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Journal articles on the topic 'Fractional powers of operators'

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Published: 4 June 2021

Last updated: 7 February 2022

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1

MARTINEZ, Celso, Miguel SANZ, and Luis MARCO. "Fractional powers of operators." Journal of the Mathematical Society of Japan 40, no. 2 (April 1988): 331–47. http://dx.doi.org/10.2969/jmsj/04020331.

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2

Schiavone, S. E. "Fractional powers of operators and Riesz fractional integrals." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 112, no. 3-4 (1989): 237–47. http://dx.doi.org/10.1017/s0308210500018709.

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SynopsisIn this paper, a theory of fractional powers of operators due to Balakrishnan, which is valid for certain operators on Banach spaces, is extended to Fréchet spaces. The resultingtheory is shown to be more general than that developed in an earlier approach by Lamb, and is applied to obtain mapping properties of certain Riesz fractional integral operators on spaces of test functions.

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3

Kostic, Marko. "Complex powers of operators." Publications de l'Institut Math?matique (Belgrade) 83, no. 97 (2008): 15–25. http://dx.doi.org/10.2298/pim0897015k.

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We define the complex powers of a densely defined operator A whose resolvent exists in a suitable region of the complex plane. Generally, this region is strictly contained in an angle and there exists ? ? [0,?) such that the resolvent of A is bounded by O((1 + |?|)?) there. We prove that for some particular choices of a fractional number b, the negative of the fractional power (-A)b is the c.i.g. of an analytic semigroup of growth order r > 0.

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4

Ashyralyev, Allaberen, and Ayman Hamad. "A note on fractional powers of strongly positive operators and their applications." Fractional Calculus and Applied Analysis 22, no. 2 (April 24, 2019): 302–25. http://dx.doi.org/10.1515/fca-2019-0020.

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Abstract The present paper deals with fractional powers of positive operators in a Banach space. The main theorem concerns the structure of fractional powers of positive operators in fractional spaces. As applications, the structure of fractional powers of elliptic operators is studied.

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5

Kostic, Marko. "Complex powers of nondensely defined operators." Publications de l'Institut Math?matique (Belgrade) 90, no. 104 (2011): 47–64. http://dx.doi.org/10.2298/pim1104047k.

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The power (?A)b, b ? C is defined for a closed linear operator A whose resolvent is polynomially bounded on the region which is, in general, strictly contained in an acute angle. It is proved that all structural properties of complex powers of densely defined operators with polynomially bounded resolvent remain true in the newly arisen situation. The fractional powers are considered as generators of analytic semigroups of growth order r > 0 and applied in the study of corresponding incomplete abstract Cauchy problems. In the last section, the constructed powers are incorporated in the analysis of the existence and growth of mild solutions of operators generating fractionally integrated semigroups and cosine functions.

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6

Gomilko, A. M. "Purely imaginary fractional powers of operators." Functional Analysis and Its Applications 25, no. 2 (1991): 148–50. http://dx.doi.org/10.1007/bf01079601.

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7

TARASOV, VASILY E. "FRACTIONAL DERIVATIVE AS FRACTIONAL POWER OF DERIVATIVE." International Journal of Mathematics 18, no. 03 (March 2007): 281–99. http://dx.doi.org/10.1142/s0129167x07004102.

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Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of selfadjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

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8

deLaubenfels, Ralph, Fuyuan Yao, and Shengwang Wang. "Fractional Powers of Operators of Regularized Type." Journal of Mathematical Analysis and Applications 199, no. 3 (May 1996): 910–33. http://dx.doi.org/10.1006/jmaa.1996.0182.

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9

Lamb, W. "A distributional theory of fractional calculus." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 99, no. 3-4 (1985): 347–57. http://dx.doi.org/10.1017/s0308210500014360.

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SynopsisIn this paper, a theory of fractional calculus is developed for certain spacesD′p,μof generalised functions. The theory is based on the construction of fractionalpowers of certain simple differential and integral operators. With the parameter μ suitably restricted, these fractional powers are shown to coincide with the Riemann-Liouville and Weyl operators of fractional integration and differentiation. Standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators; see [6], [7]. Some spectral properties are also obtained.

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10

Dalsen, Marié Grobbelaar-Van. "Fractional powers of a closed pair of operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 102, no. 1-2 (1986): 149–58. http://dx.doi.org/10.1017/s0308210500014566.

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SynopsisIn this paper we introduce the concept of fractional powers of a pair of operators between two Banach spaces. The operators need not be closed, but form a closed pair. The properties of the fractional powers are studied. An application of the theory is briefly discussed.

