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 0 (not necessarily α j ∈ ℕ) in the sense of Caputo. The methods are based on numerical integration techniques applied to an equivalent nonlinear and weakly singular Volterra integral equation. The classical approach leads to an algorithm with very high arithmetic complexity. Therefore we derive an alternative that leads to lower complexity without sacrificing too much precision. Y1 - 2003 U6 - https://doi.org/10.1007/s00607-003-0033-3 VL - 71 SP - 305 EP - 319 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 - CHAP A1 - Diethelm, Kai ED - Lipitakis, E. A. T1 - Predictor-corrector strategies for single- and multi-term fractional differential equations T2 - Proceedings of the 5th Hellenic-European Conference on Computer Mathematics and its Applications Y1 - 2002 SP - 117 EP - 122 PB - LEA Press CY - Athen ER - TY - JOUR A1 - Diethelm, Kai A1 - Weilbeer, Marc T1 - A numerical approach for Joulin's model of a point source initiated flame JF - Fractional Calculus and Applied Analysis Y1 - 2004 VL - 7 SP - 191 EP - 212 ER - TY - JOUR A1 - Diethelm, Kai T1 - A fractional version of the Peano-Sard theorem JF - Numerical Functional Analysis and Optimization N2 - 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. Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20577 UR - https://doi.org/10.1080/01630569708816789 N1 - Accepted version des Artikels. This is an Accepted Manuscript of an article published by Taylor & Francis in Numerical Functional Analysis and Optimization on 15 May 2007, available at: https://www.tandfonline.com/doi/abs/10.1080/01630569708816789. Published source: Kai Diethelm (1997) A fractional version of the peano-sard theorem, Numerical Functional Analysis and Optimization, 18:7-8, 745-757, DOI: 10.1080/01630569708816789 VL - 18 IS - 7-8 SP - 745 EP - 757 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 - 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 - BOOK A1 - Diethelm, Kai T1 - The analysis of fractional differential equations BT - An application-oriented exposition using differential operators of Caputo type T3 - Lecture Notes in Mathematics N2 - 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. KW - fractional calculus KW - fractional differential equation KW - Caputo derivative KW - Riemann-Liouville integral KW - Riemann-Liouville derivative Y1 - 2010 SN - 978-3-642-14573-5 U6 - https://doi.org/10.1007/978-3-642-14574-2 SN - 1617-9692 PB - Springer CY - Berlin ET - 1. Auflage ER - TY - JOUR A1 - Diethelm, Kai T1 - On the separation of solutions of fractional differential equations JF - Fractional Calculus and Applied Analysis N2 - Consider two different solutions of a first-order differential equation. Under rather general conditions we know that these two functions are separated from each other, i.e. their graphs never meet or even cross each other. We ask whether such a result is true for Caputo-type fractional differential equations as well. We can give a partial answer that is positive in some situations and negative under different assumptions. For the remaining cases we state a conjecture and explain why we believe in it. A key ingredient of the analysis is a result concerning the existence of zeros of the solutions of a class of Volterra equations. Y1 - 2008 UR - https://www.diogenes.bg/fcaa/volume11/fcaa113/KDiethelm_fcaa113.pdf VL - 11 SP - 259 EP - 268 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 T1 - Smoothness properties of solutions of Caputo-type fractional differential equations JF - Fractional Calculus and Applied Analysis N2 - We consider ordinary fractional differential equations with Caputo-type differential operators with smooth right-hand sides. In various places in the literature one can find the statement that such equations cannot have smooth solutions. We prove that this is wrong, and we give a full charac-terization of the situations where smooth solutions exist. The results can be extended to a class of weakly singular Volterra integral equations. Y1 - 2007 UR - https://www.diogenes.bg/fcaa/volume10/fcaa102/Diethelm_102.pdf VL - 10 IS - 2 SP - 151 EP - 160 ER - TY - JOUR A1 - Murawski, Kris A1 - Diethelm, Kai T1 - Randomly generated spectrum of the solar f-mode JF - Astronomy and Astrophysics N2 - We show that a random flow that depends on space and time generates a mode which is damped and move slower than the coherent f-mode as well as possesses properties which are consistent with the recent SOHO/MDI data (Antia & Basu 1999). Y1 - 2000 UR - https://ui.adsabs.harvard.edu/link_gateway/2000A%26A...358..753M/ADS_PDF VL - 358 SP - P753 EP - P758 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 - Garrappa, Roberto A1 - Stynes, Martin T1 - Why fractional derivatives with nonsingular kernels should not be used JF - Fractional Calculus and Applied Analysis N2 - In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new. Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-20666 N1 - Accepted version des Artikels. Published source: Diethelm, K., Garrappa, R., Giusti, A. et al. Why Fractional Derivatives with Nonsingular Kernels Should Not Be Used. Fract Calc Appl Anal 23, 610–634 (2020). https://doi.org/10.1515/fca-2020-0032 VL - 23 SP - 610 EP - 634 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 - 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 - BOOK A1 - Diethelm, Kai T1 - Gemeinschaftliches Entscheiden BT - Untersuchung von Entscheidungsverfahren mit mathematischen Hilfsmitteln T3 - Mathematik im Fokus N2 - Eine Gruppe, deren Mitglieder sich zwischen mehreren zur Wahl stehenden Alternativen entscheiden müssen, hat eine große Anzahl von Möglichkeiten, aus den Präferenzen der Einzelnen eine von der Gemeinschaft getragene Entscheidung zu ermitteln. Wie lässt 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 für die gesamte Gruppe zusammenfassen, und die Darstellung der wichtigsten Eigenschaften dieser Verfahren sind Inhalt dieses Buches. Neben theoretischen Überlegungen steht dabei gleichberechtigt die Betrachtung zahlreicher Beispiele, die oft unerwartete Eigenschaften erkennen lassen. KW - Wahlverfahren Y1 - 2016 SN - 978-3-662-48780-8 U6 - https://doi.org/10.1007/978-3-662-48780-8 SN - 2570-463X PB - Springer CY - Berlin ET - 1. Auflage 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 - Diethelm, Kai T1 - Multi-term fractional differential equations, multi-order fractional differential systems and their numerical solution JF - Journal Européen des Systèmes Automatisés N2 - Fractional differential equations containing only one fractional derivative are a well understood and frequently used tool for the mathematical description of many physical processes, but they are not always sufficient to reflect all the relevant phenomena. It is sometimes necessary to use models with more than one fractional derivative. We describe two important mathematical ways to use this concept: multi-term equations and multi-order systems. First we show the relations between these two concepts. Then we investigate their most important analytical properties, and finally we look at numerical methods for their approximate solution. Y1 - 2008 U6 - https://doi.org/10.3166/jesa.42.665-676 VL - 42 IS - 6 SP - 665 EP - 676 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 - 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 - an Mey, Dieter A1 - Biersdorf, Scott A1 - Bischof, Christian A1 - Diethelm, Kai A1 - Eschweiler, Dominic A1 - Gerndt, Michael A1 - Knüpfer, Andreas A1 - Lorenz, Daniel A1 - Malony, Allen A1 - Nagel, Wolfgang E. A1 - Oleynik, Yury A1 - Rössel, Christian A1 - Saviankou, Pavel A1 - Schmidl, Dirk A1 - Shende, Sameer A1 - Wagner, Michael A1 - Wesarg, Bert A1 - Wolf, Felix ED - Bischof, Christian ED - Hegering, Heinz-Gerd ED - Nagel, Wolfgang E. ED - Wittum, Gabriel T1 - Score-P: A Unified Performance Measurement System for Petascale Applications JF - Competence in High Performance Computing 2010 N2 - The rapidly growing number of cores on modern supercomputers imposes scalability demands not only on applications but also on the software tools needed for their development. At the same time, increasing application and system complexity makes the optimization of parallel codes more difficult, creating a need for scalable performance-analysis technology with advanced functionality. However, delivering such an expensive technology can hardly be accomplished by single tool developers and requires higher degrees of collaboration within the HPC community. The unified performance-measurement system Score-P is a joint effort of several academic performance-tool builders, funded under the BMBF program HPC-Software für skalierbare Parallelrechner in the SILC project (Skalierbare Infrastruktur zur automatischen Leistungsanalyse paralleler Codes). It is being developed with the objective of creating a common basis for several complementary optimization tools in the service of enhanced scalability, improved interoperability, and reduced maintenance cost. Y1 - 2012 SN - 978-3-642-24024-9 SN - 978-3-642-24025-6 U6 - https://doi.org/10.1007/978-3-642-24025-6_8 SP - 85 EP - 97 PB - Springer CY - Berlin, Heidelberg ER - TY - JOUR A1 - Diethelm, Kai T1 - An efficient parallel algorithm for the numerical solution of fractional differential equations JF - Fractional Calculus and Applied Analysis N2 - The numerical solution of differential equations of fractional order is known to be a computationally very expensive problem due to the nonlocal nature of the fractional differential operators. We demonstrate that parallelization may be used to overcome these difficulties. To this end we propose to implement the fractional version of the second-order Adams-Bashforth-Moulton method on a parallel computer. According to many recent publications, this algorithm has been successfully applied to a large number of fractional differential equations arising from a variety of application areas. The precise nature of the parallelization concept is discussed in detail and some examples are given to show the viability of our approach. Y1 - 2011 U6 - https://doi.org/10.2478/s13540-011-0029-1 VL - 14 SP - 