Trapezoidal methods for fractional differential equations: Theoretical and computational aspects

Mathematics and Computers in Simulation - Tập 110 - Trang 96-112 - 2015
Roberto Garrappa1
1Università degli Studi di Bari, Dipartimento di Matematica, Via E. Orabona n. 4, 70125 Bari, Italy#TAB#

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, Discretized fractional calculus, SIAM J. Math. Anal., 17, 704, 10.1137/0517050

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

Yuste, 2012, A finite difference method with non-uniform timesteps for fractional diffusion equations, Comput. Phys. Commun., 183, 2594, 10.1016/j.cpc.2012.07.011