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11

Ashyralyev, A. "A note on fractional derivatives and fractional powers of operators." Journal of Mathematical Analysis and Applications 357, no. 1 (September 2009): 232–36. http://dx.doi.org/10.1016/j.jmaa.2009.04.012.

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12

Arendt, W., A. F. M. Ter Elst, and M. Warma. "Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator." Communications in Partial Differential Equations 43, no. 1 (December 22, 2017): 1–24. http://dx.doi.org/10.1080/03605302.2017.1363229.

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13

Diagana, Toka. "Fractional powers of hyponormal operators of Putnam type." International Journal of Mathematics and Mathematical Sciences 2005, no. 12 (2005): 1925–32. http://dx.doi.org/10.1155/ijmms.2005.1925.

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We are concerned with fractional powers of the so-called hyponormal operators of Putnam type. Under some suitable assumptions it is shown that ifA,Bare closed hyponormal linear operators of Putnam type acting on a complex Hilbert spaceℍ, thenD((A+B¯)α)=D(Aα)∩D(Bα)=D((A+B¯)∗α)for eachα∈(0,1). As an application, a large class of the Schrödinger's operator with a complex potentialQ∈Lloc1(ℝd)+L∞(ℝd)is considered.

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14

Hauer, Daniel, Yuhan He, and Dehui Liu. "Fractional Powers of Monotone Operators in Hilbert Spaces." Advanced Nonlinear Studies 19, no. 4 (November 1, 2019): 717–55. http://dx.doi.org/10.1515/ans-2019-2053.

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AbstractThe aim of this article is to provide a functional analytical framework for defining the fractional powers{A^{s}} for {-1<s<1} of maximal monotone (possibly multivalued and nonlinear) operators A in Hilbert spaces. We investigate the semigroup {\{e^{-A^{s}t}\}_{t\geq 0}} generated by {-A^{s}}, prove comparison principles and interpolations properties of {\{e^{-A^{s}t}\}_{t\geq 0}} in Lebesgue and Orlicz spaces. We give sufficient conditions implying that {A^{s}} has a sub-differential structure. These results extend earlier ones obtained in the case {s=1/2} for maximal monotone operators [H. Brézis, Équations d’évolution du second ordre associées à des opérateurs monotones, Israel J. Math. 12 1972, 51–60], [V. Barbu, A class of boundary problems for second order abstract differential equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 19 1972, 295–319], [V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International, Leiden, 1976], [E. I. Poffald and S. Reich, An incomplete Cauchy problem, J. Math. Anal. Appl. 113 1986, 2, 514–543], and the recent advances for linear operators A obtained in [L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 2007, 7–9, 1245–1260], [P. R. Stinga and J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 2010, 11, 2092–2122].

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15

Kittaneh, Fuad. "Norm inequalities for fractional powers of positive operators." Letters in Mathematical Physics 27, no. 4 (April 1993): 279–85. http://dx.doi.org/10.1007/bf00777375.

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16

Duan, Beiping, Raytcho D. Lazarov, and Joseph E. Pasciak. "Numerical approximation of fractional powers of elliptic operators." IMA Journal of Numerical Analysis 40, no. 3 (March 28, 2019): 1746–71. http://dx.doi.org/10.1093/imanum/drz013.

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Abstract In this paper, we develop and study algorithms for approximately solving linear algebraic systems: ${{\mathcal{A}}}_h^\alpha u_h = f_h$, $ 0< \alpha <1$, for $u_h, f_h \in V_h$ with $V_h$ a finite element approximation space. Such problems arise in finite element or finite difference approximations of the problem $ {{\mathcal{A}}}^\alpha u=f$ with ${{\mathcal{A}}}$, for example, coming from a second-order elliptic operator with homogeneous boundary conditions. The algorithms are motivated by the method of Vabishchevich (2015, Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys., 282, 289–302) that relates the algebraic problem to a solution of a time-dependent initial value problem on the interval $[0,1]$. Here we develop and study two time-stepping schemes based on diagonal Padé approximation to $(1+x)^{-\alpha }$. The first one uses geometrically graded meshes in order to compensate for the singular behaviour of the solution for $t$ close to $0$. The second algorithm uses uniform time stepping, but requires smoothness of the data $f_h$ in discrete norms. For both methods, we estimate the error in terms of the number of time steps, with the regularity of $f_h$ playing a major role for the second method. Finally, we present numerical experiments for ${{\mathcal{A}}}_h$ coming from the finite element approximations of second-order elliptic boundary value problems in one and two spatial dimensions.

