TY - BOOK A1 - Baleanu, Dumitru A1 - Diethelm, Kai A1 - Scalas, Enrico A1 - Trujillo, Juan J. T1 - Fractional Calculus BT - Models and numerical methods T3 - Series on Complexity, Nonlinearity and Chaos: Volume 5 N2 - 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. KW - fractional calculus KW - mathematical model KW - numerical method Y1 - 2016 SN - 978-981-3140-03-5 U6 - https://doi.org/https://doi.org/10.1142/10044 PB - World Scientific CY - Singapore ET - 2. Auflage 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 - 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 - TY - JOUR A1 - Diethelm, Kai T1 - Interpolatory product quadratures for Cauchy principal value integrals with Freud weights JF - Numerische Mathematik N2 - 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. Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20583 N1 - Accepted version des Artikels. Published source: Damelin, S., Diethelm, K. Interpolatory product quadratures for Cauchy principal value integrals with Freud weights. Numer. Math. 83, 87–105 (1999). https://doi.org/10.1007/s002110050440 VL - 83 SP - 87 EP - 105 ER - 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 - 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 - 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 A1 - Kiryakova, Virginia A1 - Luchko, Yuri A1 - Tenreiro Machado, José A. A1 - Tarasov, Vasily E. T1 - Trends, directions for further research, and some open problems of fractional calculus JF - Nonlinear Dynamics N2 - The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research. KW - fractional calculus Y1 - 2022 U6 - https://doi.org/https://doi.org/10.1007/s11071-021-07158-9 SN - 1573-269X VL - 107 SP - 3245 EP - 3270 ER - TY - JOUR A1 - Diethelm, Kai A1 - Tuan, Hoang The T1 - Upper and lower estimates for the separation of solutions to fractional differential equations JF - Fractional Calculus and Applied Analysis N2 - Given a fractional differential equation of order α∈(0,1] with Caputo derivatives, we investigate in a quantitative sense how the associated solutions depend on their respective initial conditions. Specifically, we look at two solutions x1 and x2, say, of the same differential equation, both of which are assumed to be defined on a common interval [0, T], and provide upper and lower bounds for the difference x1(t)−x2(t) for all t∈[0,T] that are stronger than the bounds previously described in the literature. Y1 - 2022 U6 - https://doi.org/10.1007/s13540-021-00007-x SN - 1314-2224 VL - 25 SP - 166 EP - 180 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 - BOOK A1 - Baleanu, Dimitru A1 - Diethelm, Kai A1 - Scalas, Enrico A1 - Trujillo, Juan J. T1 - Fractional Calculus BT - Models and numerical methods T3 - Series on Complexity, Nonlinearity and Chaos: Volume 3 N2 - 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. Y1 - 2012 SN - 978-981-4355-20-9 U6 - https://doi.org/10.1142/8180 PB - World Scientifc CY - Singapore ET - 1. ER - TY - JOUR A1 - Diethelm, Kai T1 - Modifed compound quadrature rules for strongly singular integrals JF - Computing N2 - 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. N2 - Wir zeigen, daß der Fehlerterm jedes modifizierten zusammengesetzten Quadraturverfahrens für Cauchy-Hauptwert-Integrale mit Exaktheitsgrads die bestmögliche Größenordnung in den KlassenC k[−1,1],k=1,2,...,s hat, aber nich inC s+1[−1,1]. Explizite obere Schranken für die Fehlerkonstanten der modifizierten Mittelpunkt-, Trapez- und Simpson-Verfahren werden angegeben. Des weiteren werden die Ergebnisse auf die entsprechenden Verfahren für Finite-Part-Integrale vom Hadamardschen Typ verallgemeinert. Y1 - 1994 U6 - https://doi.org/10.1007/BF02276881 VL - 52 SP - 337 EP - 354 ER - TY - JOUR A1 - Diethelm, Kai T1 - Uniform convergence of optimal order quadrature rules for Cauchy principal value integrals JF - Journal of Computational and Applied Mathematics N2 - For the numerical evaluation of Cauchy principal value integrals of the form , λε(−1, 1), f εCs[− 1, 1], we consider a quadrature method based on spline interpolation of odd degree 2k + 1,k ∈N0. We show that these rules converge uniformly for λ ∈ (− 1, 1). In particular, we calculate the exact order of magnitude of the error and show that it is equal to the order of the optimal remainder in the class of functions with bounded sth derivative if s ε s;;2k + 1, 2k + 2};. Finally, we compare the rule to the well-known quadrature rule of Elliott and Paget which only converges pointwise. Y1 - 1994 U6 - https://doi.org/10.1016/0377-0427(94)90086-8 VL - 56 IS - 3 SP - 321 EP - 329 ER - TY - THES A1 - Diethelm, Kai T1 - Numerische Approximation von Cauchy-Hauptwert-Integralen unter theoretischen und rechnerorientierten Aspekten Y1 - 1994 ER - TY - JOUR A1 - Diethelm, Kai T1 - Non-optimality of certain quadrature rules for Cauchy principal value integrals JF - Zeitschrift für angewandte Mathematik und Mechanik Y1 - 1994 VL - 74 IS - 6 SP - T689 EP - T690 ER - TY - JOUR A1 - Diethelm, Kai T1 - The order of convergence of modified interpolatory quadratures for singular integrals of Cauchy type JF - Zeitschrift für angewandte Mathematik und Mechanik Y1 - 1995 VL - 75 SP - 621 EP - 622 ER - TY - JOUR A1 - Diethelm, Kai T1 - Gaussian quadrature formulae of the third kind for Cauchy principal value integrals: Basic properties and error estimates JF - Journal of Computational and Applied Mathematics Y1 - 1995 U6 - https://doi.org/10.1016/0377-0427(95)00103-4 VL - 65 IS - 1-3 SP - 97 EP - 114 ER - TY - JOUR A1 - Diethelm, Kai T1 - Generalized compound quadrature formulae for finite-part integrals JF - IMA Journal of Numerical Analysis N2 - We investigate the error term of the dth degree compound quadrature formulae for finite-part integrals of the form ∫10x−pf(x) dx where p∈ and p ≥1. We are mainly interested in error bounds of the form |R[f]|≤c∥∥f(s)∥∥∞ with best possible constants c. It is shown that, for p∉ and n uniformly distributed nodes, the error behaves as O(np–s–1 for f∈Cs[0,1]⁠, p–1