Relevant bibliographies by topics / Fractional derivatives at zero

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

Academic literature on the topic 'Fractional derivatives at zero'

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Published: 18 January 2024

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Journal articles on the topic "Fractional derivatives at zero"

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Bouzeffour, Fethi. "Advancing Fractional Riesz Derivatives through Dunkl Operators." Mathematics 11, no. 19 (September 25, 2023): 4073. http://dx.doi.org/10.3390/math11194073.

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The aim of this work is to introduce a novel concept, Riesz–Dunkl fractional derivatives, within the context of Dunkl-type operators. A particularly noteworthy revelation is that when a specific parameter κ equals zero, the Riesz–Dunkl fractional derivative smoothly reduces to both the well-known Riesz fractional derivative and the fractional second-order derivative. Furthermore, we introduce a new concept: the fractional Sobolev space. This space is defined and characterized using the versatile framework of the Dunkl transform.
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HILFER, R. "FOUNDATIONS OF FRACTIONAL DYNAMICS." Fractals 03, no. 03 (September 1995): 549–56. http://dx.doi.org/10.1142/s0218348x95000485.

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Time flow in dynamical systems is reconsidered in the ultralong time limit. The ultralong time limit is a limit in which a discretized time flow is iterated infinitely often and the discretization time step is infinite. The new limit is used to study induced flows in ergodic theory, in particular for subsets of measure zero. Induced flows on subsets of measure zero require an infinite renormalization of time in the ultralong time limit. It is found that induced flows are given generically by stable convolution semigroups and not by the conventional translation groups. This could give new insight into the origin of macroscopic irreversibility. Moreover, the induced semigroups are generated by fractional time derivatives of orders less than unity, and not by a first order time derivative. Invariance under the induced semiflows therefore leads to a new form of stationarity, called fractional stationarity. Fractionally stationary states are dissipative. Fractional stationarity also provides the dynamical foundation for a previously proposed generalized equilibrium concept.
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Farr, Ricky E., Sebastian Pauli, and Filip Saidak. "zero-free region for the fractional derivatives of the Riemann zeta function." New Zealand Journal of Mathematics 50 (September 4, 2020): 1–9. http://dx.doi.org/10.53733/42.

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For any , we denote by the α-th Grunwald-Letnikov fractional derivative of the Riemann zeta function ζ(s). For these derivatives we show: inside the region | s − 1 | < 1. This result, the first of its kind, is proved by a careful analysis of integrals involving Bernoulli polynomials and bounds for fractional Stieltjes constants.
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NABER, MARK. "DISTRIBUTED ORDER FRACTIONAL SUB-DIFFUSION." Fractals 12, no. 01 (March 2004): 23–32. http://dx.doi.org/10.1142/s0218348x04002410.

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A distributed order fractional diffusion equation is considered. Distributed order derivatives are fractional derivatives that have been integrated over the order of the derivative within a given range. In this paper, sub-diffusive cases are considered. That is, the order of the time derivative ranges from zero to one. The equation is solved for Dirichlet, Neumann and Cauchy boundary conditions. The time dependence for each of the three cases is found to be a functional of the diffusion parameter. This functional is shown to have decay properties. Upper and lower bounds are computed for the functional. Examples are also worked out for comparative decay rates.
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Agarwal, Ravi P., Snezhana Hristova, and Donal O’Regan. "Asymptotic Behavior of Delayed Reaction-Diffusion Neural Networks Modeled by Generalized Proportional Caputo Fractional Partial Differential Equations." Fractal and Fractional 7, no. 1 (January 11, 2023): 80. http://dx.doi.org/10.3390/fractalfract7010080.

