Relevant bibliographies by topics / Fractional powers of operators

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  2. Dissertations / Theses
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Academic literature on the topic 'Fractional powers of operators'

Author: Grafiati
Published: 4 June 2021
Last updated: 7 February 2022

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Journal articles on the topic "Fractional powers of operators"

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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Fractional powers of operators"

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McBride, Adam C. "Fractional calculus, fractional powers of operators and Mellin multiplier transforms." Thesis, University of Edinburgh, 1994. http://hdl.handle.net/1842/15310.

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We shall present a theory of fractional calculus for generalised functions on (0,∞) and use this theory as a basis for extensions to some related areas. In the first section, appropriate spaces of test-functions and generalised functions on (0,∞) are introduced and the properties of operators of fractional calculus obtained relative to these spaces. Applications are given to hypergeometric integral equations, Hankel transforms and dual integral equations of Titchmarsh type. In the second section, the Mellin transform is used to define fractional powers of a very general class of operators. These definitions include standard operators as special cases. Of particular interest are powers of differential operators of Bessel or hyper-Bessel type which are related to integral operators with special functions, notably G-functions, as kernels. In the third section, we examine operators whose Mellin multipliers involve products and/or quotients of Γ-functions. There is a detailed study of the range and invertibility of such operators in weighted LP-spaces and in appropriate spaces of smooth functions. The Laplace and Stieltjes transforms give two particular examples. In the final section, we show how our theory of fractional calculus on (0,∞) can be used to develop a corresponding theory on IRn in the presence of radial symmetry. In this framework the mapping properties of multidimensional radial integrals and Riesz potentials are obtained very precisely.
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Peat, Rhona Margaret. "Fractional powers of operators and mellin multipliers." Thesis, University of Strathclyde, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.366801.

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Meichsner, Jan [Verfasser]. "Fractional powers of linear operators in locally convex vector spaces / Jan Meichsner." Hamburg : Universitätsbibliothek der Technischen Universität Hamburg-Harburg, 2021. http://d-nb.info/1237749166/34.

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Bongarti, Marcelo Adriano dos Santos [UNESP]. "Domínios de potências fracionárias de operadores matriciais segundo Lasiecka-Triggiani." Universidade Estadual Paulista (UNESP), 2016. http://hdl.handle.net/11449/136433.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
Sejam X um espaço de Banach,\alpha um número complexo tal que Re\alpha > 0 e A um operador linear fechado, não negativo, com domínio e imagem em X. O objetivo deste trabalho é definir o objeto A^\alpha de modo que as propriedades de potência de números complexos sejam preservadas, ou seja, (i) A ^\alpha A^\beta = A^(\alpha+\beta) ; (aditividade) (ii) A^1 = A; (iii) (A^\alpha )^\beta = A (quando o primeiro membro faz sentido). Como aplicação da teoria, caracterizamos o dom ínio da potência fracionária de um operador de nido matricialmente a partir da seguinte Equação Diferencial Parcial abstrata em espaço de Hilbert, prototipo utilizado para modelar sistemas elásticos com forte (ou estrutural) amortecimento: x '' + A^\alpha x' + Ax = 0; 0 < \alpha <= 1; com A sendo um operador positivo e autoadjunto.
Let X be a Banach space, \alpha a complex number such that Re \alpha > 0 and A a non-negative closed linear operator with domain and range in X. The purpose of this work is to de fine the object A^\alpha in a way that the properties of powers of complex numbers be preserved, i.e, (i) A ^\alpha A^\beta = A^(\alpha+\beta) ; (additivity) (ii) A^1 = A; (iii) (A^\alpha )^\beta = A (when the fi rst member makes sense). As an application of theory, we characterized the domain of fractional power of a matrix-valued operator from the abstract Partial Di erential Equation in Hilbert space, prototype used to model elastic systems with strong/structural damping: x'' + A^\alpha x' + Ax = 0; 0<\alpha <= 1; with A being a positive self-adjoint operator.
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Milica, Žigić. "Primene polugrupa operatora u nekim klasama Košijevih početnih problema." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2014. https://www.cris.uns.ac.rs/record.jsf?recordId=90322&source=NDLTD&language=en.

