Relevant bibliographies by topics / Fractional calculus operator

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Fractional calculus operator'

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

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Journal articles on the topic "Fractional calculus operator"

1

Patanarapeelert, Nichaphat, and Thanin Sitthiwirattham. "On Fractional Symmetric Hahn Calculus." Mathematics 7, no. 10 (September 20, 2019): 873. http://dx.doi.org/10.3390/math7100873.

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In this paper, we study fractional symmetric Hahn difference calculus. The new idea of the symmetric Hahn difference operator, the fractional symmetric Hahn integral, and the fractional symmetric Hahn operators of Riemann–Liouville and Caputo types are presented. In addition, we formulate some fundamental properties based on these fractional symmetric Hahn operators.
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Li, Ming, and Wei Zhao. "Essay on Fractional Riemann-Liouville Integral Operator versus Mikusinski’s." Mathematical Problems in Engineering 2013 (2013): 1–3. http://dx.doi.org/10.1155/2013/635412.

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This paper presents the representation of the fractional Riemann-Liouville integral by using the Mikusinski operators. The Mikusinski operators discussed in the paper may yet provide a new view to describe and study the fractional Riemann-Liouville integral operator. The present result may be useful for applying the Mikusinski operational calculus to the study of fractional calculus in mathematics and to the theory of filters of fractional order in engineering.
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Yilmazer, R., and O. Ozturk. "N-Fractional Calculus Operator Method to the Euler Equation." Issues of Analysis 25, no. 2 (December 2018): 144–52. http://dx.doi.org/10.15393/j3.art.2018.5730.

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Xu, Zhisong, and Mingqiu Li. "Rational Implementation of Fractional Calculus Operator Based on Quadratic Programming." Mathematical Problems in Engineering 2021 (February 12, 2021): 1–12. http://dx.doi.org/10.1155/2021/6646718.

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When fractional calculus operators and models are implemented rationally, there exist some problems such as low approximation accuracy of rational approximation function, inability to specify arbitrary approximation frequency band, or poor robustness. Based on the error criterion of the least squares method, a quadratic programming method based on the frequency-domain error is proposed. In this method, the frequency-domain error minimization between the fractional operator s ± r and its rational approximation function is transformed into a quadratic programming problem. The results show that the construction method of the optimal rational approximation function of fractional calculus operator is effective, and the optimal rational approximation function constructed can effectively approximate the fractional calculus operator and model for the specified approximation frequency band.
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Momenzadeh, Mohammad, and Sajedeh Norozpou. "Alternative fractional derivative operator on non-newtonian calculus and its approaches." Nexo Revista Científica 34, no. 02 (June 9, 2021): 906–15. http://dx.doi.org/10.5377/nexo.v34i02.11616.

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Nowadays, study on fractional derivative and integral operators is one of the hot topics of mathematics and lots of investigations and studies make their attentions in this field. Most of these concerns raised from the vast application of these operators in study of phenomena’s models. These operators interpreted by Newtonian calculus, however different types of calculi are existed and we introduce the fractional derivative operators focused on Bi-geometric calculus and also their fractional differential equations are studied.
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EL-NABULSI, RAMI AHMAD. "FRACTIONAL FIELD THEORIES FROM MULTI-DIMENSIONAL FRACTIONAL VARIATIONAL PROBLEMS." International Journal of Geometric Methods in Modern Physics 05, no. 06 (September 2008): 863–92. http://dx.doi.org/10.1142/s0219887808003119.

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Fractional calculus has recently attracted considerable attention. In particular, various fractional differential equations are used to model nonlinear wave theory that arises in many different areas of physics such as Josephson junction theory, field theory, theory of lattices, etc. Thus one may expect fractional calculus, in particular fractional differential equations, plays an important role in quantum field theories which are expected to satisfy fractional generalization of Klein–Gordon and Dirac equations. Until now, in high-energy physics and quantum field theories the derivative operator has only been used in integer steps. In this paper, we want to extend the idea of differentiation to arbitrary non-integers steps. We will address multi-dimensional fractional action-like problems of the calculus of variations where fractional field theories and fractional differential Dirac operators are constructed.
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Tilahun, Kelelaw, Hagos Tadessee, and D. L. Suthar. "The Extended Bessel-Maitland Function and Integral Operators Associated with Fractional Calculus." Journal of Mathematics 2020 (June 23, 2020): 1–8. http://dx.doi.org/10.1155/2020/7582063.