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In this paper, a delayed reaction-diffusion neural network model of fractional order and with several constant delays is considered. Generalized proportional Caputo fractional derivatives with respect to the time variable are applied, and this type of derivative generalizes several known types in the literature for fractional derivatives such as the Caputo fractional derivative. Thus, the obtained results additionally generalize some known models in the literature. The long term behavior of the solution of the model when the time is increasing without a bound is studied and sufficient conditions for approaching zero are obtained. Lyapunov functions defined as a sum of squares with their generalized proportional Caputo fractional derivatives are applied and a comparison result for a scalar linear generalized proportional Caputo fractional differential equation with several constant delays is presented. Lyapunov functions and the comparison principle are then combined to establish our main results.
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Diethelm, Kai, Roberto Garrappa, Andrea Giusti, and Martin Stynes. "Why fractional derivatives with nonsingular kernels should not be used." Fractional Calculus and Applied Analysis 23, no. 3 (June 25, 2020): 610–34. http://dx.doi.org/10.1515/fca-2020-0032.

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AbstractIn recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.
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Luchko, Yuri. "General Fractional Integrals and Derivatives with the Sonine Kernels." Mathematics 9, no. 6 (March 10, 2021): 594. http://dx.doi.org/10.3390/math9060594.

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In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the n-fold general fractional integrals and derivatives and study their properties.
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Mugbil, Ahmad, and Nasser-Eddine Tatar. "Hadamard-Type Fractional Integro-Differential Problem: A Note on Some Asymptotic Behavior of Solutions." Fractal and Fractional 6, no. 5 (May 15, 2022): 267. http://dx.doi.org/10.3390/fractalfract6050267.

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As a follow-up to the inherent nature of Hadamard-Type Fractional Integro-differential problem, little is known about some asymptotic behaviors of solutions. In this paper, an integro-differential problem involving Hadamard fractional derivatives is investigated. The leading derivative is of an order between one and two whereas the nonlinearities may contain fractional derivatives of an order between zero and one as well as some non-local terms. Under some reasonable conditions, we prove that solutions are asymptotic to logarithmic functions. Our approach is based on a generalized version of Bihari–LaSalle inequality, which we prove. In addition, several manipulations and crucial estimates have been used. An example supporting our findings is provided.
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Prodanov, Dimiter. "Generalized Differentiability of Continuous Functions." Fractal and Fractional 4, no. 4 (December 10, 2020): 56. http://dx.doi.org/10.3390/fractalfract4040056.

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Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.
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Area, I., J. Losada, and J. J. Nieto. "On Fractional Derivatives and Primitives of Periodic Functions." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/392598.

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Dissertations / Theses on the topic "Fractional derivatives at zero"

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Wang, Zhibo. "Estimations non-asymptotiques et robustes basées sur des fonctions modulatrices pour les systèmes d'ordre fractionnaire." Electronic Thesis or Diss., Bourges, INSA Centre Val de Loire, 2023. http://www.theses.fr/2023ISAB0003.