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Doktorska disertacija je posvećena primeni teorije polugrupa operatora na rešavanje dve klase Cauchy-jevih početnih problema. U prvom delu smoispitivali parabolične stohastičke parcijalne diferencijalne jednačine (SPDJ-ne), odredjene sa dva tipa operatora: linearnim zatvorenim operatorom kojigeneriše C0−polugrupu i linearnim ograničenim operatorom kombinovanimsa Wick-ovim proizvodom. Svi stohastički procesi su dati Wiener-Itô-ovomhaos ekspanzijom. Dokazali smo postojanje i jedinstvenost rešenja ove klaseSPDJ-na. Posebno, posmatrali smo i stacionarni slučaj kada je izvod povremenu jednak nuli. U drugom delu smo konstruisali kompleksne stepeneC-sektorijalnih operatora na sekvencijalno kompletnim lokalno konveksnimprostorima. Kompleksne stepene operatora smo posmatrali kao integralnegeneratore uniformno ograničenih analitičkih C-regularizovanih rezolventnihfamilija, i upotrebili dobijene rezultate na izučavanje nepotpunih Cauchy-jevih problema viš3eg ili necelog reda.
The doctoral dissertation is devoted to applications of the theoryof semigroups of operators on two classes of Cauchy problems. In the firstpart, we studied parabolic stochastic partial differential equations (SPDEs),driven by two types of operators: one linear closed operator generating aC0−semigroup and one linear bounded operator with Wick-type multipli-cation. All stochastic processes are considered in the setting of Wiener-Itôchaos expansions. We proved existence and uniqueness of solutions for thisclass of SPDEs. In particular, we also treated the stationary case when thetime-derivative is equal to zero. In the second part, we constructed com-plex powers of C−sectorial operators in the setting of sequentially completelocally convex spaces. We considered these complex powers as the integralgenerators of equicontinuous analytic C−regularized resolvent families, andincorporated the obtained results in the study of incomplete higher or frac-tional order Cauchy problems.
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Uyanik, Meltem. "Analysis of Discrete Fractional Operators and Discrete Fractional Rheological Models." TopSCHOLAR®, 2015. http://digitalcommons.wku.edu/theses/1491.

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This thesis is comprised of two main parts: Monotonicity results on discrete fractional operators and discrete fractional rheological constitutive equations. In the first part of the thesis, we introduce and prove new monotonicity concepts in discrete fractional calculus. In the remainder, we carry previous results about fractional rheological models to the discrete fractional case. The discrete method is expected to provide a better understanding of the concept than the continuous case as this has been the case in the past. In the first chapter, we give brief information about the main results. In the second chapter, we present some fundamental definitions and formulas in discrete fractional calculus. In the third chapter, we introduce two new monotonicity concepts for nonnegative or nonpositive valued functions defined on discrete domains, and then we prove some monotonicity criteria based on the sign of the fractional difference operator of a function. In the fourth chapter, we emphasize the rheological models: We start by giving a brief introduction to rheological models such as Maxwell and Kelvin-Voigt, and then we construct and solve discrete fractional rheological constitutive equations. Finally, we finish this thesis by describing the conclusion and future work.
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Adams, Jay L. "Hankel Operators for Fractional-Order Systems." University of Akron / OhioLINK, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=akron1248198109.

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Abatangelo, Nicola. "Large solutions for fractional Laplacian operators." Thesis, Amiens, 2015. http://www.theses.fr/2015AMIE0019/document.

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La thèse étudie les problèmes de Dirichlet linéaires et semilinéaires pour différents opérateurs du type Laplacien fractionnaire. Les données peuvent être des fonctions régularières [régulières] ou plus généralement des mesures de Radon. Le but est de classifier les solutions qui présentent une singularité au bord du domaine prescrit. Nous remarquons d'abord l'existence de toute une gamme de fonctions harmoniques explosant au bord et nous les caractérisons selon une nouvelle notion de trace au bord. A l'aide d'une nouvelle formule d'intégration par parties, nous élaborons ensuite une théorie faible de type Stampacchia pour étendre la théorie linéaire à un cadre qui comprend ces fonctions : nous étudions les questions classiques d'existence, d'unicité, de dépendance à l'égard des données, la régularité et le comportement asymptotique au bord. Puis, nous développons la théorie des problèmes sémilinéaires, en généralisant la méthode des sous- et sursolutions. Cela nous permet de construire l'analogue fractionnaire des grandes solutions dans la théorie des EDPs elliptiques nonlinéaires, en donnant des conditions suffisantes pour l'existence. La thèse se termine par la définition et l'étude d'une notion de courbures directionnelles nonlocales
The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The goal is to classify the solutions which have a singularity on the boundary of the prescribed domain. We first remark the existence of a large class of harmoni functions with a boundary blow-up and we characterize them in termsof a new notion of degenerate boundary trace. Via some integration by parts formula, we then provide a weak theory of Stampacchia's sort to extend the linear theory to a setting including these functions: we study the classical questions of existence, uniqueness, continuous dependence on the data, regularity and asymptotic behaviour at the boundary. Afterwards we develop the theory of semilinear problems, by adapting and generalizing some sub- and supersolution methods. This allows us to build the fractional counterpart of large solutions in the elliptic PDE theory of nonlinear equations, giving sufficient conditions for the existence. The thesis is concluded with the definition and the study of a notion of nonlocal directional curvatures
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Lin, Lijing. "Roots of stochastic matrices and fractional matrix powers." Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/roots-of-stochastic-matrices-and-fractional-matrix-powers(3f7dbb69-7c22-4fe9-9461-429c25c0db85).html.

