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 - 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 - 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 - 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 - 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 - 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 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 - 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 - 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 - Uhlig, Frank T1 - A New Approach to Shooting Methods for Terminal Value Problems of Fractional Differential Equations JF - Journal of Scientific Computing N2 - For terminal value problems of fractional differential equations of order α∈(0,1) that use Caputo derivatives, shooting methods are a well developed and investigated approach. Based on recently established analytic properties of such problems, we develop a new technique to select the required initial values that solves such shooting problems quickly and accurately. Numerical experiments indicate that this new proportional secting technique converges very quickly and accurately to the solution. Run time measurements indicate a speedup factor of between 4 and 10 when compared to the standard bisection method. KW - Fractional differential equation KW - Caputo derivative KW - Terminal condition KW - Terminal value problem KW - Shooting method KW - Proportional secting KW - Secant method Y1 - 2023 UR - doi.org/10.1007/s10915-023-02361-9 SN - 0885-7474 VL - 97 IS - 2 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 - JOUR A1 - Chaudhary, Renu A1 - Diethelm, Kai A1 - Hashemishahraki, Safoura T1 - On the separation of solutions to fractional differential equations of order α ∈ (1, 2) JF - Applied Numerical Mathematics N2 - Given a Caputo-type fractional differential equation with order between 1 and 2, we consider two distinct solutions to this equation subject to different sets of initial conditions. In this framework, we discuss nontrivial upper and lower bounds for the difference between these solutions. The main emphasis is on describing how such bounds are related to the differences of the associated initial values. Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-57338 VL - 203 SP - Article No. 38 ER - TY - JOUR 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 JF - Journal of Mathematical Analysis and Applications N2 - This paper is devoted to studying the asymptotic behaviour of solutions to generalized incommensurate 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 Šanca et al., we formulate the concept of a rational approximation for the fractional spectrum of incommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a incommensurate 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. It is effective and widely applicable in studying the asymptotic behaviour of solutions to linear incommensurate fractional differential systems with constant coefficient matrices and linearized stability theory for nonlinear incommensurate fractional differential systems. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results. Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-57547 VL - 540 ER - TY - JOUR A1 - Chaudhary, Renu A1 - Diethelm, Kai T1 - Revisiting diffusive representations for enhanced numerical approximation of fractional integrals JF - IFAC-PapersOnLine 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 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-51989 VL - 58 IS - 12 SP - 418 EP - 423 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 A1 - Hashemishahraki, Safoura A1 - Thai, Ha Duc A1 - Tuan, Hoang The T1 - A constructive approach for investigating the stability of incommensurate fractional differential systems JF - Journal of Mathematical Analysis and Applications 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 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-51954 VL - 540 IS - 2 ER - TY - JOUR 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 JF - IFAC-PapersOnLine 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 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-51960 VL - 58 IS - 12 SP - 231 EP - 236 ER - TY - JOUR A1 - Chaudhary, Renu A1 - Diethelm, Kai T1 - Novel variants of diffusive representation of fractional integrals: Construction and numerical computation JF - IFAC-PapersOnLine N2 - In this paper, we revisit the diffusive representations of fractional integrals established in Diethelm (2023a) to explore novel variants of such representations which provide highly efficient numerical algorithms for the approximate numerical evaluation of fractional integrals. Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-51975 VL - 58 IS - 12 SP - 412 EP - 417 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 - 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 - 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 - 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 - 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 - 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 -