@article{DiethelmFord, author = {Diethelm, Kai and Ford, Neville J.}, title = {Numerical analysis for distributed-order differential equations}, series = {Journal of Computational and Applied Mathematics}, volume = {225}, journal = {Journal of Computational and Applied Mathematics}, number = {1}, doi = {10.1016/j.cam.2008.07.018}, pages = {96 -- 104}, language = {en} } @inproceedings{DiethelmKassemMantheyThiloetal., author = {Diethelm, Kai and Kassem-Manthey, Koutaiba and Thilo, Frank and Grauer, Manfred}, title = {On the application of computational optimization methods to increase the efficiency of metal forming processes}, series = {Proceedings of the IVth European Conference on Computational Mechanics}, booktitle = {Proceedings of the IVth European Conference on Computational Mechanics}, address = {Paris}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {An efficient parallel algorithm for the numerical solution of fractional differential equations}, series = {Fractional Calculus and Applied Analysis}, volume = {14}, journal = {Fractional Calculus and Applied Analysis}, doi = {10.2478/s13540-011-0029-1}, pages = {475 -- 490}, abstract = {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.}, language = {en} } @article{anMeyBiersdorfBischofetal., author = {an Mey, Dieter and Biersdorf, Scott and Bischof, Christian and Diethelm, Kai and Eschweiler, Dominic and Gerndt, Michael and Kn{\"u}pfer, Andreas and Lorenz, Daniel and Malony, Allen and Nagel, Wolfgang E. and Oleynik, Yury and R{\"o}ssel, Christian and Saviankou, Pavel and Schmidl, Dirk and Shende, Sameer and Wagner, Michael and Wesarg, Bert and Wolf, Felix}, title = {Score-P: A Unified Performance Measurement System for Petascale Applications}, series = {Competence in High Performance Computing 2010}, journal = {Competence in High Performance Computing 2010}, editor = {Bischof, Christian and Hegering, Heinz-Gerd and Nagel, Wolfgang E. and Wittum, Gabriel}, publisher = {Springer}, address = {Berlin, Heidelberg}, isbn = {978-3-642-24024-9}, doi = {10.1007/978-3-642-24025-6_8}, pages = {85 -- 97}, abstract = {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 f{\"u}r 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.}, language = {en} } @article{DiethelmFord, author = {Diethelm, Kai and Ford, Neville J.}, title = {Volterra integral equations and fractional calculus: Do neighbouring solutions intersect?}, series = {Journal of Integral Equations and Applications}, volume = {24}, journal = {Journal of Integral Equations and Applications}, number = {1}, doi = {10.1216/JIE-2012-24-1-25}, pages = {25 -- 37}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {The mean value theorems and a Nagumo-type uniqueness theorem for Caputo's fractional calculus}, series = {Fractional Calculus and Applied Analysis}, volume = {15}, journal = {Fractional Calculus and Applied Analysis}, doi = {10.2478/s13540-012-0022-3}, pages = {304 -- 313}, abstract = {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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {Erratum: The mean value theorems and a Nagumo-type uniqueness theorem for Caputo's fractional calculus}, series = {Fractional Calculus and Applied Analysis}, volume = {20}, journal = {Fractional Calculus and Applied Analysis}, doi = {10.1515/fca-2017-0082}, pages = {1567 -- 1570}, abstract = {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.}, language = {en} } @article{KnuepferRoesselanMeyetal., author = {Kn{\"u}pfer, Andreas and R{\"o}ssel, Christian and an Mey, Dieter and Diethelm, Kai and Biersdorf, Scott and Eschweiler, Dominic and Geimer, Markus and Gerndt, Michael and Lorenz, Daniel and Malony, Allen and Nagel, Wolfgang E. and Oleynik, Yury and Philippen, Peter and Saviankou, Pavel and Schmidl, Dirk and Shende, Sameer and Tsch{\"u}ter, Ronny and Wagner, Michael and Wesarg, Bert and Wolf, Felix}, title = {Score-P: A Joint Performance Measurement Run-Time Infrastructure for Periscope, Scalasca, TAU, and Vampir}, series = {Tools for High Performance Computing 2011}, journal = {Tools for High Performance Computing 2011}, editor = {Brunst, Holger and M{\"u}ller, Matthias S. and Nagel, Wolfgang E. and Resch, Michael M.}, publisher = {Springer}, address = {Berlin, Heidelberg}, isbn = {978-3-642-31475-9}, doi = {10.1007/978-3-642-31476-6_7}, pages = {79 -- 91}, abstract = {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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {Error bounds for the numerical integration of functions with limited smoothness}, series = {SIAM Journal on Numerical Analysis}, volume = {52}, journal = {SIAM Journal on Numerical Analysis}, number = {2}, doi = {10.1137/130921246}, pages = {877 -- 879}, abstract = {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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {An extension of the well-posedness concept for fractional differential equations of Caputo's type}, series = {Applicable Analysis}, volume = {93}, journal = {Applicable Analysis}, number = {10}, doi = {10.1080/00036811.2013.872776}, pages = {2126 -- 2135}, abstract = {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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {Monotonicity of Functions and Sign Changes of Their Caputo Derivatives}, series = {Fractional Calculus and Applied Analysis}, volume = {19}, journal = {Fractional Calculus and Applied Analysis}, doi = {10.1515/fca-2016-0029}, pages = {561 -- 566}, abstract = {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.