 5, 145–158.
  • Matignon, D., & d’Andrea-Novel, D. (1997). Some results on controllability and observability of finite-dimensional fractional differential systems. In Proceedings of Computational Engineering in Systems Applications (pp. 952–956). Lille, France.
  • Oustaloup, A., Mathieu, B., & Lanusse, P. (1995). The CRONE control of resonant plants: Application to a flexible transmission. European Journal of Control, 1, 113–121.
  • Oustaloup, A., Moreau, X., & Nouillant, M. (1996). The CRONE suspension. Control Engineering Practice, 4(8), 1101–1108.
  • Podlubny, I. (1998). Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. San Diego, CA: Academic Press.
  • Podlubny, I. (1999). Fractional-order systems and PIλDμ-controllers. IEEE Transactions on Automatic Control, 44(1), 208–214.
  • Podlubny, I. (2000). Matrix approach to discrete fractional calculus. Fractional Calculus and Applied Analysis, 3(4), 359–386.
  • Torvik, P.J., & Bagley, R.L. (1984). On the appereance of the frcational derivative in the behavior of real materials. Journal of Applied Mechanics, 51(2), 294–298.
  • Zhao, C.N., & Xue, D. (2008). Closed-form solutions to fractional-order linear differential equations. Frantiers of Electrical and Electronic Engineering in China, 3(2), 214–217.

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