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The aim of this paper is to introduce a presumably and remarkably altered integral operator involving the extended generalized Bessel-Maitland function. Particular properties are considered for the extended generalized Bessel-Maitland function connected with fractional integral and differential operators. The integral operator connected with operators of the fractional calculus is also observed. We point out important links to known findings from some individual cases with our key outcomes.
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Razzaghi, Mohsen. "Hybrid approximations for fractional calculus." ITM Web of Conferences 29 (2019): 01001. http://dx.doi.org/10.1051/itmconf/20192901001.

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In this paper, a numerical method for solving the fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting ofblock-pulse functions and Taylor polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the initial value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.
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Ibrahim, Rabha W., and Hamid A. Jalab. "The Fractional Complex Step Method." Discrete Dynamics in Nature and Society 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/515973.

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It is well known that the complex step method is a tool that calculates derivatives by imposing a complex step in a strict sense. We extended the method by employing the fractional calculus differential operator in this paper. The fractional calculus can be taken in the sense of the Caputo operator, Riemann-Liouville operator, and so forth. Furthermore, we derived several approximations for computing the fractional order derivatives. Stability of the generalized fractional complex step approximations is demonstrated for an analytic test function.
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Faraj, Ahmad, Tariq Salim, Safaa Sadek, and Jamal Ismail. "Generalized Mittag-Leffler Function Associated with Weyl Fractional Calculus Operators." Journal of Mathematics 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/821762.

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This paper is devoted to study further properties of generalized Mittag-Leffler functionEα,β,pγ,δ,qassociated with Weyl fractional integral and differential operators. A new integral operatorℰα,β,p,w,∞γ,δ,qdepending on Weyl fractional integral operator and containingEα,β,pγ,δ,q(z)in its kernel is defined and studied, namely, its boundedness. Also, composition of Weyl fractional integral and differential operators with the new operatorℰα,β,p,w,∞γ,δ,qis established.
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Dissertations / Theses on the topic "Fractional calculus operator"

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Amsheri, Somia M. A. "Fractional calculus operator and its applications to certain classes of analytic functions. A study on fractional derivative operator in analytic and multivalent functions." Thesis, University of Bradford, 2013. http://hdl.handle.net/10454/6320.

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The main object of this thesis is to obtain numerous applications of fractional derivative operator concerning analytic and -valent (or multivalent) functions in the open unit disk by introducing new classes and deriving new properties. Our finding will provide interesting new results and indicate extensions of a number of known results. In this thesis we investigate a wide class of problems. First, by making use of certain fractional derivative operator, we define various new classes of -valent functions with negative coefficients in the open unit disk such as classes of -valent starlike functions involving results of (Owa, 1985a), classes of -valent starlike and convex functions involving the Hadamard product (or convolution) and classes of -uniformly -valent starlike and convex functions, in obtaining, coefficient estimates, distortion properties, extreme points, closure theorems, modified Hadmard products and inclusion properties. Also, we obtain radii of convexity, starlikeness and close-to-convexity for functions belonging to those classes. Moreover, we derive several new sufficient conditions for starlikeness and convexity of the fractional derivative operator by using certain results of (Owa, 1985a), convolution, Jack¿s lemma and Nunokakawa¿ Lemma. In addition, we obtain coefficient bounds for the functional of functions belonging to certain classes of -valent functions of complex order which generalized the concepts of starlike, Bazilevi¿ and non-Bazilevi¿ functions. We use the method of differential subordination and superordination for analytic functions in the open unit disk in order to derive various new subordination, superordination and sandwich results involving the fractional derivative operator. Finally, we obtain some new strong differential subordination, superordination, sandwich results for -valent functions associated with the fractional derivative operator by investigating appropriate classes of admissible functions. First order linear strong differential subordination properties are studied. Further results including strong differential subordination and superordination based on the fact that the coefficients of the functions associated with the fractional derivative operator are not constants but complex-valued functions are also studied.
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Amsheri, Somia Muftah Ahmed. "Fractional calculus operator and its applications to certain classes of analytic functions : a study on fractional derivative operator in analytic and multivalent functions." Thesis, University of Bradford, 2013. http://hdl.handle.net/10454/6320.