475 EP - 490 ER - TY - JOUR A1 - Diethelm, Kai T1 - The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus JF - Fractional Calculus and Applied Analysis N2 - We generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz. the Riemann-Liouville operator, instead of a classical (firstorder) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations. Y1 - 2012 U6 - https://doi.org/10.2478/s13540-012-0022-3 N1 - Zu diesem Artikel wurde 2017 ein Erratum veröffentlicht VL - 15 SP - 304 EP - 313 ER - TY - JOUR A1 - Diethelm, Kai T1 - Erratum: The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus JF - Fractional Calculus and Applied Analysis N2 - In some of the statements of the author’s paper “The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus” (Fract. Calc. Appl. Anal. 15, No 2 (2012), pp. 304–313), a factor α was missing. This note provides the correct formulations. Y1 - 2017 U6 - https://doi.org/10.1515/fca-2017-0082 N1 - Der orginale Artikel findet sich hier: https://doi.org/10.2478/s13540-012-0022-3 VL - 20 SP - 1567 EP - 1570 ER - TY - CHAP A1 - Diethelm, Kai A1 - Kassem-Manthey, Koutaiba A1 - Thilo, Frank A1 - Grauer, Manfred T1 - On the application of computational optimization methods to increase the efficiency of metal forming processes T2 - Proceedings of the IVth European Conference on Computational Mechanics Y1 - 2010 CY - Paris ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Neville J. T1 - Volterra integral equations and fractional calculus: Do neighbouring solutions intersect? JF - Journal of Integral Equations and Applications Y1 - 2012 U6 - https://doi.org/10.1216/JIE-2012-24-1-25 VL - 24 IS - 1 SP - 25 EP - 37 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 - TY - JOUR A1 - Knüpfer, Andreas A1 - Rössel, Christian A1 - an Mey, Dieter A1 - Diethelm, Kai A1 - Biersdorf, Scott A1 - Eschweiler, Dominic A1 - Geimer, Markus A1 - Gerndt, Michael A1 - Lorenz, Daniel A1 - Malony, Allen A1 - Nagel, Wolfgang E. A1 - Oleynik, Yury A1 - Philippen, Peter A1 - Saviankou, Pavel A1 - Schmidl, Dirk A1 - Shende, Sameer A1 - Tschüter, Ronny A1 - Wagner, Michael A1 - Wesarg, Bert A1 - Wolf, Felix ED - Brunst, Holger ED - Müller, Matthias S. ED - Nagel, Wolfgang E. ED - Resch, Michael M. T1 - Score-P: A Joint Performance Measurement Run-Time Infrastructure for Periscope, Scalasca, TAU, and Vampir JF - Tools for High Performance Computing 2011 N2 - This paper gives an overview about the Score-P performance measurement infrastructure which is being jointly developed by leading HPC performance tools groups. It motivates the advantages of the joint undertaking from both the developer and the user perspectives, and presents the design and components of the newly developed Score-P performance measurement infrastructure. Furthermore, it contains first evaluation results in comparison with existing performance tools and presents an outlook to the long-term cooperative development of the new system. Y1 - 2012 SN - 978-3-642-31475-9 SN - 978-3-642-31476-6 U6 - https://doi.org/10.1007/978-3-642-31476-6_7 SP - 79 EP - 91 PB - Springer CY - Berlin, Heidelberg ER - TY - JOUR A1 - Diethelm, Kai T1 - Error bounds for the numerical integration of functions with limited smoothness JF - SIAM Journal on Numerical Analysis N2 - Recently, Trefethen (SIAM Rev., 50 (2008), pp. 67--87) and Xiang and Bornemann (SIAM J. Numer. Anal., 50 (2012), pp. 2581--2587) investigated error bounds for n-point Gauss and Clenshaw--Curtis quadrature for the Legendre weight with integrands having limited smoothness properties. Putting their results into the context of classical quadrature theory, we find that the observed behavior is by no means surprising and that it can essentially be proved for a very large class of quadrature formulas with respect to a broad set of weight functions. Y1 - 2014 U6 - https://doi.org/10.1137/130921246 VL - 52 IS - 2 SP - 877 EP - 879 ER - TY - JOUR A1 - Schuchart, Joseph A1 - Gerndt, Michael A1 - Kjeldsberg, Per Gunnar A1 - Diethelm, Kai A1 - Lysaght, Michael A1 - Horák, David A1 - Říha, Lubomír A1 - Gocht, Andreas A1 - Sourouri, Mohammed A1 - Kumaraswamy, Madhura A1 - Chowdhury, Anamika A1 - Jahre, Magnus A1 - Bouizi, Othman A1 - Sabir Mian, Umbreen A1 - Kružík, Jakub A1 - Sojka, Radim A1 - Beseda, Martin A1 - Kannan, Venkatesh A1 - Bendifallah, Zakaria A1 - Hackenberg, Daniel A1 - Nagel, Wolfgang E. T1 - The READEX formalism for automatic tuning for energy efficiency JF - Computing N2 - Energy efficiency is an important aspect of future exascale systems, mainly due to rising energy cost. Although High performance computing (HPC) applications are compute centric, they still exhibit varying computational characteristics in different regions of the program, such as compute-, memory-, and I/O-bound code regions. Some of today’s clusters already offer mechanisms to adjust the system to the resource requirements of an application, e.g., by controlling the CPU frequency. However, manually tuning for improved energy efficiency is a tedious and painstaking task that is often neglected by application developers. The