@article{BartschBierDietrich, author = {Bartsch, Hendrik and Bier, Markus and Dietrich, Siegfried}, title = {Interface structures in ionic liquid crystals}, series = {Soft Matter}, volume = {15}, journal = {Soft Matter}, pages = {4109 -- 4126}, abstract = {Ionic liquid crystals (ILCs) are anisotropic mesogenic molecules which additionally carry charges. This combination gives rise to a complex interplay of the underlying (anisotropic) contributions to the pair interactions. It promises interesting and distinctive structural and orientational properties to arise in systems of ILCs, combining properties of liquid crystals and ionic liquids. While previous theoretical studies have focused on the phase behavior of ILCs and the structure of the respective bulk phases, in the present study we provide new results, obtained within density functional theory, concerning (planar) free interfaces between an isotropic liquid L and two types of smectic-A phases (SA or SAW). We discuss the structural and orientational properties of these interfaces in terms of the packing fraction profile η(r) and the orientational order parameter profile S2(r) concerning the tilt angle α between the (bulk) smectic layer normal and the interface normal. The asymptotic decay of η(r) and of S2(r) towards their values in the isotropic bulk is discussed, too.}, language = {en} } @article{ZarubinBierDietrich, author = {Zarubin, Grigorii and Bier, Markus and Dietrich, S.}, title = {A ferronematic slab in external magnetic fields}, series = {Soft Matter}, volume = {14}, journal = {Soft Matter}, pages = {9806 -- 9818}, abstract = {The behavior of a uniformly magnetized ferronematic slab is investigated numerically in a situation in which an external magnetic field is applied parallel and antiparallel, respectively, to its initial magnetization direction. The employed numerical method allows one to determine hysteresis curves from which a critical magnetic field strength (i.e., the one at which the ferronematic sample becomes distorted) as a function of the system parameters can be inferred. Two possible mechanisms of switching the magnetization by applying a magnetic field in the antiparallel direction are observed and characterized in terms of the coupling constant between the magnetization and the nematic director and in terms of the coupling strength of the nematic liquid crystal and the walls of the slab. Suitably prepared walls allow one to combine both switching mechanisms in one setup, such that one can construct a cell, the magnetization of which can be reversibly switched off.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {Numerical Methods for the Fractional Differential Equations of Viscoelasticity}, series = {Encyclopedia of Continuum Mechanics}, journal = {Encyclopedia of Continuum Mechanics}, editor = {Altenbach, Holm and {\"O}chsner, Andreas}, publisher = {Springer}, address = {Berlin}, isbn = {978-3-662-53605-6}, doi = {10.1007/978-3-662-53605-6_89-1}, pages = {1 -- 12}, abstract = {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.}, language = {en} } @article{DiethelmFord, author = {Diethelm, Kai and Ford, Neville J.}, title = {A note on the well-posedness of terminal value problems for fractional differential equations}, series = {Journal of Integral Equations and Applications}, volume = {30}, journal = {Journal of Integral Equations and Applications}, number = {3}, doi = {10.1216/JIE-2018-30-3-371}, pages = {371 -- 376}, abstract = {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.}, language = {en} } @inproceedings{Diethelm, author = {Diethelm, Kai}, title = {A New Diffusive Representation for Fractional Derivatives, Part I: Construction, Implementation and Numerical Examples}, series = {Fractional Differential Equations: Modeling, Discretization, and Numerical Solvers}, booktitle = {Fractional Differential Equations: Modeling, Discretization, and Numerical Solvers}, editor = {Cardone, Angelamaria and Donatelli, Marco and Durastante, Fabio and Garrappa, Roberto and Mazza, Mariarosa and Popolizio, Marina}, publisher = {Springer}, address = {Singapore}, isbn = {978-981-19-7715-2}, doi = {10.1007/978-981-19-7716-9_1}, pages = {1 -- 15}, abstract = {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.}, language = {en} } @inproceedings{Diethelm, author = {Diethelm, Kai}, title = {Diffusive Representations for the Numerical Evaluation of Fractional Integrals}, series = {Proceeding of the 2023 International Conference on Fractional Differentiation and its Applications (ICFDA)}, booktitle = {Proceeding of the 2023 International Conference on Fractional Differentiation and its Applications (ICFDA)}, publisher = {IEEE}, address = {Piscataway}, doi = {10.48550/arXiv.2301.11931}, pages = {6}, abstract = {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.}, language = {en} } @inproceedings{Diethelm, author = {Diethelm, Kai}, title = {Fast Solution Methods for Fractional Differential Equations in the Modeling of Viscoelastic Materials}, series = {2021 9th International Conference on Systems and Control (ICSC)}, booktitle = {2021 9th International Conference on Systems and Control (ICSC)}, publisher = {IEEE}, doi = {10.1109/ICSC50472.2021.9666636}, pages = {455 -- 460}, abstract = {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.}, language = {en} } @article{DamelinDiethelm, author = {Damelin, Steven B. and Diethelm, Kai}, title = {An Analytic and Numerical Analysis of Weighted Singular Cauchy Integrals with Exponential Weights on ℝ}, series = {Numerical Functional Analysis and Optimization}, volume = {43}, journal = {Numerical Functional Analysis and Optimization}, number = {13}, doi = {10.1080/01630563.2022.2112051}, pages = {1538 -- 1577}, abstract = {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.}, language = {en} } @article{Diethelm, author = {Diethelm, Kai}, title = {A New Diffusive Representation for Fractional Derivatives, Part II: Convergence Analysis of the Numerical Scheme}, series = {Mathematics}, volume = {10}, journal = {Mathematics}, number = {8}, doi = {10.3390/math10081245}, abstract = {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.}, language = {en} } @incollection{Diethelm, author = {Diethelm, Kai}, title = {Fundamental approaches for the numerical handling of fractional operators and time-fractional differential equations}, series = {Handbook of Fractional Calculus with Applications, Vol. 3: Numerical Methods}, booktitle = {Handbook of Fractional Calculus with Applications, Vol. 3: Numerical Methods}, editor = {Karniadakis, George Em}, publisher = {De Gruyter}, address = {Berlin}, doi = {10.1515/9783110571684-001}, pages = {1 -- 22}, abstract = {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.}, language = {en} }