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 - 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 - 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 - 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 - Ball, Joseph ED - Tylli, Hans-Olav ED - Virtanen, Jani A. T1 - Shooting methods for fractional Dirichlet-type boundary value problems of order α ∈ (1, 2) with Caputo derivatives T2 - Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis N2 - For the numerical solution of Dirichlet-type boundary value problems associated to nonlinear fractional differential equations of order 𝛼 ∈ (1, 2) that use Caputo derivatives, we suggest to employ shooting methods. In particular, we demonstrate that the so-called proportional secting technique for selecting the required initial values leads to numerical schemes that converge to high accuracy in a very small number of shooting iterations, and we provide an explanation of the analytical background for this favourable numerical behaviour. KW - fractional differential equation KW - Caputo derivative KW - boundary condition KW - boundary value problem KW - shooting method KW - proportional secting KW - secant method Y1 - 2025 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:863-opus-59665 SN - 978-3-032-00154-2 PB - Birkhäuser CY - Cham ER -