TY - CHAP A1 - Diethelm, Kai A1 - Freed, Alan D. ED - Heinzel, S. ED - Plesser, T. T1 - The FracPECE subroutine for the numerical solution of differential equations of fractional order T2 - Forschung und Wissenschaftliches Rechnen: Beiträge zum Heinz-Billing-Preis 1998 Y1 - 1999 SP - 57 EP - 71 PB - Gesellschaft für wissenschaftliche Datenverarbeitung CY - Göttingen ER - TY - CHAP A1 - Diethelm, Kai ED - Chui, C. K. ED - Schumaker, L. L. T1 - Fractional error constants for quadrature formulas T2 - Approximation Theory IX, Vol. 1: Theoretical Aspects Y1 - 1998 SP - 113 EP - 118 PB - Vanderbilt University Press CY - Nashville ER - TY - CHAP A1 - Diethelm, Kai A1 - Freed, Alan D. ED - Keil, Frerich ED - Mackens, Wolfgang ED - Voß, Heinrich ED - Werther, Joachim T1 - On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. T2 - Scientific Computing in Chemical Engineering II - Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties N2 - The authors have recently developed a mathematical model for the description of the behavior of viscoplastic materials. The model is based on a nonlinear differential equation of order β, where β is a material constant typically in the range 0 < β < 1. This equation is coupled with a first-order differential equation. In the present paper, we introduce and discuss a numerical scheme for the numerical solution of these equations. The algorithm is based on a PECE-type approach. Y1 - 1999 SN - 978-3-642-64295-1 SN - 978-3-642-60185-9 U6 - https://doi.org/10.1007/978-3-642-60185-9_24 SP - 217 EP - 224 PB - Springer CY - Heidelberg ER - TY - CHAP A1 - Diethelm, Kai ED - Alefeld, G. ED - Herzberger, J. T1 - Error bounds for compound quadratures for Hadamard-type finite-part integrals T2 - Numerical Methods and Error Bounds Y1 - 1996 SP - 58 EP - 63 PB - Akademie- Verlag CY - Berlin 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 - 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 -