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The main object of this thesis is to obtain numerous applications of fractional derivative operator concerning analytic and ρ-valent (or multivalent) functions in the open unit disk by introducing new classes and deriving new properties. Our finding will provide interesting new results and indicate extensions of a number of known results. In this thesis we investigate a wide class of problems. First, by making use of certain fractional derivative operator, we define various new classes of ρ-valent functions with negative coefficients in the open unit disk such as classes of ρ-valent starlike functions involving results of (Owa, 1985a), classes of ρ-valent starlike and convex functions involving the Hadamard product (or convolution) and classes of κ-uniformly ρ-valent starlike and convex functions, in obtaining, coefficient estimates, distortion properties, extreme points, closure theorems, modified Hadmard products and inclusion properties. Also, we obtain radii of convexity, starlikeness and close-to-convexity for functions belonging to those classes. Moreover, we derive several new sufficient conditions for starlikeness and convexity of the fractional derivative operator by using certain results of (Owa, 1985a), convolution, Jack's lemma and Nunokakawa' Lemma. In addition, we obtain coefficient bounds for the functional |αρ+2-θα²ρ+1| of functions belonging to certain classes of p-valent functions of complex order which generalized the concepts of starlike, Bazilevič and non-Bazilevič functions. We use the method of differential subordination and superordination for analytic functions in the open unit disk in order to derive various new subordination, superordination and sandwich results involving the fractional derivative operator. Finally, we obtain some new strong differential subordination, superordination, sandwich results for ρ-valent functions associated with the fractional derivative operator by investigating appropriate classes of admissible functions. First order linear strong differential subordination properties are studied. Further results including strong differential subordination and superordination based on the fact that the coefficients of the functions associated with the fractional derivative operator are not constants but complex-valued functions are also studied.
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Khan, Mumtaz Ahmad, and Bhagwat Swaroop Sharma. "A study of three variable analogues of certain fractional integral operators." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/95821.

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The paper deals with a three variable analogues of certain fractional integral operators introduced by M. Saigo. Resides giving three variable analogues of earlier known fractional integral operators of one variable as a special cases of newly defined operators, the paper establishes certain results in the form of theorems including integration by parts.
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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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Kisela, Tomáš. "Basics of Qualitative Theory of Linear Fractional Difference Equations." Doctoral thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2012. http://www.nusl.cz/ntk/nusl-234025.

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Tato doktorská práce se zabývá zlomkovým kalkulem na diskrétních množinách, přesněji v rámci takzvaného (q,h)-kalkulu a jeho speciálního případu h-kalkulu. Nejprve jsou položeny základy teorie lineárních zlomkových diferenčních rovnic v (q,h)-kalkulu. Jsou diskutovány některé jejich základní vlastnosti, jako např. existence, jednoznačnost a struktura řešení, a je zavedena diskrétní analogie Mittag-Lefflerovy funkce jako vlastní funkce operátoru zlomkové diference. Dále je v rámci h-kalkulu provedena kvalitativní analýza skalární a vektorové testovací zlomkové diferenční rovnice. Výsledky analýzy stability a asymptotických vlastností umožňují vymezit souvislosti s jinými matematickými disciplínami, např. spojitým zlomkovým kalkulem, Volterrovými diferenčními rovnicemi a numerickou analýzou. Nakonec je nastíněno možné rozšíření zlomkového kalkulu na obecnější časové škály.
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Waddell, Chris. "Fractional calculus and scales of spaces." Thesis, University of Strathclyde, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.288637.

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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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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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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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Hu, Ke. "On an equation being a fractional differential equation with respect to time and a pseudo-differential equation with respect to space related to Lévy-type processes." Thesis, Swansea University, 2012. https://cronfa.swan.ac.uk/Record/cronfa43021.

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Books on the topic "Fractional calculus operator"

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

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

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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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Bologna, Mauro, Bruce J. West, and Paolo Grigolini. Physics of Fractal Operators. Springer, 2011.

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Physics of Fractal Operators. Springer, 2003.

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Operators of Fractional Calculus and Their Applications. MDPI, 2019. http://dx.doi.org/10.3390/books978-3-03897-341-6.

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

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Atangana, Abdon. Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology. Elsevier Science & Technology Books, 2017.

