Fractional integration and differentiation of variable order: an overview

DOI10.1007/s11071-012-0485-0zbMath1268.34025OpenAlexW2109367300MaRDI QIDQ354973

Stefan G. Samko

Publication date: 22 July 2013

Published in: Nonlinear Dynamics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s11071-012-0485-0

zbMATH Keywords

fractional integralsLaplaciansRiesz potentialsquasimetric measure spacesvariable orders

Mathematics Subject Classification ID

Research exposition (monographs, survey articles) pertaining to difference and functional equations (39-02) Fractional ordinary differential equations (34A08)

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Cites Work

• Variable-order fractional derivatives and their numerical approximations
• Fractional integrals and hypersingular integrals in variable order Hölder spaces on homogeneous spaces
• On potentials in generalized Hölder spaces over uniform domains in \(\mathbb R^{n}\)
• Fractional powers of closed operators and the semigroups generated by them
• Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators. II
• Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent
• The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type
• Lipschitz functions on spaces of homogeneous type
• The fractional calculus. Theory and applications of differentiation and integration to arbitrary order
• Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
• Fractional differentiation and integration of variable order
• Statistical physics of dynamical systems with variable memory
• Fractional integration and differentiation of variable order
• Fractional integration operator of variable order in the Hölder spaces \(H^{\lambda (x)}\)
• Pointwise inequalities in variable Sobolev spaces and applications
• Spherical convolution operators in spaces of variable Hölder order
• On some weighted inequalities for fractional integrals on nonhomogeneous spaces
• Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators
• Weighted Sobolev theorem in Lebesgue spaces with variable exponent
• Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces
• Spherical fractional and hypersingular integrals of variable order in generalized Hölder spaces with variable characteristic
• Convolution and potential type operators inL^p(x)(Rⁿ)
• Mechanics with variable-order differential operators
• Two-Weight Estimates For Singular Integrals Defined On Spaces Of Homogeneous Type
• On a progress in the theory of lebesgue spaces with variable exponent: maximal and singular operators
• Spherical potentials of complex order in the variable order holder spaces
• Boundedness properties of fractional integral operators associated to non-doubling measures
• Integration and differentiation to a variable fractional order
• Fractional integrals and singular integrals in the holder classes of variable order
• Maximal functions and potentials in variable exponent Morrey spaces with non-doubling measure
• The characterization of functions arising as potentials
• Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent
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