TY - JOUR A1 - Diethelm, Kai A1 - Damelin, Steven B. T1 - Boundedness and uniform numerical approximation of the weighted Hilbert transform on the real line JF - Numerical Functional Analysis and Optimization N2 - We establish the uniform boundedness of the weighted Hilbert transform in function spaces associated with a class of even weights on the real line with varying rates of smooth decay near∞. We then consider the numerical approximation of the weighted Hilbert transform and to this end we establish convergence results and error estimates which we prove are sharp. Our formulae are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration, augmented by two carefully chosen extra points. Typical examples of weights that are studied are: (a) w α (x) : = exp(− | x|α , α> 1, x ∈ R; (b) w k,β x: =exp (−expk (| x 7verbar;β)), β> 0, k > 1, x ∈ R. Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20593 N1 - Accepted version des Artikels. This is an Accepted Manuscript of an article published by Taylor & Francis in Numerical Functional Analysis and Optimization on 31 Jan 2014, available at: https://www.tandfonline.com/doi/abs/10.1081/NFA-100103786 Published source: Kai Diethelm (2001) Boundedness and uniform numerical approximation of the weighted Hilbert transform on the real line, Numerical Functional Analysis and Optimization, 22:1-2, 13-45, DOI: 10.1081/nfa-100103786 VL - 22 IS - 1-2 SP - 13 EP - 54 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 T1 - An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives JF - Numerical Algorithms N2 - Traditional methods for the numerical approximation of fractional derivatives have a number of drawbacks due to the non-local nature of the fractional differential operators. The main problems are the arithmetic complexity and the potentially high memory requirements when they are implemented on a computer. In a recent paper, Yuan and Agrawal have proposed an approach for operators of order α ∈ (0,1) that differs substantially from the standard methods. We extend the method to arbitrary α > 0, α∉N, and give an analysis of the main properties of this approach. In particular it turns out that the original algorithm converges rather slowly. Based on our analysis we are able to identify the source of this slow convergence and propose some modifications leading to a much more satisfactory behaviour. Similar results are obtained for a closely related method proposed by Chatterjee. Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20616 N1 - Accepted version des Artikels. Published source: Diethelm, K. An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives. Numer Algor 47, 361–390 (2008). https://doi.org/10.1007/s11075-008-9193-8 VL - 47 SP - 361 EP - 390 ER - TY - JOUR A1 - Danca, Marius-F. A1 - Diethelm, Kai T1 - Fractional-order attractors synthesis via parameter switchings JF - Communications in Nonlinear Science and Numerical Simulation N2 - In this paper we provide numerical evidence, via graphics generated with the help of computer simulations, that switching the control parameter of a dynamical system belonging to a class of fractional-order systems in a deterministic way, one obtains an attractor which belongs to the class of all admissible attractors of the considered system. For this purpose, while a multistep numerical method for fractional-order differential equations approximates the solution to the mathematical model, the control parameter is switched periodically every few integration steps. The switch is made inside of a considered set of admissible parameter values. Moreover, the synthesized attractor matches the attractor obtained with the control parameter replaced with the averaged switched parameter values. The results are verified in this paper on a representative system, the fractional-order Lü system. In this way we were able to extend the applicability of the algorithm presented in earlier papers using a numerical method for fractional differential equations. Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20627 N1 - Accepted version des Artikels VL - 15 IS - 12 SP - 3745 EP - 3753 ER - TY - JOUR A1 - Diethelm, Kai T1 - The limits of reproducibility in numerical simulation JF - Computing in Science and Engineering N2 - Modern computational simulation's increasing and mainly speed-oriented use of HPC systems often conflicts with the goal of making research reproducible. Indeed, the simulations that result from HPC use often behave reproducibly in only a limited way. As a discussion of this phenomenon's technical background describes, the problems entailed will be very difficult to overcome. Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20637 N1 - © 2012 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Published source: K. Diethelm, "The Limits of Reproducibility in Numerical Simulation," in Computing in Science & Engineering, vol. 14, no. 1, pp. 64-72, Jan.