European Union’s Horizon 2020 project READEX (Runtime Exploitation of Application Dynamism for Energy-efficient eXascale computing) aims at developing a tools-aided approach for improved energy efficiency of current and future HPC applications. To reach this goal, the READEX project combines technologies from two ends of the compute spectrum, embedded systems and HPC, constituting a split design-time/runtime methodology. From the HPC domain, the Periscope Tuning Framework (PTF) is extended to perform dynamic auto-tuning of fine-grained application regions using the systems scenario methodology, which was originally developed for improving the energy efficiency in embedded systems. This paper introduces the concepts of the READEX project, its envisioned implementation, and preliminary results that demonstrate the feasibility of this approach. Y1 - 2017 U6 - https://doi.org/10.1007/s00607-016-0532-7 VL - 99 SP - 727 EP - 745 ER - TY - CHAP A1 - Diethelm, Kai ED - Spasic, D. T. ED - Grahovac, N. ED - Zigic, M. ED - Rapaic, M. ED - Atanackovic, T. M. T1 - Properties of the solutions to "fractionalized" ODE systems, with applications to processes arising in the life sciences T2 - Proceedings of the International Conference on Fractional Differentiation and its Applications 2016, Vol. 1 N2 - Traditional mathematical models for many phenomena in various different fields of science are based on the use of integer order derivatives. These models are usually well understood from an analytic point of view, in particular regarding the qualitative behaviour of their solutions. The availability of such information is important for evaluating whether the mathematical model really reflects the actual properties that the process in question has, and thus for showing that the set of equations is indeed a suitable model for the concrete process. In many cases, fractional order generalizations of the integer order models allow to obtain better quantitative agreement with experimental data, but the knowledge about qualitative properties is frequently lacking. Thus, the question whether the fractional order model is in fact able to correctly reproduce the behaviour that the underlying process must exhibit frequently remains unanswered. In this paper we want to use a specific example from the life sciences, viz. a model describing a fermentation process, in order to initiate a discussion of this matter. Y1 - 2016 VL - 1 SP - 32 EP - 44 PB - Faculty of Technical Sciences CY - Novi Sad ER - TY - CHAP A1 - Diethelm, Kai T1 - Tools for assessing and optimizing the energy requirements of high performance scientific computing software T2 - Proceedings in Applied Mathematics and Mechanics N2 - Score-P is a measurement infrastructure originally designed for the analysis and optimization of the performance of HPC codes. Recent extensions of Score-P and its associated tools now also allow the investigation of energy-related properties and support the user in the implementation of corresponding improvements. Since it would be counterproductive to completely ignore performance issues in this connection, the focus should not be laid exclusively on energy. We therefore aim to optimize software with respect to an objective function that takes into account energy and run time. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) Y1 - 2016 U6 - https://doi.org/10.1002/pamm.201610407 VL - 16 IS - 1 SP - 837 EP - 838 ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Neville J. T1 - A note on the well-posedness of terminal value problems for fractional differential equations JF - Journal of Integral Equations and Applications N2 - This note is intended to clarify some important points about the well-posedness of terminal value problems for fractional differential equations. It follows the recent publication of a paper by Cong and Tuan in this journal, in which a counter-example calls into question the earlier results in a paper by this note's authors. Here, we show in the light of these new insights, that a wide class of terminal value problems of fractional differential equations is well posed, and we identify those cases where the well-posedness question must be regarded as open. Y1 - 2018 U6 - https://doi.org/10.1216/JIE-2018-30-3-371 VL - 30 IS - 3 SP - 371 EP - 376 ER - TY - JOUR A1 - Diethelm, Kai A1 - Siegmund, Stefan A1 - Tuan, Hoang The T1 - Asymptotic behavior of solutions of linear multi-order fractional differential systems JF - Fractional Calculus and Applied Analysis N2 - In this paper, we investigate some aspects of the qualitative theory for multi-order fractional differential equation systems. First, we obtain a fundamental result on the existence and uniqueness for multi-order fractional differential equation systems. Next, a representation of solutions of homogeneous linear multi-order fractional differential equation systems in series form is provided. Finally, we give characteristics regarding the asymptotic behavior of solutions to some classes of linear multi-order fractional differential equation systems. Y1 - 2017 U6 - https://doi.org/10.1515/fca-2017-0062 VL - 20 SP - 1165 EP - 1195 ER - TY - JOUR A1 - Diethelm, Kai T1 - Monotonicity of Functions and Sign Changes of Their Caputo Derivatives JF - Fractional Calculus and Applied Analysis N2 - It is well known that a continuously differentiable function is monotone in an interval [a, b] if and only if its first derivative does not change its sign there. We prove that this is equivalent to requiring that the Caputo derivatives of all orders α ∈ (0, 1) with starting point a of this function do not have a change of sign there. In contrast to what is occasionally conjectured, it is not sufficient if the Caputo derivatives have a constant sign for a few values of α ∈ (0, 1) only. Y1 - 2016 U6 - https://doi.org/10.1515/fca-2016-0029 VL - 19 SP - 561 EP - 566 ER - TY - CHAP A1 - Kjeldsberg, Per Gunnar A1 - Schöne, Robert A1 - Gerndt, Michael A1 - Diethelm, Kai A1 - Říha, Lubomír A1 - Kannan, Venkatesh A1 - Sawley, Marie-Christine A1 - Zapletal, Jan A1 - Gocht, Andreas A1 - Reissmann, Nico A1 - Vysocky, Ondrei A1 - Kumaraswamy, Madhura A1 - Nagel, Wolfgang E. ED - Catthoor, Francky ED - Basten, Twan ED - Zompakis, Nikolaos ED - Geilen, Marc ED - Kjeldsberg, Per Gunnar T1 - Run-Time Exploitation of Application Dynamism for Energy-Efficient Exascale Computing T2 - System-Scenario-based Design Principles and Applications N2 - As in the embedded systems domain, energy efficiency has recently become one of the main design criteria in high performance computing. The European Union Horizon 2020 project READEX (Run-time Exploitation of Application Dynamism for Energy-efficient eXascale computing) has developed a tools-aided auto-tuning methodology inspired by system scenario based design. Applying similar concepts as those presented in earlier chapters of this book, the dynamic behavior of HPC applications is exploited to achieve improved energy efficiency and performance. Driven by a consortium of European experts from academia, HPC resource providers, and industry, the READEX project has developed the first generic framework of its kind for split design-time and run-time tuning while targeting heterogeneous systems at the Exascale level. Using a real-life boundary element application, energy savings of more than 30% can be shown. Y1 - 2020 SN - 978-3-030-20342-9 SN - 978-3-030-20343-6 U6 - https://doi.org/10.1007/978-3-030-20343-6_6 SP - 113 EP - 126 PB - Springer CY - Cham ER - TY - JOUR A1 - Diethelm, Kai A1 - Garrappa, Roberto A1 - Stynes, Martin T1 - Good (and Not So Good) Practices in Computational Methods for Fractional Calculus JF - Mathematics N2 - The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results. Y1 - 2020 U6 - https://doi.org/10.3390/math8030324 VL - 8 ER - TY - CHAP A1 - Diethelm, Kai T1 - Diffusive Representations for the Numerical Evaluation of Fractional Integrals T2 - Proceeding of the 2023 International Conference on Fractional Differentiation and its Applications (ICFDA) N2 - Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of such representations have been proposed. Concentrating on Riemann-Liouville integrals whose order is in (0,1), we here present a general approach that comprises most of these variants as special cases and that allows a detailed investigation of the analytic properties of each variant. The availability of this information allows to choose concrete numerical methods for handling the representations that exploit the specific properties, thus allowing to construct very efficient overall methods. Y1 - 2023 UR - https://arxiv.org/abs/2301.11931 U6 - https://doi.org/10.48550/arXiv.2301.11931 PB - IEEE CY - Piscataway ER - TY - JOUR A1 - Diethelm, Kai T1 - Monotonicity results for a compound quadrature method for finite-part integrals JF - Journal of Inequalities in Pure and Applied Mathematics N2 - Given a function f ∈ C^3[0, 1] and some q ∈ (0, 1), we look at the approximation for the Hadamard finite-part integral = R1/0x^−q−1f(x)dx based on a piecewise linear interpolant for f at n equispaced nodes (i.e., the product trapezoidal rule). The main purpose of this paper is to give sufficient conditions for the sequence of approximations to converge against the correct value of the integral in a monotonic way. An application of the results yields detailed information on the error term of a backward differentiation formula for a fractional differential equation. Y1 - 2004 UR - http://emis.icm.edu.pl/journals/JIPAM/article400.html?sid=400 VL - 5 ER - TY - JOUR A1 - Diethelm, Kai A1 - Luchko, Yuri T1 - Numerical solution of linear multi-term initial value problems of fractional order JF - Journal of Computational Analysis and Applications N2 - In this paper, a new algorithm for the numerical solution of the initial value problems for general linear multi-term differential equations of frac-tional order with constant coefficients and fractional derivatives defined in the Caputo sense is presented. The algorithm essentially uses some ideas from the convolution quadrature and discretized operational calculus. An-other basic element of the method is the formulas for analytical solution of the problem under consideration given in terms of the Mittag-Leffler type functions. Error estimates and numerical examples are presented. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods. Y1 - 2004 VL - 6 IS - 3 SP - 243 EP - 263 ER - TY - JOUR A1 - Diethelm, Kai T1 - A note on the midpoint rectangle formula for Riemann-Stieltjes integrals JF - Journal of Statistical Computation and Simulation N2 - The midpoint rectangle formula for Riemann-Stieltjes integrals is known to provide an O(h2) error bound if the integrator is linear, but only an O(h) bound in the general case. Recently Xie et al. (J. Stat. Comput. Simulation 73 (2003), 59-70) have provided sufficient conditions for O(h2) bounds also for nonlinear integrators. We now provide additional insight by giving alternative sufficient conditions that may be used in cases where those of Xie et al. are not applicable. Specifically, our bounds have a simpler structure. Y1 - 2004 VL - 74 IS - 12 SP - 920 EP - 922 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 - CHAP A1 - Damelin, Steven B. A1 - Diethelm, Kai T1 - Weighted polynomial approximation and Hilbert transforms: Their connections to the numerical solution of singular integral equations T2 - Proceedings of the 4th International Conference on Dynamic Systems and Applications Y1 - 2004 SP - 20 EP - 26 PB - Dynamic Publishers CY - Atlanta ER - TY - JOUR A1 - Damelin, Steven B. A1 - Diethelm, Kai T1 - Numerical approximation and stability of singular integral equations for Freud exponential weights on the line JF - Journal of Integral Equations Applications Y1 - 2004 U6 - https://doi.org/10.1216/jiea/1181075285 VL - 16 IS - 3 SP - 273 EP - 292 ER - TY - CHAP A1 - Diethelm, Kai A1 - Radtke, Klaus ED - Pöhlandt, K. T1 - Rückfederungsberechnung mit INDEED T2 - 8. Workshop über Simulation in der Umformtechnik N2 - Tiefziehprozesse lassen sich heute durch viele der auf dem Markt befindlichen Umformsimulationsprogramme mit ausreichender Genauigkeit simulieren. Im Hinblick auf Rückfederungsprognosen muß die Qualität der Simulationssoftware jedoch weiter verbessert werden, um auch hier zuverlässige Ergebnisse zu ermöglichen. In der Industrie und bei Anbietern kommerzieller Finite Element Programme werden daher auf diesem Gebiet erhebliche Anstregungen unternommen. Aufgrund der großen Anzahl von Einflussfaktoren ist die Problemstellung überaus komplex. Entsprechend groß ist die Anzahl von Untersuchungen zu diesem Thema. Diese Untersuchungen beleuchten aber zumeist nur Teilaspekte des Problems. Dies ist verständlich, da Betrachtungen über das Zusammenwirken aller Einflussfaktoren einen enormen Aufwand bedeuten und durch einzelne Softwarehäuser nicht geleistet werden können. Darüber hinaus verhindern auch bestimmte Vorgaben durch die verwendete kommerzielle Software detailliertere Untersuchungen. So werden z.B. Untersuchungen zu Materialmodellen, mit denen eine verbesserte Beschreibung des Verfestigungsverhaltens von Metallen erreicht werden soll, häufig mit Elementtypen durchgeführt, deren Eignung für die Anforderungen einer zuverlässigen Rückfederungssimulation durchaus in Frage gestellt werden muss. GNS nimmt derzeit mit dem Umformsimulationsprogramm INDEED neben anderen Softwarehäusern an einem Projekt zur Verbesserung von Rückfederungsprogosen teil. Einige der dabei gewonnenen Erkenntnisse sollen im folgenden am Beispiel eines dabei aufgetretenen Problems vorgestellt werden. Insbesondere soll die große Bedeutung der verwendeten Elementtypen für die Rückfederungssimulation betont werden. Y1 - 2005 UR - https://www.researchgate.net/publication/229422350_Ruckfederungsberechnung_mit_INDEED 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 - 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 - TY - JOUR A1 - Diethelm, Kai T1 - An extension of the well-posedness concept for fractional differential equations of Caputo’s type JF - Applicable Analysis N2 - It is well known that, under standard assumptions, initial value problems for fractional ordinary differential equations involving Caputo-type derivatives are well posed in the sense that a unique solution exists and that this solution continuously depends on the given function, the initial value, and the order of the derivative. Here, we extend this well-posedness concept to the extent that we also allow the location of the starting point of the differential operator to be changed, and we prove that the solution depends on this parameter in a continuous way too if the usual assumptions are satisfied. Similarly, the solution to the corresponding terminal value problems depends on the location of the starting point and of the terminal point in a continuous way too. Y1 - 2014 U6 - https://doi.org/10.1080/00036811.2013.872776 VL - 93 IS - 10 SP - 2126 EP - 2135 ER - TY - CHAP A1 - Diethelm, Kai ED - Kochubei, Anatoli ED - Luchko, Yuri T1 - General theory of Caputo-type fractional differential equations T2 - Handbook of Fractional Calculus with Applications, Vol. 2: Fractional Differential Equations N2 - This