}, language = {en} } @inproceedings{Diethelm, author = {Diethelm, Kai}, title = {Properties of the solutions to "fractionalized" ODE systems, with applications to processes arising in the life sciences}, series = {Proceedings of the International Conference on Fractional Differentiation and its Applications 2016, Vol. 1}, volume = {1}, booktitle = {Proceedings of the International Conference on Fractional Differentiation and its Applications 2016, Vol. 1}, editor = {Spasic, D. T. and Grahovac, N. and Zigic, M. and Rapaic, M. and Atanackovic, T. M.}, publisher = {Faculty of Technical Sciences}, address = {Novi Sad}, pages = {32 -- 44}, abstract = {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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {A note on the midpoint rectangle formula for Riemann-Stieltjes integrals}, series = {Journal of Statistical Computation and Simulation}, volume = {74}, journal = {Journal of Statistical Computation and Simulation}, number = {12}, pages = {920 -- 922}, abstract = {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.}, language = {en} } @article{DiethelmLuchko, author = {Diethelm, Kai and Luchko, Yuri}, title = {Numerical solution of linear multi-term initial value problems of fractional order}, series = {Journal of Computational Analysis and Applications}, volume = {6}, journal = {Journal of Computational Analysis and Applications}, number = {3}, pages = {243 -- 263}, abstract = {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.}, language = {en} } @inproceedings{DamelinDiethelm, author = {Damelin, Steven B. and Diethelm, Kai}, title = {Weighted polynomial approximation and Hilbert transforms: Their connections to the numerical solution of singular integral equations}, series = {Proceedings of the 4th International Conference on Dynamic Systems and Applications}, booktitle = {Proceedings of the 4th International Conference on Dynamic Systems and Applications}, publisher = {Dynamic Publishers}, address = {Atlanta}, pages = {20 -- 26}, language = {en} } @article{DiethelmFordFordetal., author = {Diethelm, Kai and Ford, Judith M. and Ford, Neville J. and Weilbeer, Marc}, title = {A comparison of backward differentiation approaches for ordinary and partial differential equations of fractional order}, series = {Fractional Differentiation and its Applications}, journal = {Fractional Differentiation and its Applications}, editor = {Le Mehaute, A. and Tenreiro Machado, Jos{\´e} A. and Trigeassou, J. C. and Sabatier, J.}, publisher = {Ubooks}, address = {Neus{\"a}ß}, pages = {557 -- 569}, language = {en} } @article{FreedDiethelm, author = {Freed, Alan D. and Diethelm, Kai}, title = {Tensor fields for use in fractional-order viscoelasticity}, series = {Fractional Differentiation and its Applications}, journal = {Fractional Differentiation and its Applications}, editor = {Le Mehaute, A. and Tenreiro Machado, Jos{\´e} A. and Trigeassou, J. C. and Sabatier, J.}, publisher = {Ubooks}, address = {Neus{\"a}ß}, pages = {169 -- 182}, language = {en} } @article{DamelinDiethelm, author = {Damelin, Steven B. and Diethelm, Kai}, title = {Numerical approximation and stability of singular integral equations for Freud exponential weights on the line}, series = {Journal of Integral Equations Applications}, volume = {16}, journal = {Journal of Integral Equations Applications}, number = {3}, doi = {10.1216/jiea/1181075285}, pages = {273 -- 292}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {The order of convergence of modified interpolatory quadratures for singular integrals of Cauchy type}, series = {Zeitschrift f{\"u}r angewandte Mathematik und Mechanik}, volume = {75}, journal = {Zeitschrift f{\"u}r angewandte Mathematik und Mechanik}, pages = {621 -- 622}, language = {en} } @article{DiethelmKitzingPicardetal., author = {Diethelm, Kai and Kitzing, Konrad and Picard, Rainer and Siegmund, Stefan and Trostorff, Sascha and Waurick, Marcus}, title = {A Hilbert Space Approach to Fractional Differential Equations}, series = {Journal of Dynamics and Differential Equations}, volume = {34}, journal = {Journal of Dynamics and Differential Equations}, doi = {10.1007/s10884-020-09932-6}, pages = {481 -- 504}, abstract = {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.}, language = {en} } @article{DiethelmKiryakovaLuchkoetal., author = {Diethelm, Kai and Kiryakova, Virginia and Luchko, Yuri and Tenreiro Machado, Jos{\´e} A. and Tarasov, Vasily E.}, title = {Trends, directions for further research, and some open problems of fractional calculus}, series = {Nonlinear Dynamics}, volume = {107}, journal = {Nonlinear Dynamics}, issn = {1573-269X}, doi = {https://doi.org/10.1007/s11071-021-07158-9}, pages = {3245 -- 3270}, abstract = {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.}, language = {en} }