-Feb. 2012, doi: 10.1109/MCSE.2011.21. VL - 14 IS - 1 SP - 64 EP - 72 ER - TY - JOUR A1 - Diethelm, Kai T1 - A fractional calculus based model for the simulation of an outbreak of dengue fever JF - Nonlinear Dynamics N2 - We propose a new mathematical model for the simulation of the dynamics of a dengue fever outbreak. Our model differs from the classical model in that it involves nonlinear differential equations of fractional, not integer, order. Using statistics from the 2009 outbreak of the disease in the Cape Verde islands, we demonstrate that our model is capable of providing numerical results that agree very well with the real data. KW - dengue fever KW - disease simulation Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20644 UR - https://doi.org/10.1007/s11071-012-0475-2 N1 - Accepted version des Artikels. Published source: Diethelm, K. A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn 71, 613–619 (2013). https://doi.org/10.1007/s11071-012-0475-2 VL - 71 SP - 613 EP - 619 ER - TY - JOUR A1 - Diethelm, Kai T1 - Increasing the efficiency of shooting methods for terminal value problems of fractional order JF - Journal of Computational Physics N2 - Shooting methods are a well established tool for the numerical solution of terminal value problems of fractional order. However, they can be computationally quite expensive because of their iterative nature in which (a) each single iteration may be costly, and (b) the number of iterations can be large. In this paper we propose algorithmic strategies for improving the efficiency of such methods. Our strategies are aimed at simultaneously reducing the cost of each iteration and reducing the number of required iterations. Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20658 N1 - Accepted version des Artikels VL - 293 SP - 135 EP - 141 ER - TY - JOUR A1 - Diethelm, Kai T1 - An Improvement of a Nonclassical Numerical Method for the Computation of Fractional Derivatives JF - Journal of Vibration and Acoustics N2 - Standard methods for the numerical calculation of fractional derivatives can be slow and memory consuming due to the nonlocality of the differential operators. Yuan and Agrawal (2002, “A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives,” ASME J. Vibr. Acoust., 124, pp. 321–324) have proposed a more efficient approach for operators whose order is between 0 and 1 that differs substantially from the traditional concepts. It seems, however, that the accuracy of the results can be poor. We modify the approach, adapting it better to the properties of the problem, and show that this leads to a significantly improved quality. Our idea also works for operators of order greater than 1. KW - differential equations KW - numerical analysis KW - fractional derivative Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20688 UR - https://asmedigitalcollection.asme.org/vibrationacoustics/article-abstract/131/1/014502/471073/An-Improvement-of-a-Nonclassical-Numerical-Method N1 - Accepted Version des Artikels. © ASME VL - 131 IS - 1 ER - TY - CHAP A1 - Diethelm, Kai T1 - Error estimates for a quadrature rule for Cauchy principal value integrals T2 - Proceedings of Symposia in Applied Mathematics Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20550 UR - http://www.ams.org/books/psapm/048/1314858 N1 - Submitted version des Artikels VL - 48 SP - 287 EP - 291 ER - TY - JOUR A1 - Diethelm, Kai T1 - Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals JF - Numerische Mathematik N2 - We show that, iff∈Ck[−1,1] (k≥2), the error term of every modified positive interpolatory quadrature rule for Cauchy principal value integrals of the type∫−1−1w(x)f(x)x−λdx ,λ∈(−1,1), fulfills Rn[f;λ]=O(n−klnn) uniformly for allλ∈(−1,1), and hence it is of optimal order of magnitude in the classesCk[−1,1] (k=2,3,4,…). Here, w is a weight function with the property0≤w(x)1−x2−−−−−√≤C . We give explicit upper bounds for the Peano-type error constants of such rules. This improves and completes earlier results by Criscuolo and Mastroianni (Calcolo 22 (1985), 391–441 and Numer. Math. 54 (1989), 445–461) and Ioakimidis (Math. Comp. 44 (1985), 191–198). For the special case of the Gaussian rule, we show that the restrictionk≥2 can be dropped. The results are based on a new representation of the Peano kernels of these formulae via the Peano kernels of the underlying classical quadrature formulae. This representation may also be useful in connection with some different problems. Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20560 UR - https://doi.org/10.1007/s002110050183 N1 - Accepted version des Artikels. Published source: Diethelm, K. Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals . Numer. Math. 73, 53–63 (1996). https://doi.org/10.1007/s002110050183 IS - 73 SP - 53 EP - 63 ER -