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Cette thèse développe la méthode des fonctions modulatrices pour des estimations non-asymptotiques et robustes pour des pseudo-états des systèmes nonlinéaires d'ordre fractionnaire, des systèmes linéaires d'ordre fractionnaire avec des accélérations en sortie, et des systèmes à retards d'ordre fractionnaire. Les estimateurs conçus sont fournis en termes de formules intégrales algébriques, ce qui assure une convergence non-asymptotique. Comme une caractéristique essentielle des algorithmes d'estimation conçus, les mesures de sorties bruitées ne sont impliquées que dans les termes intégraux, ce qui confère aux estimateurs une robustesse contre les bruits. Premièrement, pour les systèmes nonlinéaires d'ordre fractionnaire et partiellement inconnu, l'estimation de la dérivée fractionnaire du pseudo-état est abordée via la méthode des fonctions modulatrices. Grâce à la loi de l'indice additif des dérivées fractionnaires, l'estimation est décomposée en une estimation des dérivées fractionnaires de la sortie et une estimation des valeurs initiales fractionnaires. Pendant ce temps, la partie inconnue est estimée via une stratégie innovante de fenêtre glissante. Deuxièmement, pour les systèmes linéaires d'ordre fractionnaire avec des accélérations comme sortie, l'estimation de l'intégrale fractionnaire de l'accélération est d'abord considérée pour les systèmes mécaniques de vibration d'ordre fractionnaire, où seules des mesures d'accélération bruitées sont disponibles. Basée sur des approches numériques existantes qui traitent des intégrales fractionnaires, notre attention se limite principalement à l'estimation des valeurs initiales inconnues en utilisant la méthode des fonctions modulatrices. Sur cette base, le résultat est ensuite généralisé aux systèmes linéaires plus généraux d'ordre fractionnaire. En particulier, le comportement des dérivées fractionnaires à zéro est étudié pour des fonctions absolument continues, ce qui est assez différent de celui de l'ordre entier. Troisièment, pour les systèmes à retards d'ordre fractionnaire, l'estimation du pseudo-état est étudiée en concevant un système dynamique auxiliaire d'ordre fractionnaire, qui fournit un cadre plus général pour générer les fonctions modulatrices requises. Avec l'introduction de l'opérateur de retard et du changement de coordonnées généralisé bicausal, l'estimation du pseudo-état du système considéré peut être réduite à celle de la forme normale correspondante. Contrairement aux travaux précédents le schéma présenté permet une estimation directe du pseudo-état plutôt que d'estimer les dérivées fractionnaires de la sortie et un ensemble de valeurs initiales fractionnaires. De plus, l'efficacité et la robustesse des estimateurs proposés sont vérifiées par des simulations numériques dans cette thèse. Enfin, un résumé de ce travail et un aperçu des travaux futurs sont tirés
This thesis develops the modulating functions method for non-asymptotic and robust estimations for fractional-order nonlinear systems, fractional-order linear systems with accelerations as output, and fractional-order time-delay systems. The designed estimators are provided in terms of algebraic integral formulas, which ensure non-asymptotic convergence. As an essential feature of the designed estimation algorithms, noisy output measurements are only involved in integral terms, which endows the estimators with robustness against corrupting noises. First, for fractional-order nonlinear systems which are partially unknown, fractional derivative estimation of the pseudo-state is addressed via the modulating functions method. Thanks to the additive index law of fractional derivatives, the estimation is decomposed into the fractional derivatives estimation of the output and the fractional initial values estimation. Meanwhile, the unknown part is fitted via an innovative sliding window strategy. Second, for fractional-order linear systems with accelerations as output, fractional integral estimation of the acceleration is firstly considered for fractional-order mechanical vibration systems, where only noisy acceleration measurements are available. Based on the existing numerical approaches addressing the proper fractional integrals of accelerations, our attention is primarily restricted to estimating the unknown initial values using the modulating functions method. On this basis, the result is further generalized to more general fractional-order linear systems. In particular, the behaviour of fractional derivatives at zero is studied for absolutely continuous functions, which is quite different from that of integer order. Third, for fractional-order time-delay systems, pseudo-state estimation is studied by designing a fractional-order auxiliary modulating dynamical system, which provides a more general framework for generating the required modulating functions. With the introduction of the delay operator and the bicausal generalized change of coordinates, the pseudo-state estimation of the considered system can be reduced to that of the corresponding observer normal form. In contrast to the previous work, the presented scheme enables direct estimation for the pseudo-state rather than estimating the fractional derivatives of the output and a bunch of fractional initial values. In addition, the efficiency and robustness of the proposed estimators are verified by numerical simulations in this thesis. Finally, a summary of this work and an insight into future work were drawn
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Katugampola, Don Udita Nalin. "ON GENERALIZED FRACTIONAL INTEGRALS AND DERIVATIVES." OpenSIUC, 2011. https://opensiuc.lib.siu.edu/dissertations/387.

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In this paper we present a generalization to two existing fractional integrals and derivatives, namely, the Riemann-Liouville and Hadamard fractional operators. The existence and uniqueness results for single term fractional differential equations (FDE) have also been established. We also obtain the Mellin transforms of such generalized fractional operators which are important in solving fractional differential equations.
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Schiavone, S. E. "Distributional theories for multidimensional fractional integrals and derivatives." Thesis, University of Strathclyde, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.382492.

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Traytak, Sergey D., and Tatyana V. Traytak. "Method of fractional derivatives in time-dependent diffusion." Universitätsbibliothek Leipzig, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-193646.

