A high-order numerical scheme for the impulsive fractional ordinary differential equations

References

• J.Y. Cao and C.J. Xu, A high order schema for the numerical solution of the fractional ordinary differential equations, J. Comput. Phys. 238 (2013), pp. 154–168.
   Web of Science ®Google Scholar
• J. Dixon and S. McKee, Weakly singular discrete Gronwall inequalities, Z. Angew. Math. Mech. 66(11) (1986), pp. 535–544.
   Web of Science ®Google Scholar
• M. Fec̆kan, Y. Zhou, and J.R. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17(7) (2012), pp. 3050–3060.
   Web of Science ®Google Scholar
• G.H. Gao, Z.Z. Sun, and H.W. Zhang, A new fractional numerical differentiation formula to approximate the caputo fractional derivative and its applications, J. Comput. Phys. 259 (2014), pp. 33–50.
   Web of Science ®Google Scholar
• T.L. Guo and K.J. Zhang, Impulsive fractional partial differential equations, Appl. Math. Comput. 257 (2015), pp. 581–590.
   Web of Science ®Google Scholar
• J.F. Huang, Y.F. Tang, and L. Vázquez, Convergence analysis of a block-by-block method for fractional differential equations, Numer. Math. Theory Methods Appl. 5(2) (2012), pp. 229–241.
   Google Scholar
• P. Kumar, D.N. Pandey, and D. Bahuguna, On a new class of abstract impulsive functional differential equations of fractional order, J. Nonlinear Sci. Appl. 7 (2014), pp. 102–114.
   Web of Science ®Google Scholar
• V. Lakshmikantham, D.D. Baunov, and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
   Google Scholar
• Y.M. Lin, X.J. Li, and C.J. Xu, Finite difference/spectral approximations for the fractional cable equation, Math. Comp. 80(275) (2011), pp. 1369–1396.
   Web of Science ®Google Scholar
• Z. Lin, J.R. Wang, and W. Wei, Multipoint bvps for generalized impulsive fractional differential equations, Appl. Math. Comput. 258 (2015), pp. 608–616.
   Web of Science ®Google Scholar
• S. Liu, A. Debbouche, and J.R. Wang, On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths, J. Comput. Appl. Math. 312 (2017), pp. 47–57.
   Web of Science ®Google Scholar
• X. Liu, G.L. Zhang, and M.Z. Liu, Analytic and numerical exponential asymptotic stability of nonlinear impulsive differential equations, Appl. Numer. Math. 81 (2014), pp. 40–49.
   Web of Science ®Google Scholar
• S. Ntouyas, M. Benchohra, and J. Henderson, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York, 2006.
   Google Scholar
• N. Nyamoradi and R. Rodríguez-López, On boundary value problems for impulsive fractional differential equations, Appl. Math. Comput. 271 (2015), pp. 874–892.
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Academy Press, New York, 1999.
   Google Scholar
• A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
   Google Scholar
• J.R. Wang, Y. Zhou, and M. Fec̆kan, On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comput. Math. Appl. 64(10) (2012), pp. 3008–3020.
   Web of Science ®Google Scholar
• J.R. Wang, Y. Zhou, and M. Fec̆kan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64(10) (2012), pp. 3389–3405.
   Web of Science ®Google Scholar
• J.R. Wang, Stability of Noninstantaneous Impulsive Evolution Equations, Appl. Math. Lett. 73 (2017), pp. 157–162.
   Web of Science ®Google Scholar
• J.Y. Yang, J.F. Huang, D.M. Liang, and Y.F. Tang, Numerical solution of fractional diffusion-wave equation based on fractional multistep method, Appl. Math. Model. 38(14) (2014), pp. 3652–3661.
   Web of Science ®Google Scholar
• Z.H. Zhang and H. Liang, Collocation methods for impulsive differential equations, Appl. Math. Comput. 228 (2014), pp. 336–348.
   Web of Science ®Google Scholar
• G.L. Zhang and M.H. Song, Asymptotical stability of Runge–Kutta methods for advanced linear impulsive differential equations with piecewise constant arguments, Appl. Math. Comput. 259 (2015), pp. 831–837.
   Web of Science ®Google Scholar