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 -