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What is Fractional Integration? [PDF]

Review of Economics and Statistics, 1999
A simple construction that will be referred to as an error-duration model is shown to generate fractional integration and long memory. An error-duration representation also exists for many familiar ARMA models, making error duration an alternative to autoregression for explaining dynamic persistence in economic variables.
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On the integration of differential fractions

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, 2013
In this paper, we provide a differential algebra algorithm for integrating fractions of differential polynomials. It is not restricted to differential fractions that are the derivatives of other differential fractions. The algorithm leads to new techniques for representing differential fractions, which may help converting differential equations to ...
François Boulier, François Lemaire, Georg Regensburger, Markus Rosenkranz   +3 more
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Fractional integration: A comparative analysis of fractional integrators

Eighth International Multi-Conference on Systems, Signals & Devices, 2011
The 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 and integral colourings

Mathematical Programming, 1997
Let \(G=(V,E)\) be an undirected graph and \(c\) any vector in \(\mathbb{Z}^{V(G)}_+\). Denote by \(\chi(G_c)\) and \(\eta(G_c)\) the chromatic number and fractional chromatic number respectively, of \(G\) with respect to \(c\). In this paper graphs are studied for which \(\chi(G_c)-\lceil\eta(G_c)\rceil\leq 1\).
Kilakos, K., Marcotte, O.
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Fractional Integrals of Distributions

SIAM Journal on Mathematical Analysis, 1970
Certain 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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Approximations of the Fractional Integral and Numerical Solutions of Fractional Integral Equations

Communications on Applied Mathematics and Computation, 2021
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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On Fractional Integration by Parts

Proceedings of the London Mathematical Society, 1938
Let \((a,b)\) be a finite interval, \(f\in L(a,b)\), \(\alpha>0\), \[ f_\alpha^+\equiv f_\alpha^+(a,x) = (\Gamma(\alpha))^{-1} \int_a^x f(t)(x-t)^{\alpha-1}\,dt, \; f_\alpha^-\equiv f_\alpha^-(a,x) = (\Gamma(\alpha))^{-1} \int_x^b f(t)(t-x)^{\alpha-1}\,dt. \] As a consequence of results of Hardy and Littlewood the authors prove that if \(p>1\), \(q>1\),
Love, E. R., Young, L. C.
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Fractional Integration and Dual Integral Equations

Canadian Journal of Mathematics, 1962
In 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

2018
For 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, Gheorghe Drăgănescu, J. Tenreiro Machado   +2 more
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On Fractional Multilinear Singular Integrals

Mathematische Nachrichten, 2002
The authors consider the following type of fractional multilinear integrals with rough kernels defined by \[ T_{\Omega,\alpha}^{A}f(x)= \int_{\mathbb R^n}\frac{\Omega(x-y)}{|x-y|^{n-\alpha+m-1}} R_m(A;x,y)f(y) dy, \] where ...
Wu, Qiang, Yang, Dachun
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