@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} } @article{DiethelmThaiTuan, author = {Diethelm, Kai and Thai, Ha Duc and Tuan, Hoang The}, title = {Asymptotic behaviour of solutions to non-commensurate fractional-order planar systems}, series = {Fractional Calculus and Applied Analysis}, volume = {25}, journal = {Fractional Calculus and Applied Analysis}, issn = {1314-2224}, doi = {https://doi.org/10.1007/s13540-022-00065-9}, pages = {1324 -- 1360}, abstract = {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.}, language = {en} } @article{DiethelmTuan, author = {Diethelm, Kai and Tuan, Hoang The}, title = {Upper and lower estimates for the separation of solutions to fractional differential equations}, series = {Fractional Calculus and Applied Analysis}, volume = {25}, journal = {Fractional Calculus and Applied Analysis}, issn = {1314-2224}, doi = {10.1007/s13540-021-00007-x}, pages = {166 -- 180}, abstract = {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.}, 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} }