article describes the fundamentals of the theory of ordinary fractional differential equations of Caputo’s type. Starting from the existence and uniqueness of solutions and the well-posedness in general, the flow of topics continues via a derivation of explicit solution formulas for certain important classes of problems and the discussion of their smoothness properties to the stability properties of these solutions. The main focus is on initial value problems, but terminal value problems are briefly considered as well. In addition to dealing with standard single-order problems, the presentation also contains a short discussion of multiterm equations and multiorder systems. Y1 - 2019 U6 - https://doi.org/10.1515/9783110571660-001 SP - 1 EP - 20 PB - De Gruyter CY - Berlin ER - TY - JOUR A1 - Diethelm, Kai ED - Altenbach, Holm ED - Öchsner, Andreas T1 - Numerical Methods for the Fractional Differential Equations of Viscoelasticity JF - Encyclopedia of Continuum Mechanics N2 - Mathematical models based on differential operators of fractional order have proven to be very useful for describing the properties of viscoelastic materials. However, the associated differential equations can usually not be solved analytically. In this article, we provide a survey of the most important numerical methods. We restrict our attention to those types of fractional differential equations that are most important in the context of viscoelasticity, i.e., we discuss numerical methods for ordinary fractional differential equations and for certain types of time-fractional partial differential equations. Space-fractional partial differential equations are not discussed. Y1 - 2018 SN - 978-3-662-53605-6 U6 - https://doi.org/10.1007/978-3-662-53605-6_89-1 SP - 1 EP - 12 PB - Springer CY - Berlin ER - TY - CHAP A1 - Diethelm, Kai ED - Karniadakis, George Em T1 - Fundamental approaches for the numerical handling of fractional operators and time-fractional differential equations T2 - Handbook of Fractional Calculus with Applications, Vol. 3: Numerical Methods N2 - This article describes fundamental approaches for the numerical handling of problems arising in fractional calculus. This includes, in particular, methods for approximately computing fractional integrals and fractional derivatives, where the emphasis is placed on Caputo operators, as well as solvers for the associated differential and integral equations. Y1 - 2019 U6 - https://doi.org/10.1515/9783110571684-001 SP - 1 EP - 22 PB - De Gruyter CY - Berlin ER - TY - CHAP A1 - Diethelm, Kai ED - Cardone, Angelamaria ED - Donatelli, Marco ED - Durastante, Fabio ED - Garrappa, Roberto ED - Mazza, Mariarosa ED - Popolizio, Marina T1 - A New Diffusive Representation for Fractional Derivatives, Part I: Construction, Implementation and Numerical Examples T2 - Fractional Differential Equations: Modeling, Discretization, and Numerical Solvers N2 - Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known variants of this approach is that they require the numerical approximation of some integrals over an unbounded integral whose integrand decays rather slowly, which implies that their numerical handling is difficult and costly. We present a novel variant of such a diffusive representation. This form also requires the numerical approximation of an integral over an unbounded domain, but the integrand decays much faster. This property allows to use well established quadrature rules with much better convergence properties. Y1 - 2023 SN - 978-981-19-7715-2 SN - 978-981-19-7716-9 U6 - https://doi.org/10.1007/978-981-19-7716-9_1 SP - 1 EP - 15 PB - Springer CY - Singapore ER - TY - CHAP A1 - Diethelm, Kai T1 - Fast Solution Methods for Fractional Differential Equations in the Modeling of Viscoelastic Materials T2 - 2021 9th International Conference on Systems and Control (ICSC) N2 - Fractional order models have proven to be a very useful tool for the modeling of the mechanical behaviour of viscoelastic materials. Traditional numerical solution methods exhibit various undesired properties due to the non-locality of the fractional differential operators, in particular regarding the high computational complexity and the high memory requirements. The infinite state representation is an approach on which one can base numerical methods that overcome these obstacles. Such algorithms contain a number of parameters that influence the final result in nontrivial ways. Based on numerical experiments, we initiate a study leading to good choices of these parameters. Y1 - 2021 U6 - https://doi.org/10.1109/ICSC50472.2021.9666636 SP - 455 EP - 460 PB - IEEE ER - TY - JOUR A1 - Diethelm, Kai A1 - Kitzing, Konrad A1 - Picard, Rainer A1 - Siegmund, Stefan A1 - Trostorff, Sascha A1 - Waurick, Marcus T1 - A Hilbert Space Approach to Fractional Differential Equations JF - Journal of Dynamics and Differential Equations N2 - We study fractional differential equations of Riemann–Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on R, we define fractional operators by means of a functional calculus using the Fourier transform. Main