10,269 research outputs found
A multiscale collocation method for fractional differential problems
We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional
derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage
of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale
collocation method are proved and some numerical results are shown.We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional
derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage
of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale
collocation method are proved and some numerical results are shown
No Violation of the Leibniz Rule. No Fractional Derivative
We demonstrate that a violation of the Leibniz rule is a characteristic
property of derivatives of non-integer orders. We prove that all fractional
derivatives D^a, which satisfy the Leibniz rule D^(fg)=(D^a f) g + f (D^a g),
should have the integer order a=1, i.e. fractional derivatives of non-integer
orders cannot satisfy the Leibniz rule.Comment: 6 page
Isoperimetric problems of the calculus of variations with fractional derivatives
In this paper we study isoperimetric problems of the calculus of variations
with left and right Riemann-Liouville fractional derivatives. Both situations
when the lower bound of the variational integrals coincide and do not coincide
with the lower bound of the fractional derivatives are considered.Comment: Submitted 02-Oct-2009; revised 30-Jun-2010; accepted 10-May-2011; for
publication in the journal Acta Mathematica Scienti
Extension of Mikhlin Multiplier Theorem to Fractional Derivatives and Stable Processes
In this paper, we prove a new generalized Mikhlin multiplier theorem whose
conditions are given with respect to fractional derivatives in integral forms
with two different integration intervals. We also discuss the connection
between fractional derivatives and stable processes and prove a version of
Mikhlin theorem under a condition given in terms of the infinitesimal generator
of symmetric stable process. The classical Mikhlin theorem is shown to be a
corollary of this new generalized version in this paper.Comment: 23 page
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