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In Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic $p$th root of astochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of stochastic $p$th roots. Our contributions include characterization of when a real matrix hasa real $p$th root, a classification of $p$th roots of a possibly singular matrix,a sufficient condition for a $p$th root of a stochastic matrix to have unit row sums,and the identification of two classes of stochastic matrices that have stochastic $p$th roots for all $p$. We also delineate a wide variety of possible configurationsas regards existence, nature (primary or nonprimary), and number of stochastic roots,and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix. On the computational side, we emphasize finding an approximate stochastic root: perturb the principal root $A^{1/p}$ or the principal logarithm $\log(A)$ to the nearest stochastic matrix or the nearest intensity matrix, respectively, if they are not valid ones;minimize the residual $\normF{X^p-A}$ over all stochastic matrices $X$ and also over stochastic matrices that are primary functions of $A$. For the first two nearness problems, the global minimizers are found in the Frobenius norm. For the last two nonlinear programming problems, we derive explicit formulae for the gradient and Hessian of the objective function $\normF{X^p-A}^2$ and investigate Newton's method, a spectral projected gradient method (SPGM) and the sequential quadratic programming method to solve the problem as well as various matrices to start the iteration. Numerical experiments show that SPGM starting with the perturbed $A^{1/p}$to minimize $\normF{X^p-A}$ over all stochastic matrices is method of choice.Finally, a new algorithm is developed for computing arbitrary real powers $A^\a$ of a matrix $A\in\mathbb{C}^{n\times n}$. The algorithm starts with a Schur decomposition,takes $k$ square roots of the triangular factor $T$, evaluates an $[m/m]$ Pad\'e approximant of $(1-x)^\a$ at $I - T^$, and squares the result $k$ times. The parameters $k$ and $m$ are chosen to minimize the cost subject to achieving double precision accuracy in the evaluation of the Pad\'e approximant, making use of a result that bounds the error in the matrix Pad\'e approximant by the error in the scalar Pad\'e approximant with argument the norm of the matrix. The Pad\'e approximant is evaluated from the continued fraction representation in bottom-up fashion, which is shown to be numerically stable. In the squaring phase the diagonal and first superdiagonal are computed from explicit formulae for $T^$, yielding increased accuracy. Since the basic algorithm is designed for $\a\in(-1,1)$, a criterion for reducing an arbitrary real $\a$ to this range is developed, making use of bounds for the condition number of the $A^\a$ problem. How best to compute $A^k$ for a negative integer $k$ is also investigated. In numerical experiments the new algorithm is found to be superior in accuracy and stability to several alternatives,including the use of an eigendecomposition, a method based on the Schur--Parlett\alg\ with our new algorithm applied to the diagonal blocks and approaches based on the formula $A^\a = \exp(\a\log(A))$.
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Bologna, Mauro. "The Dynamic Foundation of Fractal Operators." Thesis, University of North Texas, 2003. https://digital.library.unt.edu/ark:/67531/metadc4235/.