tools are extrapolation- and interpolation spaces. Main results are the existence and uniqueness of solutions and the causality of solution operators for non-linear fractional differential equations. Y1 - 2022 U6 - https://doi.org/10.1007/s10884-020-09932-6 VL - 34 SP - 481 EP - 504 ER - TY - JOUR A1 - Damelin, Steven B. A1 - Diethelm, Kai T1 - An Analytic and Numerical Analysis of Weighted Singular Cauchy Integrals with Exponential Weights on ℝ JF - Numerical Functional Analysis and Optimization N2 - This article concerns an analytic and numerical analysis of a class of weighted singular Cauchy integrals with exponential weights w:= exp (−Q) with finite moments and with smooth external fields Q:R→[0,∞), with varying smooth convex rate of increase for large argument. Our analysis relies in part on weighted polynomial interpolation at the zeros of orthonormal polynomials with respect to w2. We also study bounds for the first derivatives of a class of functions of the second kind for w2. Y1 - 2022 U6 - https://doi.org/10.1080/01630563.2022.2112051 VL - 43 IS - 13 SP - 1538 EP - 1577 ER - TY - JOUR A1 - Diethelm, Kai T1 - A New Diffusive Representation for Fractional Derivatives, Part II: Convergence Analysis of the Numerical Scheme JF - Mathematics N2 - Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately evident that the method is fast and memory-efficient. Moreover, the method’s design is such that good convergence properties may be expected. In this paper, we commence a systematic investigation of these convergence properties. Y1 - 2022 U6 - https://doi.org/10.3390/math10081245 VL - 10 IS - 8 ER - TY - JOUR A1 - Diethelm, Kai A1 - Thai, Ha Duc A1 - Tuan, Hoang The T1 - Asymptotic behaviour of solutions to non-commensurate fractional-order planar systems JF - Fractional Calculus and Applied Analysis N2 - This paper is devoted to studying non-commensurate fractional order planar systems. Our contributions are to derive sufficient conditions for the global attractivity of non-trivial solutions to fractional-order inhomogeneous linear planar systems and for the Mittag-Leffler stability of an equilibrium point to fractional order nonlinear planar systems. To achieve these goals, our approach is as follows. Firstly, based on Cauchy’s argument principle in complex analysis, we obtain various explicit sufficient conditions for the asymptotic stability of linear systems whose coefficient matrices are constant. Secondly, by using Hankel type contours, we derive some important estimates of special functions arising from a variation of constants formula of solutions to inhomogeneous linear systems. Then, by proposing carefully chosen weighted norms combined with the Banach fixed point theorem for appropriate Banach spaces, we get the desired conclusions. Finally, numerical examples are provided to illustrate the effect of the main theoretical results. KW - non-commensurate fractional order planar systems KW - asymptotic behaviour of solutions Y1 - 2022 U6 - https://doi.org/https://doi.org/10.1007/s13540-022-00065-9 SN - 1314-2224 VL - 25 SP - 1324 EP - 1360 ER - TY - JOUR A1 - Diethelm, Kai A1 - Walz, Guido T1 - Numerical solution of fractional order differential equations by extrapolation JF - Numerical Algorithms N2 - We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples. Y1 - 1997 U6 - https://doi.org/10.1023/A:1019147432240 VL - 16 SP - 231 EP - 253 ER - TY - INPR A1 - Diethelm, Kai A1 - Hashemishahraki, Safoura A1 - Thai, Ha Duc A1 - Tuan, Hoang The T1 - A constructive approach for investigating the stability of incommensurate fractional differential systems N2 - This paper is devoted to studying the asymptotic behaviour of solutions to generalized noncommensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional-order pseudospectrum definition proposed by Sanca et al., we formulate the concept of a rational approximation for the fractional spectrum of a noncommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a noncommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results. Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-51954 ER - TY - INPR A1 - Chaudhary, Renu A1 - Diethelm, Kai T1 - Revisiting diffusive representations for enhanced numerical approximation of fractional integrals N2 - This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of fractional integrals. The approach seamlessly aligns with established techniques used in addressing problems involving integer-order operators, contributing to a unified framework for numerical solutions. Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-51989 ER - TY - INPR A1 - Diethelm, Kai A1 - Hashemishahraki, Safoura A1 - Thai, Ha Duc A1 - Tuan, Hoang The T1 - Stability properties of multi-order fractional differential systems in 3D N2 - This paper is devoted to studying three-dimensional non-commensurate fractional order differential equation systems with Caputo derivatives. Necessary and sufficient conditions are for the asymptotic stability of such systems are obtained. Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-51960 ER -