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Fractional integration: A comparative analysis of fractional integrators
Eighth International Multi-Conference on Systems, Signals & Devices, 2011The fractional integrator is certainly the key operator of fractional calculus, because of its fundamental applications in Fractional Differential Equation simulation and for the definition of fractional initial conditions. Fractional integration is defined by the classical Riemman-Liouville integral, derived from repeated integration. Three approaches
J.-C Trigeassou, A Oustaloup
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Fractional Integrals of Distributions
SIAM Journal on Mathematical Analysis, 1970Certain operators of fractional integration arising in connection with singular differential operators, Hankel transforms, and dual integral equations involve integration of fractional order with respect to $r^2$ and multiplication of functions by fractional powers of the independent variable. Such operations are not meaningful for distributions.
Erdélyi, Arthur, McBride, A. C.
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Fractional Integration and Dual Integral Equations
Canadian Journal of Mathematics, 1962In the analysis of mixed boundary value problems by the use of Hankel transforms we often encounter pairs of dual integral equations which can be written in the symmetrical form(1.1)Equations of this type seem to have been formulated first by Weber in his paper (1) in which he derives (by inspection) the solution for the case in which α — β = ½, v = 0,
Erdélyi, Arthur, Sneddon, I. N.
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Fractional Derivative and Fractional Integral
2018For every α > 0 and a local integrable function f(t), the right FI of order α is defined: $$\displaystyle{ }_aI_t^\alpha f(t) = \displaystyle\frac {1}{\Gamma (\alpha )}\displaystyle\int _a^t(t - u)^{\alpha - 1}f(u)du,\qquad-\infty \le a < t < \infty .$$
Constantin Milici +2 more
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Probability Theory and Related Fields, 1999
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Dasgupta, A., Kallianpur, G.
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Dasgupta, A., Kallianpur, G.
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Multilinear Singular and Fractional Integrals
Acta Mathematica Sinica, English Series, 2006zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Ding, Yong, Lu, Shanzhen, Yabuta, Kôzô
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2002
A new class of fractional integrals connected with balls in R n was introduced and investigated by B. Rubin in [246] (see also [247]). The special interest in ball fractional integrals (BFI’s) arises from the fact that Riesz potentials I a f over a ball B may be represented by a composition of such integrals.
David E. Edmunds +2 more
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A new class of fractional integrals connected with balls in R n was introduced and investigated by B. Rubin in [246] (see also [247]). The special interest in ball fractional integrals (BFI’s) arises from the fact that Riesz potentials I a f over a ball B may be represented by a composition of such integrals.
David E. Edmunds +2 more
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An overview of real‐world data sources for oncology and considerations for research
Ca-A Cancer Journal for Clinicians, 2022Lynne Penberthy +2 more
exaly
2013
This chapter introduces the reader to a collection of problems that are rarely seen: the evaluation of exotic integrals involving a fractional part term, called fractional part integrals. The problems were motivated by the interesting formula \(\int _{0}^{1}\left \{1/x\right \}\mathrm{d}x = 1-\gamma ,\) which connects an exotic integral to the Euler ...
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This chapter introduces the reader to a collection of problems that are rarely seen: the evaluation of exotic integrals involving a fractional part term, called fractional part integrals. The problems were motivated by the interesting formula \(\int _{0}^{1}\left \{1/x\right \}\mathrm{d}x = 1-\gamma ,\) which connects an exotic integral to the Euler ...
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Fractionally Differenced and Fractionally Integrated Processes
2016The adjective “fractional” appears frequently in the names of processes related to long-range dependence; two immediate examples are the fractional Brownian motion of Example 3.5.1 and the fractional Gaussian noise introduced in Section 5 ...
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