[1] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag: Berlin, 2010.
[2] A.A. Kilbas, H. M Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science
B.V., Amsterdam, 2006.
[3] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.
[4] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons,
Inc., New York, 1993.
[5] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
[6] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers,
Yverdon 1993.
[7] A. Young, Approximate product-integration, Proc. R. Soc. Lond. Ser. A., 224 (1954) 552–561.
[8] A. Young, The application of approximate product integration to the numerical solution of integral equations, Proc. R. Soc. Lond.
Ser. A., 224 (1954) 561–573.
[9] K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential
equations, Nonlinear Dyn., 29 (2002) 3–22.
[10] R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations, Int. J. Comput. Math., 87
(2010) 2281–2290.
[11] Y. Yan, K. Pal and N. J. Ford, Higher order numerical methods for solving fractional differential equations, BIT Numer. Math., 54
(2014) 555–584.
[12] Z. Li, Z. Liang and Y. Yan, High-order numerical methods for solving time fractional partial differential equations, J. Sci. Comput.,
71 (2017) 785–803.
[13] J. Dixon, On the order of the error in discretization methods for weakly singular second kind Volterra integral equations with
nonsmooth solutions, BIT Numer. Math., 25 (1985) 624–634.
[14] K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004)
31–52.
[15] K. B. Oldham and J. Spanier, Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press,
New York, 1974.
[16] V. E. Lynch, B. A. Carreras, D. del Castillo-Negrete, K. M. Ferreira-Mejias and H. R. Hicks, Numerical methods for the solution
of partial differential equations of fractional order, J. put. Phys., 192 (2003) 406–421.
[17] C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986) 704–719.
[18] C. Lubich, Convolution quadrature and discretized operational calculus I, Numer. Math., 52 (1988) 129–145.
[19] C. Lubich, Convolution quadrature and discretized operational calculus II, Numer. Math., 52 (1988) 413–425.
[20] C. Lubich, Convolution quadrature revisited, BIT, 44 (2004) 503–514.
[21] R. Garrappa, Trapezoidal methods for fractional differential equations: theoretical and computational aspects, Math. Comput.
Simul., 110 (2015) 96–112.
[22] K. Diethelm, J. M. Ford, N. J. Ford and M. Weilbeer, Pitfalls in fast numerical solvers for fractional differential equations, J.
Comput. Appl. Math., 186 (2006) 482–503.
[23] M. Stynes, Singularities, In Handbook of Fractional Calculus With Applications, De Gruyter, Berlin, 3 2019 287–305.
[24] R. K. Miller and A. Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math.
Anal., 2 (1971) 242–258.
[25] C. Lubich, Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comput., 41 (1983) 87–102.
[26] A. Hanyga, A comment on a controversial issue: A generalized fractional derivative cannot have a regular kernel, Fract. Calc.
Appl. Anal., 23 (2020) 211–223.
[27] A. Giusti, General fractional calculus and Prabhakar’s theory, Commun Nonlinear Sci. Numer. Simul., 83 (2019) 105–114.
[28] A. Hanyga, Physically acceptable viscoelastic models, Trends in applications of mathematics to mechanics, 125–136, Ber. Math.,
Shaker Verlag, Aachen, 2005.
[29] M. Stynes, E. O’Riordan, and J.L. Gracia, Necessary conditions for convergence of difference schemes for fractional-derivative
two-point boundary value problems, BIT, 56 (2016) 1455–1477.
[30] S. Sarv Ahrabi and A. Momenzadeh, On failed methods of fractional differential equations: the case of multi-step generalized
differential transform method, Mediterr. J. Math., 15 (2018) pp. 149.
[31] R. Garrappa, Neglecting nonlocality leads to unreliable numerical methods for fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 70 (2019) 302–306.
[32] ٌW. H. Deng, Short memory principle and a predictor-corrector approach for fractional differential equations, J. Comput. Appl.
Math., 206 (2007) 174–188.
[33] N. J. Ford and A. C. Simpson, The numerical solution of fractional differential equations: Speed versus accuracy, Numer. Algorithms, 26 (2001), 333–346.
[34] K. Diethelm and A. D. Freed, An Efficient Algorithm for the Evaluation of Convolution Integrals, Comput. Math. Appl., 51 (2006)
51–72.
[35] E. Hairer, C. Lubich and M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations, SIAM J. Sci. Statist.
Comput., 6 (1985) 532–541.
[36] E. Hairer, C. Lubich and M. Schlichte, Fast numerical solution of weakly singular Volterra integral equations, J. Comput. Appl.
Math., 23 (1988) 87–98.
[37] P. Henrici, Fast Fourier methods in computational complex analysis, SIAM Rev., 21 (1979) 481–527.
[38] R. Garrappa, Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial, Mathematics, 6 (2018)
pp. 16.
[39] R. Garrappa, Mathworks Author’s Profile, 2020.
[40] D. Baffet, A Gauss-Jacobi kernel compression scheme for fractional differential equations, J. Sci. Comput., 79 (2019) 227–248.
[41] D. Baffet and J. S. Hesthaven, A kernel compression scheme for fractional differential equations, SIAM J. Numer. Anal., 55 (2017)
496–520.
[42] D. Baffet and J. S. Hesthaven, High-order accurate adaptive kernel compression time-stepping schemes for fractional differential
equations, J. Sci. Comput., 72 (2017) 1169–1195.
[43] K. Diethelm, An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives, Numer. Algorithms, 47 (2008) 361–90.
[44] M. López-Fernández, C. Lubich, and A. Schädle, Adaptive, fast, and oblivious convolution in evolution equations with memory,
SIAM J. Sci. Comput., 30 (2008), pp. 1015–1037.
[45] C. Lubich and A. Schädle,Fast convolution for nonreflecting boundary conditions, SIAM J. Sci. Comput., 24 (2002) 161–182.
[46] A. Schädle, M. López-Fernández and C. Lubich, Fast and oblivious convolution quadrature, SIAM J. Sci. Comput., 28 (2006)
421–38.
[47] L. Banjai and M. López-Fernández, Efficient high order algorithms for fractional integrals and fractional differential equations,
Numer. Math., 141 (2019) 289–317.
[48] M. Fischer, Fast and parallel Runge-Kutta approximation of fractional evolution equations, SIAM J. Sci. Comput., 41 (2019)
A927–A947.
[49] S. Jiang, J. Zhang, Q. Zhang, and Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional
diffusion equations, Commun. Comput. Phys., 21 (2017) 650–678.
[50] J. R. Li, A fast time stepping method for evaluating fractional integrals, SIAM J. Sci. Comput., 31 (2010) 4696–4714.
[51] F. Zeng, I. Turner and K. Burrage, A stable fast time-stepping method for fractional integral and derivative operators, J. Sci.
Comput., 77 (2018) 283–307.
[52] L. Guo, F. Zeng, I. Turner, K. Burrage and G. E. M. Karniadakis, Efficient multistep methods for tempered fractional calculus:
Algorithms and simulations, SIAM J. Sci. Comput., 41 (2019) 2510–2535.
[53] M. Stynes, Too much regularity may force too much uniqueness, Fract. Calc. Appl. Anal., 19 (2016) 1554–1562.
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