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The fractal operators discussed in this dissertation are introduced in the form originally proposed in an earlier book of the candidate, which proves to be very convenient for physicists, due to its heuristic and intuitive nature. This dissertation proves that these fractal operators are the most convenient tools to address a number of problems in condensed matter, in accordance with the point of view of many other authors, and with the earlier book of the candidate. The microscopic foundation of the fractal calculus on the basis of either classical or quantum mechanics is still unknown, and the second part of this dissertation aims at this important task. This dissertation proves that the adoption of a master equation approach, and so of probabilistic as well as dynamical argument yields a satisfactory solution of the problem, as shown in a work by the candidate already published. At the same time, this dissertation shows that the foundation of Levy statistics is compatible with ordinary statistical mechanics and thermodynamics. The problem of the connection with the Kolmogorov-Sinai entropy is a delicate problem that, however, can be successfully solved. The derivation from a microscopic Liouville-like approach based on densities, however, is shown to be impossible. This dissertation, in fact, establishes the existence of a striking conflict between densities and trajectories. The third part of this dissertation is devoted to establishing the consequences of the conflict between trajectories and densities in quantum mechanics, and triggers a search for the experimental assessment of spontaneous wave-function collapses. The research work of this dissertation has been the object of several papers and two books.
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Books on the topic "Fractional powers of operators"

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Colombo, Fabrizio, and Jonathan Gantner. Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16409-6.

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Carracedo, Celso Martínez. The theory of fractional powers of operators. Amsterdam: Elsevier, 2001.

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Beghin, Luisa, Francesco Mainardi, and Roberto Garrappa, eds. Nonlocal and Fractional Operators. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-69236-0.

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Allahviranloo, Tofigh. Fuzzy Fractional Differential Operators and Equations. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-51272-9.

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Rubin, B. Fractional integrals, hypersingular operators, and inversion problem for potential. New York: Longman, 1995.

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Anastassiou, George A. Nonlinearity: Ordinary and Fractional Approximations by Sublinear and Max-Product Operators. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-89509-3.

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1954-, Sickel Winfried, ed. Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. Berlin: Walter de Gruyter, 1996.

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The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Heidelberg: Springer-Verlag, 2010.

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Anastassiou, George A. Frontiers in approximation theory. New Jersey: World Scientific, 2015.

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Martinez, C., and M. Sanz. Theory of Fractional Powers of Operators. Elsevier Science & Technology Books, 2001.

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Book chapters on the topic "Fractional powers of operators"

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Haase, Markus. "Fractional Powers and Semigroups." In The Functional Calculus for Sectorial Operators, 61–89. Basel: Birkhäuser Basel, 2006. http://dx.doi.org/10.1007/3-7643-7698-8_3.

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Colombo, Fabrizio, and Jonathan Gantner. "Fractional powers of quaternionic linear operators." In Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes, 213–50. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16409-6_8.

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Colombo, Fabrizio, and Jonathan Gantner. "The Quaternionic Evolution Operator." In Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes, 105–31. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16409-6_4.

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Gelfand, Izrail Moiseevich. "Fractional powers of operators and Hamiltonian systems." In Collected Papers, 610–24. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/978-3-642-61705-8_32.

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Samko, Stefan G. "Approximative Approach to Fractional Powers of Operators." In Proceedings of the Second ISAAC Congress, 1163–70. Boston, MA: Springer US, 2000. http://dx.doi.org/10.1007/978-1-4613-0271-1_41.

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Samko, Stefan. "Fractional Powers of Operators Via Hypersingular Integrals." In Semigroups of Operators: Theory and Applications, 259–72. Basel: Birkhäuser Basel, 2000. http://dx.doi.org/10.1007/978-3-0348-8417-4_27.

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Colombo, Fabrizio, and Jonathan Gantner. "Applications to fractional diffusion." In Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes, 267–84. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16409-6_10.

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Colombo, Fabrizio, and Jonathan Gantner. "Introduction." In Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes, 1–15. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16409-6_1.

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Colombo, Fabrizio, and Jonathan Gantner. "Historical notes and References." In Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes, 285–300. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16409-6_11.

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Colombo, Fabrizio, and Jonathan Gantner. "Appendix: Principles of functional Analysis." In Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes, 301–6. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16409-6_12.

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Conference papers on the topic "Fractional powers of operators"

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Trebels, Walter, and Ursula Westphal. "Fractional powers of operators, K-functionals, Ulyanov inequalities." In Linear and Non-Linear Theory of Generalized Functions and its Applications. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2010. http://dx.doi.org/10.4064/bc88-0-22.

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Ashyralyev, Allaberen, and Ayman Hamad. "Fractional powers of strongly positive operators and their applications." In INTERNATIONAL CONFERENCE “FUNCTIONAL ANALYSIS IN INTERDISCIPLINARY APPLICATIONS” (FAIA2017). Author(s), 2017. http://dx.doi.org/10.1063/1.5000638.

