Author
Kai Diethelm
Other affiliations: University of Hildesheim
Bio: Kai Diethelm is an academic researcher from Braunschweig University of Technology. The author has contributed to research in topics: Fractional calculus & Cauchy principal value. The author has an hindex of 27, co-authored 70 publications receiving 10826 citations. Previous affiliations of Kai Diethelm include University of Hildesheim.
Papers published on a yearly basis
Papers
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TL;DR: In this paper, the authors discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order, and investigate the dependence of the solution on the order of the differential equation and on the initial condition.
3,047 citations
Book•
02 Dec 2010
TL;DR: In this paper, the existence and uniqueness results for Riemann-Liouville Fractional Differential Equations are presented. But they do not cover the special cases of fractional calculus.
Abstract: Fundamentals of Fractional Calculus.- Riemann-Liouville Differential and Integral Operators.- Caputo's Approach.- Mittag-Leffler Functions.- Theory of Fractional Differential Equations.- Existence and Uniqueness Results for Riemann-Liouville Fractional Differential Equations.- Single-Term Caputo Fractional Differential Equations: Basic Theory and Fundamental Results.- Single-Term Caputo Fractional Differential Equations: Advanced Results for Special Cases.- Multi-Term Caputo Fractional Differential Equations.
2,263 citations
Book•
07 Aug 2013TL;DR: In this paper, an Adams-type predictor-corrector method for the numerical solution of fractional differential equations is discussed, which may be used both for linear and nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator) too.
Abstract: We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.
1,617 citations
TL;DR: The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations and a detailed error analysis is given.
Abstract: We investigate a method for the numerical solution of the nonlinear fractional differential equation D
*
α
y(t)=f(t,y(t)), equipped with initial conditions y
(k)(0)=y
0
(k), k=0,1,...,⌈α⌉−1. Here α may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.
787 citations
TL;DR: In this article, a collection of numerical algorithms for the solution of various problems arising in fractional models is presented, which will give the engineer the necessary tools required to work with fractional model in an efficient way.
592 citations
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01 Jan 2015
3,828 citations
TL;DR: In this paper, the authors discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order, and investigate the dependence of the solution on the order of the differential equation and on the initial condition.
3,047 citations
01 Jan 2015
TL;DR: In this article, the authors present a new definition of fractional derivative with a smooth kernel, which takes on two different representations for the temporal and spatial variable, for which it is more convenient to work with the Fourier transform.
Abstract: In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.
1,972 citations
Book•
07 Aug 2013TL;DR: In this paper, an Adams-type predictor-corrector method for the numerical solution of fractional differential equations is discussed, which may be used both for linear and nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator) too.
Abstract: We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.
1,617 citations
TL;DR: It is proved that the full discretization is unconditionally stable, and the numerical solution converges to the exact one with order O(@Dt^2^-^@a+N^- ^m), where @Dt,N and m are the time step size, polynomial degree, and regularity of the exact solution respectively.
1,420 citations