@article{DiethelmDamelin, author = {Diethelm, Kai and Damelin, Steven B.}, title = {Boundedness and uniform numerical approximation of the weighted Hilbert transform on the real line}, series = {Numerical Functional Analysis and Optimization}, volume = {22}, journal = {Numerical Functional Analysis and Optimization}, number = {1-2}, doi = {https://doi.org/10.1081/nfa-100103786}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20593}, pages = {13 -- 54}, abstract = {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.}, language = {en} } @article{DiethelmFord, author = {Diethelm, Kai and Ford, Neville J.}, title = {Multi-order fractional differential equations and their numerical solution}, series = {Applied Mathematics and Computation}, volume = {154}, journal = {Applied Mathematics and Computation}, number = {3}, doi = {https://doi.org/10.1016/s0096-3003(03)00739-2}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20604}, pages = {621 -- 640}, abstract = {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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives}, series = {Numerical Algorithms}, volume = {47}, journal = {Numerical Algorithms}, doi = {https://doi.org/10.1007/s11075-008-9193-8}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20616}, pages = {361 -- 390}, abstract = {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.}, language = {en} } @article{DancaDiethelm, author = {Danca, Marius-F. and Diethelm, Kai}, title = {Fractional-order attractors synthesis via parameter switchings}, series = {Communications in Nonlinear Science and Numerical Simulation}, volume = {15}, journal = {Communications in Nonlinear Science and Numerical Simulation}, number = {12}, doi = {https://doi.org/10.1016/j.cnsns.2010.01.011}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20627}, pages = {3745 -- 3753}, abstract = {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 L{\"u} 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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {The limits of reproducibility in numerical simulation}, series = {Computing in Science and Engineering}, volume = {14}, journal = {Computing in Science and Engineering}, number = {1}, doi = {https://doi.org/10.1109/mcse.2011.21}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20637}, pages = {64 -- 72}, abstract = {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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {A fractional calculus based model for the simulation of an outbreak of dengue fever}, series = {Nonlinear Dynamics}, volume = {71}, journal = {Nonlinear Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20644}, pages = {613 -- 619}, abstract = {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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {Increasing the efficiency of shooting methods for terminal value problems of fractional order}, series = {Journal of Computational Physics}, volume = {293}, journal = {Journal of Computational Physics}, doi = {https://doi.org/10.1016/j.jcp.2014.10.054}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20658}, pages = {135 -- 141}, abstract = {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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {An Improvement of a Nonclassical Numerical Method for the Computation of Fractional Derivatives}, series = {Journal of Vibration and Acoustics}, volume = {131}, journal = {Journal of Vibration and Acoustics}, number = {1}, doi = {10.1115/1.2981167}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20688}, pages = {4}, abstract = {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.}, language = {en} } @inproceedings{Diethelm, author = {Diethelm, Kai}, title = {Error estimates for a quadrature rule for Cauchy principal value integrals}, series = {Proceedings of Symposia in Applied Mathematics}, volume = {48}, booktitle = {Proceedings of Symposia in Applied Mathematics}, doi = {https://doi.org/10.1090/psapm/048/1314858}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20550}, pages = {287 -- 291}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals}, series = {Numerische Mathematik}, journal = {Numerische Mathematik}, number = {73}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20560}, pages = {53 -- 63}, abstract = {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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {A fractional version of the Peano-Sard theorem}, series = {Numerical Functional Analysis and Optimization}, volume = {18}, journal = {Numerical Functional Analysis and Optimization}, number = {7-8}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20577}, pages = {745 -- 757}, abstract = {Sard's classical generalization of the Peano kernel theorem provides an extremely useful method for expressing and calculating sharp bounds for approximation errors. The error is expressed in terms of a derivative of the underlying function. However, we can apply the theorem only if the approximation is exact on a certain set of polynomials. In this paper, we extend the Peano-Sard theorem to the case that the approximation is exact for a class of generalized polynomials (with non-integer exponents). As a result, we obtain an expression for the remainder in terms of a fractional derivative of the function under consideration. This expression permits us to give sharp error bounds as in the classical situation. An application of our results to the classical functional (vanishing on polynomials) gives error bounds of a new type involving weighted Sobolev-type spaces. In this way, we may state estimates for functions with weaker smoothness properties than usual. The standard version of the Peano-Sard theory is contained in our results as a special case.