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Traytak, Sergey D., and Tatyana V. Traytak. "Method of fractional derivatives in time-dependent diffusion." Diffusion fundamentals 6 (2007) 38, S. 1-2, 2007. https://ul.qucosa.de/id/qucosa%3A14215.

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Munkhammar, Joakim. "Riemann-Liouville Fractional Derivatives and the Taylor-Riemann Series." Thesis, Uppsala University, Department of Mathematics, 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-121418.

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Haveroth, Thais Clara da Costa. "On the use of fractional derivatives for modeling nonlinear viscoelasticity." Universidade do Estado de Santa Catarina, 2015. http://tede.udesc.br/handle/handle/2069.

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Made available in DSpace on 2016-12-12T20:25:13Z (GMT). No. of bitstreams: 1 Thais Clara da Costa Haveroth.pdf: 3726370 bytes, checksum: 204349100247f52ea6bf4916ec49a0ab (MD5) Previous issue date: 2015-10-26
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Dentre a vasta gama de polímeros estruturais atualmente disponíveis no mercado, este trabalho está particularmente voltado ao estudo do polietileno de alta densidade. Embora este material já tenha sido investigado por diversos autores, seu típico comportamento viscoelástico não-linear apresenta dificuldades na modelagem. Visando uma nova contribuição, este trabalho propõe a descrição de tal comportamento utilizando uma abordagem baseada em derivadas fracionários. Esta formulação produz equações constitutivas fracionais que resultam em boas propriedades de ajuste de curvas com menos parâmetros a serem identificados que nos métodos tradicionais. Neste sentido, os resultados experimentais de fluência para o polietileno de alta densidade, avaliados em diferentes níveis de tensão, são ajustados por este esquema. Para estimar a deformação à níveis de tensão que não tenham sido medidos experimentalmente, o princípio da equivalência tensão-tempo é utilizado e os resultados são comparados com aqueles apresentados por uma interpolação linear dos parâmetros. Além disso, o princípio da superposição modificado é aplicado para predizer a comportamento de materiais sujeitos a níveis de tensão que mudam abruptamente ao longo do tempo. Embora a abordagem fracionária simplifique o problema de otimização inversa subjacente, é observado um grande aumento no esforço computacional. Assim, alguns algoritmos que objetivam economia computacional, são estudados. Conclui-se que, quando acurária é necessária ou quando um modelo de séries Prony requer um número muito grande de parâmetros, a abordagem fracionária pode ser uma opção interessante.
Among the wide range of structural polymers currently available in the market, this work is concerned particularly with high density polyethylene. The typical nonlinear viscoelastic behavior presented by this material is not trivial to model, and has already been investigated by many authors in the past. Aiming at a further contribution, this work proposes modeling this material behavior using an approach based on fractional derivatives. This formulation produces fractional constitutive equations that result in good curve-fitting properties with less parameters to be identified when compared to traditional methods. In this regard, experimental creep results of high density polyethylene evaluated at different stress levels are fitted by this scheme. To estimate creep at stress levels that have not been measured experimentally, the time-stress equivalence principle is used and the results are compared with those presented by a linear interpolation of the parameters. Furthermore, the modified superposition principle is applied to predict the strain for materials subject to stress levels which change abruptly from time to time. Some comparative results are presented showing that the fractional approach proposed in this work leads to better results in relation to traditional formulations described in the literature. Although the fractional approach simplifies the underlying inverse optimization problem, a major increase in computational effort is observed. Hence, some algorithms that show computational cost reduction, are studied. It is concluded that when high accuracy is mandatory or when a Prony series model requires a very large number of parameters, the fractional approach may be an interesting option.
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Shi, Chen Yang. "High order compact schemes for fractional differential equations with mixed derivatives." Thesis, University of Macau, 2017. http://umaclib3.umac.mo/record=b3691348.

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Atkins, Zoe. "Almost sharp fronts : limit equations for a two-dimensional model with fractional derivatives." Thesis, University of Warwick, 2012. http://wrap.warwick.ac.uk/55759/.

