77 research outputs found
An Extension of the Well-Posedness Concept for Fractional Differential Equations of Caputo's Type
It is well known that, under standard assumptions, initial value problems for
fractional ordinary differential equations involving Caputo-type derivatives
are well posed in the sense that a unique solution exists and that this
solution continuously depends on the given function, the initial value and the
order of the derivative. Here we extend this well-posedness concept to the
extent that we also allow the location of the starting point of the
differential operator to be changed, and we prove that the solution depends on
this parameter in a continuous way too if the usual assumptions are satisfied.
Similarly, the solution to the corresponding terminal value problems depends on
the location of the starting point and of the terminal point in a continuous
way too.Comment: 11 page
Volterra integral equations and fractional calculus: Do neighbouring solutions intersect?
This is the author's PDF version of an article published in Journal of integral equations
and applications. The definitive version is available at rmmc.asu.edu/jie/jie.html.This journal article considers the question of whether or not the solutions to two Volterra integral equations which have the same kernel but different forcing terms may intersect at some future time
A New Diffusive Representation for Fractional Derivatives, Part II: Convergence Analysis of the Numerical Scheme
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. This
paper here starts a systematic investigation of these convergence properties
Smoothness Properties of Solutions of Caputo-Type Fractional Differential Equations
Mathematics Subject Classification: 26A33, 34A25, 45D05, 45E10We consider ordinary fractional differential equations with Caputo-type
differential operators with smooth right-hand sides. In various places in
the literature one can find the statement that such equations cannot have
smooth solutions. We prove that this is wrong, and we give a full
characterization of the situations where smooth solutions exist. The results can
be extended to a class of weakly singular Volterra integral equations
Numerical Solution of Fractional Order Differential Equations by Extrapolation
We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples
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