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {Interpolatory product quadratures for Cauchy principal value integrals with Freud weights}, series = {Numerische Mathematik}, volume = {83}, journal = {Numerische Mathematik}, doi = {https://doi.org/10.1007/s002110050440}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:863-opus-20583}, pages = {87 -- 105}, abstract = {We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value integrals on the real line with Freud-type weight functions. The formulas are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration. As a by-product, we obtain new bounds for the derivative of the functions of the second kind for these weight functions.}, language = {en} } @book{Diethelm, author = {Diethelm, Kai}, title = {The analysis of fractional differential equations}, series = {Lecture Notes in Mathematics}, journal = {Lecture Notes in Mathematics}, edition = {1. Auflage}, publisher = {Springer}, address = {Berlin}, isbn = {978-3-642-14573-5}, issn = {1617-9692}, doi = {10.1007/978-3-642-14574-2}, publisher = {Hochschule f{\"u}r Angewandte Wissenschaften W{\"u}rzburg-Schweinfurt}, pages = {viii+247}, abstract = {Fractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations.}, language = {en} } @book{BaleanuDiethelmScalasetal., author = {Baleanu, Dumitru and Diethelm, Kai and Scalas, Enrico and Trujillo, Juan J.}, title = {Fractional Calculus}, series = {Series on Complexity, Nonlinearity and Chaos: Volume 5}, journal = {Series on Complexity, Nonlinearity and Chaos: Volume 5}, edition = {2. Auflage}, publisher = {World Scientific}, address = {Singapore}, isbn = {978-981-3140-03-5}, doi = {https://doi.org/10.1142/10044}, publisher = {Hochschule f{\"u}r Angewandte Wissenschaften W{\"u}rzburg-Schweinfurt}, pages = {476}, abstract = {This book will give readers the possibility of finding very important mathematical tools for working with fractional models and solving fractional differential equations, such as a generalization of Stirling numbers in the framework of fractional calculus and a set of efficient numerical methods. Moreover, we will introduce some applied topics, in particular fractional variational methods which are used in physics, engineering or economics. We will also discuss the relationship between semi-Markov continuous-time random walks and the space-time fractional diffusion equation, which generalizes the usual theory relating random walks to the diffusion equation. These methods can be applied in finance, to model tick-by-tick (log)-price fluctuations, in insurance theory, to study ruin, as well as in macroeconomics as prototypical growth models. All these topics are complementary to what is dealt with in existing books on fractional calculus and its applications. This book will keep in mind the trade-off between full mathematical rigor and the needs of readers coming from different applied areas of science and engineering. In particular, the numerical methods listed in the book are presented in a readily accessible way that immediately allows the readers to implement them on a computer in a programming language of their choice. The second edition of the book has been expanded and now includes a discussion of additional, newly developed numerical methods for fractional calculus and a chapter on the application of fractional calculus for modeling processes in the life sciences.}, language = {en} } @book{Diethelm, author = {Diethelm, Kai}, title = {Gemeinschaftliches Entscheiden}, series = {Mathematik im Fokus}, journal = {Mathematik im Fokus}, edition = {1. Auflage}, publisher = {Springer}, address = {Berlin}, isbn = {978-3-662-48780-8}, issn = {2570-463X}, doi = {10.1007/978-3-662-48780-8}, publisher = {Hochschule f{\"u}r Angewandte Wissenschaften W{\"u}rzburg-Schweinfurt}, pages = {ix+138}, abstract = {Eine Gruppe, deren Mitglieder sich zwischen mehreren zur Wahl stehenden Alternativen entscheiden m{\"u}ssen, hat eine große Anzahl von M{\"o}glichkeiten, aus den Pr{\"a}ferenzen der Einzelnen eine von der Gemeinschaft getragene Entscheidung zu ermitteln. Wie l{\"a}sst sich sicherstellen, dass diese gemeinschaftliche Entscheidung den Willen der Gruppe sinnvoll widerspiegelt? Die Untersuchung von Methoden, die die vielen individuellen Meinungen zu einer einzigen Entscheidung f{\"u}r die gesamte Gruppe zusammenfassen, und die Darstellung der wichtigsten Eigenschaften dieser Verfahren sind Inhalt dieses Buches. Neben theoretischen {\"U}berlegungen steht dabei gleichberechtigt die Betrachtung zahlreicher Beispiele, die oft unerwartete Eigenschaften erkennen lassen.