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We consider the evolution of sharp fronts and almost-sharp fronts for the ↵-equation, where for an active scalar q the corresponding velocity is defined by u = r?(−#)−(2 − ↵)/2q for 0 < ↵ < 1. This system is introduced as a model interpolating between the two-dimensional Euler equation (↵ = 0) and the surface quasi-geostrophic (SQG) equation (↵ = 1). The study of such fronts for the SQG equation was introduced as a natural extension when searching for potential singularities for the three-dimensional Euler equation due to similarities between these two systems, with sharp-fronts corresponding to vortex-lines in the Euler case (Constantin et al., 1994b). Almost-sharp fronts were introduced in C´ordoba et al. (2004) as a regularisation of a sharp front with thickness $, with interest in the study of such solutions as $ ! 0, in particular those that maintain their structure up to a time independent of $. The construction of almost-sharp front solutions to the SQG equation is the subject of current work (Fe↵erman and Rodrigo, 2012). The existence of exact solutions remains an open problem. For the ↵-equation we prove analogues of several known theorems for the SQG equations and extend these to investigate the construction of almost-sharp front solutions. Using a version of the Abstract Cauchy Kovalevskaya theorem (Safonov, 1995) we show for fixed 0 < ↵ < 1, under analytic assumptions, the existence and uniqueness of approximate solutions and exact solutions for short-time independent of $; such solutions take a form asymptotic to almost-sharp fronts. Finally, we obtain the existence and uniqueness of analytic almost-sharp front solutions.
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Jarrah, Bilal. "Fractional Order and Inverse Problem Solutions for Plate Temperature Control." Thesis, Université d'Ottawa / University of Ottawa, 2020. http://hdl.handle.net/10393/40551.

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Surface temperature control of a thin plate is investigated. Temperature is controlled on one side of the plate using the other side temperature measurements. This is a decades-old problem, reactivated more recently by the awareness that this is a fractional-order problem that justifies the investigation of the use of fractional order calculus. The approach is based on a transfer function obtained from the one-dimensional heat conduction equation solution that results in a fractional-order s-domain representation. Both the inverse problem approach and the fractional controller approach are studied here to control the surface temperature, the first one using inverse problem plus a Proportional only controller, and the second one using only the fractional controller. The direct problem defined as the ratio of the output to the input, while the inverse problem defined as the ratio of the input to the output. Both transfer functions are obtained, and the resulting fractional-order transfer functions were approximated using Taylor expansion and Zero-Pole expansion. The finite number of terms transfer functions were used to form an open-loop control scheme and a closed-loop control scheme. Simulation studies were done for both control schemes and experiments were carried out for closed-loop control schemes. For the fractional controller approach, the fractional controller was designed and used in a closed-loop scheme. Simulations were done for fractional-order-integral, fractional-order-derivative and fractional-integral-derivative controller designs. The experimental study focussed on the fractional-order-integral-derivative controller design. The Fractional-order controller results are compared to integer-order controller’s results. The advantages of using fractional order controllers were evaluated. Both Zero-Pole and Taylor expansions are used to approximate the plant transfer functions and both expansions results are compared. The results show that the use of fractional order controller performs better, in particular concerning the overshoot.
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Books on the topic "Fractional derivatives at zero"

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Uchaikin, Vladimir V. Fractional Derivatives for Physicists and Engineers. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-33911-0.

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Gómez, José Francisco, Lizeth Torres, and Ricardo Fabricio Escobar, eds. Fractional Derivatives with Mittag-Leffler Kernel. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-11662-0.

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A, Kilbas A., and Marichev O. I, eds. Fractional integrals and derivatives: Theory and applications. Switzerland: Gordon and Breach Science Publishers, 1993.

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Wang, JinRong, Shengda Liu, and Michal Fečkan. Iterative Learning Control for Equations with Fractional Derivatives and Impulses. Singapore: Springer Singapore, 2022. http://dx.doi.org/10.1007/978-981-16-8244-5.

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Brychkov, I︠U︡ A. Handbook of special functions: Derivatives, integrals, series, and other formulas. Boca Raton: CRC Press, 2008.