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17

Bonito, Andrea, and Joseph E. Pasciak. "Numerical approximation of fractional powers of elliptic operators." Mathematics of Computation 84, no. 295 (March 12, 2015): 2083–110. http://dx.doi.org/10.1090/s0025-5718-2015-02937-8.

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18

Carro, M. J., and Joan Cerdà. "Fractional Powers of Linear Operators and Complex Interpolation." Mathematische Nachrichten 151, no. 1 (1991): 199–206. http://dx.doi.org/10.1002/mana.19911510112.

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19

DʼOvidio, Mirko. "Continuous random walks and fractional powers of operators." Journal of Mathematical Analysis and Applications 411, no. 1 (March 2014): 362–71. http://dx.doi.org/10.1016/j.jmaa.2013.09.048.

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20

Baeumer, Boris, Mihály Kovács, and Harish Sankaranarayanan. "Higher order Grünwald approximations of fractional derivatives and fractional powers of operators." Transactions of the American Mathematical Society 367, no. 2 (September 4, 2014): 813–34. http://dx.doi.org/10.1090/s0002-9947-2014-05887-x.

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21

Ciaurri, Óscar, Luz Roncal, and Sundaram Thangavelu. "Hardy-Type Inequalities for Fractional Powers of the Dunkl–Hermite Operator." Proceedings of the Edinburgh Mathematical Society 61, no. 2 (April 2, 2018): 513–44. http://dx.doi.org/10.1017/s0013091517000311.

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AbstractWe prove Hardy-type inequalities for a fractional Dunkl–Hermite operator, which incidentally gives Hardy inequalities for the fractional harmonic oscillator as well. The idea is to useh-harmonic expansions to reduce the problem in the Dunkl–Hermite context to the Laguerre setting. Then, we push forward a technique based on a non-local ground representation, initially developed by Franket al. [‘Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators,J. Amer. Math. Soc.21(2008), 925–950’] in the Euclidean setting, to obtain a Hardy inequality for the fractional-type Laguerre operator. The above-mentioned method is shown to be adaptable to an abstract setting, whenever there is a ‘good’ spectral theorem and an integral representation for the fractional operators involved.

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22

Gim, M., and V. Kostin. "About fractional powers of a class of integral operators." Актуальные направления научных исследований XXI века: теория и практика 2, no. 5 (November 11, 2014): 43–46. http://dx.doi.org/10.12737/6336.

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23

Zagloul Rida, Saad. "A formula for the powers of fractional order operators." Applicable Analysis 76, no. 1-2 (October 2000): 93–102. http://dx.doi.org/10.1080/00036810008840868.

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24

Gavrilyuk, Ivan P. "An algorithmic representation of fractional powers of positive operators." Numerical Functional Analysis and Optimization 17, no. 3-4 (January 1996): 293–305. http://dx.doi.org/10.1080/01630569608816695.

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25

Diagana, Toka. "Fractional powers of the algebraic sum of normal operators." Proceedings of the American Mathematical Society 134, no. 6 (December 15, 2005): 1777–82. http://dx.doi.org/10.1090/s0002-9939-05-08183-9.

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26

Martinez, C., M. Sanz, and V. Calvo. "Fractional powers of non-negative operators in Fréchet spaces." International Journal of Mathematics and Mathematical Sciences 12, no. 2 (1989): 309–20. http://dx.doi.org/10.1155/s0161171289000360.

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27

Qiang, Jingren, Quan Zheng, and Miao Li. "The representation of fractional powers of coercive differential operators." Indian Journal of Pure and Applied Mathematics 45, no. 4 (August 2014): 461–67. http://dx.doi.org/10.1007/s13226-014-0074-7.

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28

Lanford, Oscar E., and Derek W. Robinson. "Fractional powers of generators of equicontinuous semigroups and fractional derivatives." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 46, no. 3 (June 1989): 473–504. http://dx.doi.org/10.1017/s1446788700030950.

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AbstractWe analyze fractional powers Hα, α > 0, of the generators H of uniformly bounded locally equicontinuous semigroups S. The Hα are defined as the αth derivative δα of the Dirac measure δ evaluated on S. We demonstrate that the Hα are closed operators with the natural properties of fractional powers, for example, HαHβ = Hα+β for α, β > 0, and (Hα)β = Hαβ for 1 > α > 0 and β > 0. We establish that Hα can be evaluated by the Balakrishnan-Lions-Peetre algorithm where m is an integer larger than α, Cα, m is a suitable constant, and the limit exists in the appropriate topology if, and only if, x ∈ D(Hα). Finally we prove that H∈ is the fractional derivation of S in the sense where the limit again exists if, and only if, x ∈ D(Hα).