}, language = {de} } @book{BaleanuDiethelmScalasetal., author = {Baleanu, Dimitru and Diethelm, Kai and Scalas, Enrico and Trujillo, Juan J.}, title = {Fractional Calculus}, series = {Series on Complexity, Nonlinearity and Chaos: Volume 3}, journal = {Series on Complexity, Nonlinearity and Chaos: Volume 3}, edition = {1.}, publisher = {World Scientifc}, address = {Singapore}, isbn = {978-981-4355-20-9}, doi = {10.1142/8180}, pages = {428}, abstract = {The subject of fractional calculus and its applications (that is, convolution-type pseudo-differential operators including integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, mainly due to its applications in diverse fields of science and engineering. These operators have been used to model problems with anomalous dynamics, however, they also are an effective tool as filters and controllers, and they can be applied to write complicated functions in terms of fractional integrals or derivatives of elementary functions, and so on. This book will give readers the possibility of finding very important mathematical tools for working with fractional models and solving fractional differential equations, such as a generalization of Stirling numbers in the framework of fractional calculus and a set of efficient numerical methods. Moreover, we will introduce some applied topics, in particular fractional variational methods which are used in physics, engineering or economics. We will also discuss the relationship between semi-Markov continuous-time random walks and the space-time fractional diffusion equation, which generalizes the usual theory relating random walks to the diffusion equation. These methods can be applied in finance, to model tick-by-tick (log)-price fluctuations, in insurance theory, to study ruin, as well as in macroeconomics as prototypical growth models. All these topics are complementary to what is dealt with in existing books on fractional calculus and its applications. This book was written with a trade-off in mind between full mathematical rigor and the needs of readers coming from different applied areas of science and engineering. In particular, the numerical methods listed in the book are presented in a readily accessible way that immediately allows the readers to implement them on a computer in a programming language of their choice. Numerical code is also provided.}, language = {en} } @phdthesis{Diethelm, author = {Diethelm, Kai}, title = {Numerische Approximation von Cauchy-Hauptwert-Integralen unter theoretischen und rechnerorientierten Aspekten}, language = {de} } @article{Diethelm, author = {Diethelm, Kai}, title = {Modifed compound quadrature rules for strongly singular integrals}, series = {Computing}, volume = {52}, journal = {Computing}, doi = {10.1007/BF02276881}, pages = {337 -- 354}, abstract = {We show that the error term of every modified compound quadrature rule for Cauchy principal value integrals with degree of exactnesss is of optimal order of magnitude in the classesC k[-1,1],k=1,2,...,s, but not inC s+1[-1,1]. We give explicit upper bounds for the error constants of the modified midpoint rule, the modified trapezoidal rule and the modified Simpson rule. Furthermore, the results are generalized to analogous rules for Hadamard-type finite part integrals.}, language = {en} } @incollection{DiethelmFreed, author = {Diethelm, Kai and Freed, Alan D.}, title = {The FracPECE subroutine for the numerical solution of differential equations of fractional order}, series = {Forschung und Wissenschaftliches Rechnen: Beitr{\"a}ge zum Heinz-Billing-Preis 1998}, booktitle = {Forschung und Wissenschaftliches Rechnen: Beitr{\"a}ge zum Heinz-Billing-Preis 1998}, editor = {Heinzel, S. and Plesser, T.}, publisher = {Gesellschaft f{\"u}r wissenschaftliche Datenverarbeitung}, address = {G{\"o}ttingen}, pages = {57 -- 71}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {A method for the practical evaluation of the Hilbert transform on the real line}, series = {Journal of Computational and Applied Mathematics}, volume = {112}, journal = {Journal of Computational and Applied Mathematics}, number = {1-2}, doi = {10.1016/S0377-0427(99)00212-5}, pages = {45 -- 53}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {Estimation of quadrature errors in terms of Caputo-type fractional derivatives}, series = {Fractional Calculus and Applied Analysis}, volume = {2}, journal = {Fractional Calculus and Applied Analysis}, pages = {313 -- 327}, language = {en} } @article{DiethelmKoehler, author = {Diethelm, Kai and K{\"o}hler, Peter}, title = {Asymptotic behaviour of fixed-order error constants of modified quadrature formulae for Cauchy principal value integrals}, series = {Journal of Inequalities and Applications}, volume = {5}, journal = {Journal of Inequalities and Applications}, number = {2}, doi = {10.1155/S1025583400000096}, pages = {167 -- 190}, abstract = {We consider quadrature formulae for Cauchy principal value integrals Iw,ζ[f]=∫abf(x)x-ζw(x)dx,  a<ζ