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Brychkov, I︠U︡ A. Handbook of special functions: Derivatives, integrals, series and other formulas. Boca Raton: CRC Press, 2008.

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Zero-sum game: The rise of the worlds largest derivatives exchange. Hoboken, New Jersey: Wiley, 2010.

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Yang, Xiao-Jun. General Fractional Derivatives. Taylor & Francis Group, 2019.

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Jin, Bangti. Fractional Differential Equations: An Approach Via Fractional Derivatives. Springer International Publishing AG, 2022.

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Jin, Bangti. Fractional Differential Equations: An Approach Via Fractional Derivatives. Springer International Publishing AG, 2021.

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Book chapters on the topic "Fractional derivatives at zero"

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Capelas de Oliveira, Edmundo. "Fractional Derivatives." In Studies in Systems, Decision and Control, 169–222. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-20524-9_5.

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Zhao, Xuan, and Zhi-Zhong Sun. "Time-fractional derivatives." In Numerical Methods, edited by George Em Karniadakis, 23–48. Berlin, Boston: De Gruyter, 2019. http://dx.doi.org/10.1515/9783110571684-002.

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Yang, Xiao-Jun. "Introduction." In General Fractional Derivatives, 1–37. Boca Raton : CRC Press, Taylor & Francis Group, 2019.: Chapman and Hall/CRC, 2019. http://dx.doi.org/10.1201/9780429284083-1.

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Yang, Xiao-Jun. "Fractional Derivatives of Constant Order and Applications." In General Fractional Derivatives, 39–142. Boca Raton : CRC Press, Taylor & Francis Group, 2019.: Chapman and Hall/CRC, 2019. http://dx.doi.org/10.1201/9780429284083-2.

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Yang, Xiao-Jun. "General Fractional Derivatives of Constant Order and Applications." In General Fractional Derivatives, 145–234. Boca Raton : CRC Press, Taylor & Francis Group, 2019.: Chapman and Hall/CRC, 2019. http://dx.doi.org/10.1201/9780429284083-3.

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Yang, Xiao-Jun. "Fractional Derivatives of Variable Order and Applications." In General Fractional Derivatives, 235–66. Boca Raton : CRC Press, Taylor & Francis Group, 2019.: Chapman and Hall/CRC, 2019. http://dx.doi.org/10.1201/9780429284083-4.

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Yang, Xiao-Jun. "Fractional Derivatives of Variable Order with Respect to Another Function and Applications." In General Fractional Derivatives, 267–88. Boca Raton : CRC Press, Taylor & Francis Group, 2019.: Chapman and Hall/CRC, 2019. http://dx.doi.org/10.1201/9780429284083-5.

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Uchaikin, Vladimir V. "Fractional Differentiation." In Fractional Derivatives for Physicists and Engineers, 199–255. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-33911-0_4.

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Ortigueira, Manuel Duarte. "The Causal Fractional Derivatives." In Fractional Calculus for Scientists and Engineers, 5–41. Dordrecht: Springer Netherlands, 2011. http://dx.doi.org/10.1007/978-94-007-0747-4_2.

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Ortigueira, Manuel Duarte. "Two-Sided Fractional Derivatives." In Fractional Calculus for Scientists and Engineers, 101–21. Dordrecht: Springer Netherlands, 2011. http://dx.doi.org/10.1007/978-94-007-0747-4_5.

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Conference papers on the topic "Fractional derivatives at zero"

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Agrawal, Om P. "An Analytical Scheme for Stochastic Dynamic Systems Containing Fractional Derivatives." In ASME 1999 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/detc99/vib-8238.

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Abstract This paper presents an analytical technique for the analysis of a stochastic dynamic system whose damping behavior is described by a fractional derivative of order 1/2. In this approach, an eigenvector expansion method proposed by Suarez and Shokooh is used to obtain the response of the system. The properties of Laplace transforms of convolution integrals are used to write a set of general Duhamel integral type expressions. The general response contains two parts, namely zero state and zero input. For a stochastic analysis the input force is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic characteristics of the system. Closed form stochastic response expressions are obtained for white noise. Numerical results are presented to show the stochastic response of a fractionally damped system subjected to white noise.
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Fukunaga, Masataka, and Nobuyuki Shimizu. "Initial Condition Problems of Fractional Viscoelastic Equations." In ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/detc2003/vib-48394.