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Martinez, C., and M. Sanz. The Theory of Fractional Powers of Operators (North-Holland Mathematics Studies). North Holland, 2000.

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Book chapters on the topic "Fractional calculus operator"

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Odzijewicz, T., A. B. Malinowska, and D. F. M. Torres. "Fractional Variational Calculus of Variable Order." In Advances in Harmonic Analysis and Operator Theory, 291–301. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0516-2_16.

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Rodrigues, M. M., N. Vieira, and S. Yakubovich. "Operational Calculus for Bessel’s Fractional Equation." In Advances in Harmonic Analysis and Operator Theory, 357–70. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0516-2_20.

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Anastassiou, George A. "Generalized $$\psi $$-Fractional Quantitative Approximation by Sublinear Operators." In Generalized Fractional Calculus, 135–55. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-56962-4_7.

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Anastassiou, George A. "Generalized g-Iterated Fractional Quantitative Approximation By Sublinear Operators." In Generalized Fractional Calculus, 157–78. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-56962-4_8.

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Umarov, Sabir. "Fractional calculus and fractional order operators." In Developments in Mathematics, 121–68. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-20771-1_3.

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Anastassiou, George A. "Vectorial Generalized g-Fractional Direct and Iterated Quantitative Approximation by Linear Operators." In Generalized Fractional Calculus, 223–48. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-56962-4_11.

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Colombo, Fabrizio, and Jonathan Gantner. "The Phillips functional calculus." In Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes, 151–72. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16409-6_6.

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Colombo, Fabrizio, and Jonathan Gantner. "The H∞-Functional Calculus." In Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes, 173–211. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16409-6_7.

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Jiang, Cindy X., Joan E. Carletta, and Tom T. Hartley. "Implementation of Fractional-order Operators on Field Programmable Gate Arrays." In Advances in Fractional Calculus, 333–46. Dordrecht: Springer Netherlands, 2007. http://dx.doi.org/10.1007/978-1-4020-6042-7_23.

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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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Conference papers on the topic "Fractional calculus operator"

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Galé, José E. "Some applications of fractional calculus to operator semigroups and functional calculus." In Perspectives in Operator Theory. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2007. http://dx.doi.org/10.4064/bc75-0-9.

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Ono, Yuya, and Mingcong Deng. "Operator based robust nonlinear control system for a tank process with fractional calculus." In 2017 IEEE International Conference on Systems, Man and Cybernetics (SMC). IEEE, 2017. http://dx.doi.org/10.1109/smc.2017.8123007.

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Ozturk, Okkes, and Resat Yilmazer. "Particular solutions of the radial Schrödinger equation via Nabla discrete fractional calculus operator." In INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2016). Author(s), 2017. http://dx.doi.org/10.1063/1.4992528.

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Huang, Jiacai, YangQuan Chen, and Zhuo Li. "Mathematical Model of Human Operator Using Fractional Calculus for Human-in-the-Loop Control." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-47464.

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Mathematical models of human operator play a very important role in the Human-in-the-Loop manual control system. For several decades, modeling human operator’s dynamic has been an active research area. The traditional classical human operator models are usually developed using the Quasi-linear transfer function method, the optimal control theory method, and so on. The human operator models established by the above methods have deficiencies such as complicated and over parameterized, even for basic control elements. In this paper, based on the characteristics of human brain and behaviour, two kinds of fractional order mathematical models for describing human operator behavior are proposed. Through validation and comparison by the actual data, the best_fit model with smallest root mean squared error (RMSE) is obtained, which has simple structure with only few parameters, and each parameter has definite physical meaning.
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Huang, Jiacai, Qi Shu, Xiaochun Zhu, Xinxin Shi, Lei Zhou, and Hanzhong Liu. "A fast frequency domain approximation method for variable order fractional calculus operator based on polynomial fitting." In 2018 37th Chinese Control Conference (CCC). IEEE, 2018. http://dx.doi.org/10.23919/chicc.2018.8483651.

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Huang, Jiacai, Kai Hu, Xinxin Shi, Hanzhong Liu, Lei Zhou, and Xiaochun Zhu. "Fast Numerical Implementation for Variable Order Fractional Calculus Operator Based on Polynomial Fitting Method in Time Domain." In 2018 37th Chinese Control Conference (CCC). IEEE, 2018. http://dx.doi.org/10.23919/chicc.2018.8482866.

