Showing papers by "Kai Diethelm published in 2022"
TL;DR: This paper begins a systematic investigation of the convergence properties of a new diffusive representation for fractional derivatives based on which an algorithm for their numerical computation is suggested.
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.
4 citations
TL;DR: In this article , the authors investigate how the associated solutions depend on their respective initial conditions, and provide upper and lower bounds for the difference between the two solutions of the same differential equation.
Abstract: Abstract Given a fractional differential equation of order $$\alpha \in (0,1]$$ 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 $$x_1$$ x 1 and $$x_2$$ x 2 , 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 $$x_1(t) - x_2(t)$$ x 1 ( t ) - x 2 ( t ) for all $$t \in [0,T]$$ t ∈ [ 0 , T ] that are stronger than the bounds previously described in the literature.
2 citations
TL;DR: In this article , the authors derived 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.
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 new 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.
TL;DR: A novel variant of such a diffusive representation of fractional derivatives that requires the numerical approximation of an integral over an unbounded domain, but the integrand decays much faster, which allows to use well established quadrature rules with much better convergence properties.
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 allows to use well established quadrature rules with much better convergence properties.