TY - JOUR A1 - Diethelm, Kai A1 - Ford, Judith M. A1 - Ford, Neville J. A1 - Weilbeer, Marc T1 - Pitfalls in fast numerical solvers for fractional differential equations JF - Journal of Computational and Applied Mathematics Y1 - 2006 U6 - https://doi.org/10.1016/j.cam.2005.03.023 VL - 186 IS - 2 SP - 482 EP - 503 ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Judith M. A1 - Ford, Neville J. A1 - Weilbeer, Marc ED - Le Mehaute, A. ED - Tenreiro Machado, José A. ED - Trigeassou, J. C. ED - Sabatier, J. T1 - A comparison of backward differentiation approaches for ordinary and partial differential equations of fractional order JF - Fractional Differentiation and its Applications Y1 - 2005 SP - 557 EP - 569 PB - Ubooks CY - Neusäß ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Neville J. T1 - Multi-order fractional differential equations and their numerical solution JF - Applied Mathematics and Computation N2 - We consider the numerical solution of (possibly nonlinear) fractional differential equations of the form y(α)(t)=f(t,y(t),y(β1)(t),y(β2)(t),…,y(βn)(t)) with α>βn>βn−1>⋯>β1 and α−βn⩽1, βj−βj−1⩽1, 0<β1⩽1, combined with suitable initial conditions. The derivatives are understood in the Caputo sense. We begin by discussing the analytical questions of existence and uniqueness of solutions, and we investigate how the solutions depend on the given data. Moreover we propose convergent and stable numerical methods for such initial value problems. Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20604 N1 - Accepted version des Artikels VL - 154 IS - 3 SP - 621 EP - 640 ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Neville J. T1 - Numerical solution methods for distributed order differential equations JF - Fractional Calculus and Applied Analysis Y1 - 2001 VL - 4 SP - 531 EP - 542 ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Neville J. T1 - Analysis of fractional differential equations JF - Journal of Mathematical Analysis and Applications N2 - We discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order. The differential operators are taken in the Riemann–Liouville sense and the initial conditions are specified according to Caputo's suggestion, thus allowing for interpretation in a physically meaningful way. We investigate in particular the dependence of the solution on the order of the differential equation and on the initial condition, and we relate our results to the selection of appropriate numerical schemes for the solution of fractional differential equations. Y1 - 2002 U6 - https://doi.org/10.1006/jmaa.2000.7194 VL - 265 IS - 2 SP - 229 EP - 248 ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Neville J. A1 - Freed, Alan D. T1 - A predictor-corrector approach for the numerical solution of fractional differential equations JF - Nonlinear Dynamics N2 - We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too. Y1 - 2002 U6 - https://doi.org/10.1023/A:1016592219341 VL - 29 SP - 3 EP - 22 ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Neville J. T1 - Numerical solution of the Bagley-Torvik equation JF - BIT Numerical Mathematics N2 - We consider the numerical solution of the Bagley-Torvik equation Ay″(t) + BD * 3/2 y(t) + Cy(t) = f(t), as a prototype fractional differential equation with two derivatives. Approximate solutions have recently been proposed in the book and papers of Podlubny in which the solution obtained with approximate methods is compared to the exact solution. In this paper we consider the reformulation of the Bagley-Torvik equation as a system of fractional differential equations of order 1/2. This allows us to propose numerical methods for its solution which are consistent and stable and have arbitrarily high order. In this context we specifically look at fractional linear multistep methods and a predictor-corrector method of Adams type. Y1 - 2002 U6 - https://doi.org/10.1023/A:1021973025166 VL - 42 SP - 490 EP - 507 ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Neville J. A1 - Freed, Alan D. T1 - Detailed error analysis for a fractional Adams method JF - Numerical Algorithms N2 - We investigate a method for the numerical solution of the nonlinear fractional differential equation D * α y(t)=f(t,y(t)), equipped with initial conditions y (k)(0)=y 0 (k), k=0,1,...,⌈α⌉−1. Here α may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour. Y1 - 2004 U6 - https://doi.org/10.1023/B:NUMA.0000027736.85078.be VL - 36 SP - 31 EP - 52 ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Neville J. A1 - Freed, Alan D. A1 - Luchko, Yuri T1 - Algorithms for the fractional calculus: A selection of numerical methods JF - Computer Methods in Applied Mechanics and Engineering N2 - Many recently developed models in areas like viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context. We believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way. Y1 - 2005 U6 - https://doi.org/10.1016/j.cma.2004.06.006 VL - 194 SP - 743 EP - 773 ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Neville J. T1 - Numerical analysis for distributed-order differential equations JF - Journal of Computational and Applied Mathematics Y1 - 2009 U6 - https://doi.org/10.1016/j.cam.2008.07.018 VL - 225 IS - 1 SP - 96 EP - 104 ER -