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29

Martínez, Celso, Antonia Redondo, and Miguel Sanz. "Suitable domains to define fractional integrals of Weyl via fractional powers of operators." Studia Mathematica 202, no. 2 (2011): 145–64. http://dx.doi.org/10.4064/sm202-2-3.

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30

Colombo, Fabrizio, and Jonathan Gantner. "Fractional powers of vector operators and fractional Fourier’s law in a Hilbert space." Journal of Physics A: Mathematical and Theoretical 51, no. 30 (June 19, 2018): 305201. http://dx.doi.org/10.1088/1751-8121/aac9e3.

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31

Carcione, José M. "A generalization of the Fourier pseudospectral method." GEOPHYSICS 75, no. 6 (November 2010): A53—A56. http://dx.doi.org/10.1190/1.3509472.

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The Fourier pseudospectral (PS) method is generalized to the case of derivatives of nonnatural order (fractional derivatives) and irrational powers of the differential operators. The generalization is straightforward because the calculation of the spatial derivatives with the fast Fourier transform is performed in the wavenumber domain, where the operator is an irrational power of the wavenumber. Modeling constant-[Formula: see text] propagation with this approach is highly efficient because it does not require memory variables or additional spatial derivatives. The classical acoustic wave equation is modified by including those with a space fractional Laplacian, which describes wave propagation with attenuation and velocity dispersion. In particular, the example considers three versions of the uniform-density wave equation, based on fractional powers of the Laplacian and fractional spatial derivatives.

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32

Yagi, Atsushi. "Fractional powers of operators and evolution equations of parabolic type." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 7 (1988): 227–30. http://dx.doi.org/10.3792/pjaa.64.227.

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33

Colombo, Fabrizio, Denis Deniz González, and Stefano Pinton. "Fractional powers of vector operators with first order boundary conditions." Journal of Geometry and Physics 151 (May 2020): 103618. http://dx.doi.org/10.1016/j.geomphys.2020.103618.

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34

Chen, Chuang, and Miao Li. "Characterizations of domains of fractional powers for non-negative operators." Journal of Mathematical Analysis and Applications 435, no. 1 (March 2016): 179–209. http://dx.doi.org/10.1016/j.jmaa.2015.10.032.

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35

Vabishchevich, Petr N. "Numerically solving an equation for fractional powers of elliptic operators." Journal of Computational Physics 282 (February 2015): 289–302. http://dx.doi.org/10.1016/j.jcp.2014.11.022.

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36

Aceto, Lidia, and Paolo Novati. "Rational approximations to fractional powers of self-adjoint positive operators." Numerische Mathematik 143, no. 1 (May 11, 2019): 1–16. http://dx.doi.org/10.1007/s00211-019-01048-4.

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37

Bonito, Andrea, Wenyu Lei, and Joseph E. Pasciak. "On sinc quadrature approximations of fractional powers of regularly accretive operators." Journal of Numerical Mathematics 27, no. 2 (June 26, 2019): 57–68. http://dx.doi.org/10.1515/jnma-2017-0116.

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Abstract We consider the finite element approximation of fractional powers of regularly accretive operators via the Dunford–Taylor integral approach. We use a sinc quadrature scheme to approximate the Balakrishnan representation of the negative powers of the operator as well as its finite element approximation. We improve the exponentially convergent error estimates from [A. Bonito and J. E. Pasciak, IMA J. Numer. Anal., 37 (2016), No. 3, 1245–1273] by reducing the regularity required on the data. Numerical experiments illustrating the new theory are provided.

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38

Rajagopal, A. K. "Fractional powers of operators of Tsallis ensemble and their parameter differentiation." Brazilian Journal of Physics 29, no. 1 (March 1999): 61–65. http://dx.doi.org/10.1590/s0103-97331999000100006.

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39

Colombo, Fabrizio, and Jonathan Gantner. "Fractional powers of quaternionic operators and Kato’s formula using slice hyperholomorphicity." Transactions of the American Mathematical Society 370, no. 2 (October 5, 2017): 1045–100. http://dx.doi.org/10.1090/tran/7013.

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40

Agranovich, M. S., and A. M. Selitskii. "Fractional powers of operators corresponding to coercive problems in Lipschitz domains." Functional Analysis and Its Applications 47, no. 2 (April 2013): 83–95. http://dx.doi.org/10.1007/s10688-013-0013-0.

