TY - CHAP A1 - Diethelm, Kai A1 - Freed, Alan D. ED - Heinzel, S. ED - Plesser, T. T1 - The FracPECE subroutine for the numerical solution of differential equations of fractional order T2 - Forschung und Wissenschaftliches Rechnen: Beiträge zum Heinz-Billing-Preis 1998 Y1 - 1999 SP - 57 EP - 71 PB - Gesellschaft für wissenschaftliche Datenverarbeitung CY - Göttingen 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 - CHAP A1 - Diethelm, Kai A1 - Freed, Alan D. ED - Keil, Frerich ED - Mackens, Wolfgang ED - Voß, Heinrich ED - Werther, Joachim T1 - On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. T2 - Scientific Computing in Chemical Engineering II - Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties N2 - The authors have recently developed a mathematical model for the description of the behavior of viscoplastic materials. The model is based on a nonlinear differential equation of order β, where β is a material constant typically in the range 0 < β < 1. This equation is coupled with a first-order differential equation. In the present paper, we introduce and discuss a numerical scheme for the numerical solution of these equations. The algorithm is based on a PECE-type approach. Y1 - 1999 SN - 978-3-642-64295-1 SN - 978-3-642-60185-9 U6 - https://doi.org/10.1007/978-3-642-60185-9_24 SP - 217 EP - 224 PB - Springer CY - Heidelberg 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 - Freed, Alan D. T1 - An efficient algorithm for the evaluation of convolution integrals JF - Computers & Mathematics with Applications N2 - We propose an algorithm for the numerical evaluation of convolution integrals of the form ∫0xk(x−y)f(y,x) dy, for x∈[0,X]. Our method is especially suitable in situations where the fundamental interval [0, X] is very long and the kernel function k is expensive to calculate. Separate versions are provided where the forcing function f is known in advance, and where it must be determined step-by-step along the solution path. These methods are efficient with respect to both run time and memory requirements. Y1 - 2006 U6 - https://doi.org/10.1016/j.camwa.2005.07.010 VL - 51 IS - 1 SP - 51 EP - 72 ER - TY - JOUR A1 - Freed, Alan D. A1 - Diethelm, Kai T1 - Fractional calculus in biomechanics: A 3D viscoelastic model using regularized fractional-derivative kernels with application to the human calcaneal fat pad JF - Biomechanics and Modeling in Mechanobiology N2 - A viscoelastic model of the K-BKZ (Kaye, Technical Report 134, College of Aeronautics, Cranfield 1962; Bernstein et al., Trans Soc Rheol 7: 391–410, 1963) type is developed for isotropic biological tissues and applied to the fat pad of the human heel. To facilitate this pursuit, a class of elastic solids is introduced through a novel strain-energy function whose elements possess strong ellipticity, and therefore lead to stable material models. This elastic potential – via the K-BKZ hypothesis – also produces the tensorial structure of the viscoelastic model. Candidate sets of functions are proposed for the elastic and viscoelastic material functions present in the model, including two functions whose origins lie in the fractional calculus. The Akaike information criterion is used to perform multi-model inference, enabling an objective selection to be made as to the best material function from within a candidate set. Y1 - 2006 U6 - https://doi.org/10.1007/s10237-005-0011-0 VL - 5 SP - 203 EP - 215 ER - TY - JOUR A1 - Freed, Alan D. A1 - Diethelm, Kai T1 - Caputo derivatives in viscoelasticity: A non-linear finite-deformation theory for tissue JF - Fractional Calculus and Applied Analysis N2 - The popular elastic law of Fung that describes the non-linear stress-strain behavior of soft biological tissues is extended into a viscoelastic ma-terial model that incorporates fractional derivatives in the sense of Caputo. This one-dimensional material model is then transformed into a three-dimensional constitutive model that is suitable for general analysis. The model is derived in a configuration that differs from the current, or spatial, configuration by a rigid-body rotation; it being the polar config-uration. Mappings for the fractional-order operators of integration and differentiation between the polar and spatial configurations are presented as a theorem. These mappings are used in the construction of the proposed viscoelastic model. Y1 - 2007 UR - https://www.diogenes.bg/fcaa/volume10/fcaa103/Freed_Diethelm_103.pdf VL - 10 IS - 3 SP - 219 EP - 248 ER - TY - JOUR A1 - Freed, Alan D. A1 - Diethelm, Kai ED - Le Mehaute, A. ED - Tenreiro Machado, José A. ED - Trigeassou, J. C. ED - Sabatier, J. T1 - Tensor fields for use in fractional-order viscoelasticity JF - Fractional Differentiation and its Applications Y1 - 2005 SP - 169 EP - 182 PB - Ubooks CY - Neusäß ER -