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Jiang, Cindy X., Tom T. Hartley, and Joan E. Carletta. "High Performance Low Cost Implementation of FPGA-Based Fractional-Order Operators." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84796.

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Hardware implementation of fractional-order differentiators and integrators requires careful consideration of issues of system quality, hardware cost, and speed. This paper proposes using field programmable gate arrays (FPGAs) to implement fractional-order systems, and demonstrates the advantages that FPGAs provide. As an illustration, the fundamental operators to a real power is approximated via the binomial expansion of the backward difference. The resulting high-order FIR filter is implemented in a pipelined multiplierless architecture on a low-cost Spartan-3 FPGA. Unlike common digital implementations in which all filter coefficients have the same word length, this approach exploits variable word length for each coefficient. Our system requires twenty percent less hardware than a system of comparable quality generated by Xilinx’s System Generator on its most area-efficient multiplierless setting. The work shows an effective way to implement a high quality, high throughput approximation to a fractional-order system, while maintaining less cost than traditional FPGA-based designs.
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Xu, Yufeng, and Om P. Agrawal. "Numerical Solutions of Generalized Oscillator Equations." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-12705.

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Harmonic oscillators play a fundamental role in many areas of science and engineering, such as classical mechanics, electronics, quantum physics, and others. As a result, harmonic oscillators have been studied extensively. Classical harmonic oscillators are defined using integer order derivatives. In recent years, fractional derivatives have been used to model the behaviors of damped systems more accurately. In this paper, we use three operators called K-, A- and B-operators to define the equation of motion of an oscillator. In contrast to fractional integral and derivative operators which use fractional power kernels or their variations in their definitions, the K-, A- and B-operators allow the kernel to be arbitrary. In the case when the kernel is a power kernel, these operators reduce to fractional integral and derivative operators. Thus, they are more general than the fractional integral and derivative operators. Because of the general nature of the K-, A- and B-operators, the harmonic oscillators are called the generalized harmonic oscillators. The equations of motion of a generalized harmonic oscillator are obtained using a generalized Euler-Lagrange equation presented recently. In general, the resulting equations cannot be solved in closed form. A numerical scheme is presented to solve these equations. To verify the effectiveness of the numerical scheme, a problem is considered for which a closed form solution could be found. Numerical solution for the problem is compared with the analytical solution. It is demonstrated that the numerical scheme is convergent, and the order of convergence is 2. For a special kernel, this scheme reduces to a scheme presented recently in the literature.
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Zemánek, Jaroslav. "Powers of operators." In Perspectives in Operator Theory. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2007. http://dx.doi.org/10.4064/bc75-0-25.

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Mattioli, Gabriel, and Jordi Recasens. "Powers of indistinguishability operators." In NAFIPS 2012 - 2012 Annual Meeting of the North American Fuzzy Information Processing Society. IEEE, 2012. http://dx.doi.org/10.1109/nafips.2012.6290974.

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Bronstein, Manuel, Thom Mulders, and Jacques-Arthur Weil. "On symmetric powers of differential operators." In the 1997 international symposium. New York, New York, USA: ACM Press, 1997. http://dx.doi.org/10.1145/258726.258771.

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Hristova, Miryana S. "Commutational properties of powers of operators of mixed type decreasing the powers." In 39TH INTERNATIONAL CONFERENCE APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS AMEE13. AIP, 2013. http://dx.doi.org/10.1063/1.4854781.

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Camargo, Rubens De Figueiredo, Eliana Contharteze Grigoletto, and Edmundo Capelas De Oliveira. "Fractional Differential Operators: Eigenfunctions." In CNMAC 2017 - XXXVII Congresso Nacional de Matemática Aplicada e Computacional. SBMAC, 2018. http://dx.doi.org/10.5540/03.2018.006.01.0368.

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Girejko, Ewa, and Dorota Mozyrska. "Opinion dynamics and fractional operators." In 2014 International Conference on Fractional Differentiation and its Applications (ICFDA). IEEE, 2014. http://dx.doi.org/10.1109/icfda.2014.6967403.

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Reports on the topic "Fractional powers of operators"

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D'Elia, Marta, and Hayley Olson. Analysis of Tempered Fractional Operators. Office of Scientific and Technical Information (OSTI), August 2020. http://dx.doi.org/10.2172/1647701.

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D'Elia, Marta, and Hayley Olson. Analysis of Tempered Fractional Operators. Office of Scientific and Technical Information (OSTI), August 2020. http://dx.doi.org/10.2172/1647135.

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