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41

Bonito, Andrea, Wenyu Lei, and Joseph E. Pasciak. "The approximation of parabolic equations involving fractional powers of elliptic operators." Journal of Computational and Applied Mathematics 315 (May 2017): 32–48. http://dx.doi.org/10.1016/j.cam.2016.10.016.

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42

Čiegis, Raimondas, and Petr N. Vabishchevich. "High order numerical schemes for solving fractional powers of elliptic operators." Journal of Computational and Applied Mathematics 372 (July 2020): 112627. http://dx.doi.org/10.1016/j.cam.2019.112627.

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43

Keyantuo, Valentin, and Carlos Lizama. "On a connection between powers of operators and fractional Cauchy problems." Journal of Evolution Equations 12, no. 2 (December 28, 2011): 245–65. http://dx.doi.org/10.1007/s00028-011-0131-1.

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44

Neidhardt, Hagen, and Valentin A. Zagrebnov. "Fractional powers of self-adjoint operators and Trotter-Kato product formula." Integral Equations and Operator Theory 35, no. 2 (June 1999): 209–31. http://dx.doi.org/10.1007/bf01196384.

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45

Dungey, Nick. "Asymptotic type for sectorial operators and an integral of fractional powers." Journal of Functional Analysis 256, no. 5 (March 2009): 1387–407. http://dx.doi.org/10.1016/j.jfa.2008.07.020.

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46

Ismail, M. E. H. "Fractional powers of a difference operator." Computers & Mathematics with Applications 33, no. 1-2 (January 1997): 145–50. http://dx.doi.org/10.1016/s0898-1221(96)00226-x.

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47

Závada, Petr. "Relativistic wave equations with fractional derivatives and pseudodifferential operators." Journal of Applied Mathematics 2, no. 4 (2002): 163–97. http://dx.doi.org/10.1155/s1110757x02110102.

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We study the class of the free relativistic covariant equations generated by the fractional powers of the d′Alembertian operator(□1/n). The equations corresponding ton=1and2(Klein-Gordon and Dirac equations) are local in their nature, but the multicomponent equations for arbitraryn>2are nonlocal. We show the representation of the generalized algebra of Pauli and Dirac matrices and how these matrices are related to the algebra ofSU (n)group. The corresponding representations of the Poincaré group and further symmetry transformations on the obtained equations are discussed. The construction of the related Green functions is suggested.

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48

Meichsner, Jan, and Christian Seifert. "On the harmonic extension approach to fractional powers in Banach spaces." Fractional Calculus and Applied Analysis 23, no. 4 (August 26, 2020): 1054–89. http://dx.doi.org/10.1515/fca-2020-0055.

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AbstractWe show that fractional powers of general sectorial operators on Banach spaces can be obtained by the harmonic extension approach. Moreover, for the corresponding second order ordinary differential equation with incomplete data describing the harmonic extension we prove existence and uniqueness of a bounded solution (i.e., of the harmonic extension).

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49

Colli, Pierluigi, Gianni Gilardi, and Jürgen Sprekels. "A Distributed Control Problem for a Fractional Tumor Growth Model." Mathematics 7, no. 9 (August 31, 2019): 792. http://dx.doi.org/10.3390/math7090792.

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In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three self-adjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn–Hilliard type phase field system modeling tumor growth that has been proposed by Hawkins–Daarud, van der Zee and Oden. The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in a recent work by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.

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50

Estrada, Ernesto. "d-Path Laplacians and Quantum Transport on Graphs." Mathematics 8, no. 4 (April 3, 2020): 527. http://dx.doi.org/10.3390/math8040527.

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We generalize the Schrödinger equation on graphs to include long-range interactions (LRI) by means of the Mellin-transformed d-path Laplacian operators. We find analytical expressions for the transition and return probabilities of a quantum particle at the nodes of a ring graph. We show that the average return probability in ring graphs decays as a power law with time when LRI is present. In contrast, we prove analytically that the transition and return probabilities on a complete and start graphs oscillate around a constant value. This allowed us to infer that in a barbell graph—a graph consisting of two cliques separated by a path—the quantum particle get trapped and oscillates across the nodes of the path without visiting the nodes of the cliques. We then compare the use of the Mellin-transformed d-path Laplacian operators versus the use of fractional powers of the combinatorial Laplacian to account for LRI. Apart from some important differences observed at the limit of the strongest LRI, the d-path Laplacian operators produces the emergence of new phenomena related to the location of the wave packet in graphs with barriers, which are not observed neither for the Schrödinger equation without LRI nor for the one using fractional powers of the Laplacian.

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