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Pan, Zhifeng, Xiaohong Wang, Thi Thu Giang Hoang, Ying Luo, Yangquan Chen, and Lianfang Tian. "Design and Application of Fractional Order PIλDµ Controller in Grid-Connected Inverter System." In ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/detc2017-67355.

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In comparison with the traditional controller PID, the fractional order controller PIλDμ is added with two parameters (μ and λ), that increases the flexibility of the controller. The larger adjustable space makes it more favorable to the control of the nonlinear system like the three-phase grid-connected inverter. However, since the fractional calculus operator is an irrational function on the complex plane, it can’t be implemented directly in simulation or in practical applications. In this paper, the advantages of the fractional order controller PIλDμ will be analyzed, then the fractional calculus operator is fitted by the frequency domain analysis method. Based on the vector method, the fractional order controller PIλDμ of the grid-connected inverter is designed. Simultaneously, the optimal controller parameters are searched with the ITAE and IAE as the performance index. And finally, the results are compared with those of the traditional controller PID. In consideration of the defects of fractional algorithm and single discretization method, a hybrid discretization method is proposed in order to ensure that the discretized controller can keep the same time-domain response and frequency characteristics as the designed controller. The experimental results show that the proposed method has the dynamic and static characteristics better than the traditional controller PID, which proves that the application of fractional order controller in three-phase grid-connected inverter is effective and feasible.
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Coopmans, Calvin, Ivo Petra´sˇ, and YangQuan Chen. "Analogue Fractional-Order Generalized Memristive Devices." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86861.

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Memristor is a new electrical element which has been predicted and described in 1971 by Leon O. Chua and for the first time realized by HP laboratory in 2008. Chua proved that memristor behavior could not be duplicated by any circuit built using only the other three elements (resistor, capacitor, inductor), which is why the memristor is truly fundamental. Memristor is a contraction of memory resistor, because that is exactly its function: to remember its history. The memristor is a two-terminal device whose resistance depends on the magnitude and polarity of the voltage applied to it and the length of time that voltage has been applied. The missing element—the memristor, with memristance M—provides a functional relation between charge and flux, dφ = Mdq. In this paper, for the first time, the concept of (integer-order) memristive systems is generalized to non-integer order case using fractional calculus. We also show that the memory effect of such devices can be also used for an analogue implementation of the fractional-order operator, namely fractional-order integral and fractional-order derivatives. This kind of operators are useful for realization of the fractional-order controllers. We present theoretical description of such implementation and we proposed the practical realization and did some experiments as well.
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Huang, Jiacai, Yangquan Chen, and Zhuo Li. "Human operator modeling based on fractional order calculus in the manual control system with second-order controlled element." In 2015 27th Chinese Control and Decision Conference (CCDC). IEEE, 2015. http://dx.doi.org/10.1109/ccdc.2015.7162802.

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10

Perre, Patrick, and Ian Turner. "A physical interpretation of the use of fractional operators for modelling the drying process." In 21st International Drying Symposium. Valencia: Universitat Politècnica València, 2018. http://dx.doi.org/10.4995/ids2018.2018.7885.

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Fractional order derivatives provide useful alternatives to their integer order counterparts due to their ability to model memory and other properties of the porous medium, such as nonlocal behaviour. These phenomena are driven by the constrained interactions within the complex and non-homogeneous microstructures evident at the pore scale. In this work, we investigate the suitability of time and space fractional operators for modelling drying processes and provide a physical interpretation of these operators. At first, the concept and the general formulation in the case of a 1-D finite domain is summarised. Then a selection of simulations allowed us to analyse the physical effects of these operators on the solution. In particular, we elucidate:(I )the ability of these operators to break the fundamental relationship between mean square displacement and time in the simple example of diffusion in an open space, (ii) the caution to be taken with the formulation of boundary conditions and source terms to obtain consistent balance equations, (iii) the effect of fractional in space diffusion as a way to alter the MC profiles compared to standard diffusion, therefore potentially avoiding the dependence of the diffusivity on the variable Keywords: Fractional Calculus; Transport in Porous Media; Finite Volume Method; Matrix Transfer Technique; Matrix Functions.
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