Trapezoidal methods for fractional differential equations: Theoretical and computational aspects
Tóm tắt
Từ khóa
Tài liệu tham khảo
Brunner, 1985, The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes, Math. Comput., 45, 417, 10.1090/S0025-5718-1985-0804933-3
Brunner, 2004
Bueno-Orovio, 2013
Burrage, 2012, An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput., 34, A2145, 10.1137/110847007
Cafagna, 2012, Observer-based projective synchronization of fractional systems via a scalar signal: Application to hyperchaotic Rössler systems, Nonlinear Dyn., 68, 117, 10.1007/s11071-011-0208-y
Cameron, 1985, The analysis of product integration methods for Abel's equation using discrete fractional differentiation, IMA J. Numer. Anal., 5, 339, 10.1093/imanum/5.3.339
Caponetto, 2013, Analysis and shaping of the self-sustained oscillations in relay controlled fractional-order systems, Fract. Calculus Appl. Anal., 16, 93, 10.2478/s13540-013-0007-x
Conte, 2006, Fast collocation methods for Volterra integral equations of convolution type, J. Comput. Appl. Math., 196, 652, 10.1016/j.cam.2005.10.018
de Hoog, 1974, High order methods for a class of Volterra integral equations with weakly singular kernels, SIAM J. Numer. Anal., 11, 1166, 10.1137/0711088
Dieci, 2009, Sliding motion in Filippov differential systems: theoretical results and a computational approach, SIAM J. Numer. Anal., 47, 2023, 10.1137/080724599
Dieci, 2011, Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field, Numer. Math., 117, 779, 10.1007/s00211-011-0365-4
Dieci, 2012, A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side, J. Comput. Appl. Math., 236, 3967, 10.1016/j.cam.2012.02.011
Diethelm, 2008, An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives, Numer. Algorithms, 47, 361, 10.1007/s11075-008-9193-8
Diethelm, 2010
Diethelm, 2011, An efficient parallel algorithm for the numerical solution of fractional differential equations, Fract. Calc. Appl. Anal., 14, 475, 10.2478/s13540-011-0029-1
Diethelm, 2006, Pitfalls in fast numerical solvers for fractional differential equations, J. Comput. Appl. Math., 186, 482, 10.1016/j.cam.2005.03.023
Diethelm, 2005, Algorithms for the fractional calculus: a selection of numerical methods, Comput. Methods Appl. Mech. Eng., 194, 743, 10.1016/j.cma.2004.06.006
Diethelm, 1999, The FracPECE subroutine for the numerical solution of differential equations of fractional order, 57
Dixon, 1985, On the order of the error in discretization methods for weakly singular second kind Volterra integral equations with nonsmooth solutions, BIT, 25, 624, 10.1007/BF01936141
Gafiychuk, 2008, Stability analysis and limit cycle in fractional system with Brusselator nonlinearities, Phys. Lett. A, 372, 4902, 10.1016/j.physleta.2008.05.045
Galeone, 2008, Fractional Adams-Moulton methods, Math. Comput. Simul., 79, 1358, 10.1016/j.matcom.2008.03.008
Garra, 2011, Fractional-calculus model for temperature and pressure waves in fluid-saturated porous rocks, Phys. Rev. E, 84, 036605, 10.1103/PhysRevE.84.036605
Garrappa, 2010, On linear stability of predictor–corrector algorithms for fractional differential equations, Int. J. Comput. Math., 87, 2281, 10.1080/00207160802624331
Garrappa, 2014, On some generalizations of the implicit Euler method for discontinuous fractional differential equations, Math. Comput. Simul., 95c, 213, 10.1016/j.matcom.2012.04.009
Garrappa, 2011, On accurate product integration rules for linear fractional differential equations, J. Comput. Appl. Math., 235, 1085, 10.1016/j.cam.2010.07.008
Garrappa, 2013, Evaluation of generalized Mittag–Leffler functions on the real line, Adv. Comput. Math., 39, 205, 10.1007/s10444-012-9274-z
Gorenflo, 1997, Fractional calculus: some numerical methods, 277
Hairer, 1985, Fast numerical solution of nonlinear Volterra convolution equations, SIAM J. Sci. Stat. Comput., 6, 532, 10.1137/0906037
Hairer, 1996
Henrici, 1974, vol. 1
Henrici, 1979, Fast Fourier methods in computational complex analysis, SIAM Rev., 21, 481, 10.1137/1021093
Kadem, 2010, Spectral method for solution of the fractional transport equation, Rep. Math. Phys., 66, 103, 10.1016/S0034-4877(10)80026-6
Kilbas, 2006
Lambert, 1991
Li, 2013, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim., 34, 149, 10.1080/01630563.2012.706673
Lubich, 1983, Runge–Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comput., 41, 87, 10.1090/S0025-5718-1983-0701626-6
Lubich, 1985, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comput., 45, 463, 10.1090/S0025-5718-1985-0804935-7
Lubich, 1986, A stability analysis of convolution quadratures for Abel-Volterra integral equations, IMA J. Numer. Anal., 6, 87, 10.1093/imanum/6.1.87
Machado, 2011, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16, 1140, 10.1016/j.cnsns.2010.05.027
Magin, 2010, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59, 1586, 10.1016/j.camwa.2009.08.039
Mainardi, 2010
Matignon, 1998, Stability properties for generalized fractional differential systems, 145
Meerschaert, 2004, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172, 65, 10.1016/j.cam.2004.01.033
Meerschaert, 2006, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56, 80, 10.1016/j.apnum.2005.02.008
Moret, 2013, A note on Krylov methods for fractional evolution problems, Numer. Func. Anal. Opt., 34, 539, 10.1080/01630563.2012.748669
Moret, 2011, On the convergence of Krylov subspace methods for matrix Mittag–Leffler functions, SIAM J. Numer. Anal., 49, 2144, 10.1137/080738374
Moret, 2012, The restarted shift-and-invert Krylov method for matrix functions, Numer. Linear Algebra Appl
Podlubny, 1999
Tenreiro Machado, 2013, Fractional calculus analysis of the cosmic microwave background, Romanian Rep. Phys., 65, 316
Wolkenfelt, 1979
Young, 1954, Approximate product-integration, Proc. R. Soc. Lond. Ser. A., 224, 552, 10.1098/rspa.1954.0179
Yuste, 2005, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42, 1862, 10.1137/030602666