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The 2nd order differential equation with fractional derivatives describing dynamic behavior of a single-degree-of-freedom viscoelastic oscillator, referred to as fractional viscoelastic equation (FVE), is considered. Some types of viscoelastic damped mechanical systems may be described by FVE. The differential equation with fractional derivatives is often called the fractional differential equation (FDE). FDE can be solved for zero initial values, but it can not generally be solved for non-zero initial values. How to solve the problem is one of the key issues in this field. This is called “Initial condition (value) problems” of FDE. In this paper, initial condition problems of FVE are solved by making use of the prehistory functions of unknowns which are specified before the initial instance (referred to as the initial functions) starts. Introduction of initial functions into FDE reflects the physical state in giving the initial values. In this paper, several types of initial function are used to solve unique solutions for a type of FVE (referred to as FVE-I). The solutions of FVE-I are obtained by means of both numerical and analytical methods. Implication of the solutions to viscoelastic material will also be discussed.
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Yu, Ziquan, Youmin Zhang, Yaohong Qu, and Zhewen Xing. "Adaptive Fractional-Order Fault-Tolerant Tracking Control for UAV Based on High-Gain Observer." 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/detc2017-67479.

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This paper is concerned with the fractional-order fault-tolerant tracking control design for unmanned aerial vehicle (UAV) in the presence of external disturbance and actuator fault. Based on the functional decomposition, the dynamics of UAV is divided into velocity subsystem and altitude subsystem. Altitude, flight path angle, pitch angle and pitch rate are involved in the altitude subsystem. By using an adaptive mechanism, the fractional derivative of uncertainty including external disturbance and actuator fault is estimated. Moreover, in order to eliminate the problem of explosion of complexity in back-stepping approach, the high-gain observer is utilized to estimate the derivatives of virtual control signal. Furthermore, by using a fractional-order sliding surface involved with pitch dynamics, an adaptive fractional-order fault-tolerant control scheme is proposed for UAV. It is proved that all signals of the closed-loop system are bounded and the tracking error can converge to a small region containing zero via the Lyapunov analysis. Simulation results show that the proposed controller could achieve good tracking performance in the presence of actuator fault and external disturbance.
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Agrawal, Om P. "Stochastic Analysis of a Fractionally Damped Beam." In ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/vib-21365.

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Abstract This paper presents a general analytical technique for stochastic analysis of a continuous beam whose damping characteristic is described using a fractional derivative model. In this formulation, the normal-mode approach is used to reduce the differential equation of a fractionally damped continuous beam into a set of infinite equations each of which describes the dynamics of a fractionally damped spring-mass-damper system. A Laplace transform technique is used to obtain the fractional Green’s function and a Duhamel integral type expression for the system’s response. The response expression contains two parts, namely zero state and zero input. For a stochastic analysis, the input force is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic characteristics of the system. Closed form stochastic response expressions are obtained for White noise. The approach can be extended to all those systems for which the existence of normal modes is guaranteed.
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Liu, Yaqing, Liancun Zheng, Xinxin Zhang, and Fenglei Zong. "The MHD Flows for a Heated Generalized Oldroyd-B Fluid With Fractional Derivative." In 2010 14th International Heat Transfer Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/ihtc14-22278.

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In this paper, we present a circular motion of magnetohydrodynamic (MHD) flow for a heated generalized Oldroyd-B fluid. The fractional calculus approach is introduced to establish the constitutive relationship of a viscoelastic fluid. The velocity and temperature fields of the flow are described by fractional partial differential equations. Exact analytical solutions of velocity and temperature fields are obtained by using Hankel transform and Laplace transform for fractional calculus. Results for ordinary viscous flow are deduced by making the fractional order of differential tend to one and zero. It is shown that the fractional constitutive relation model is more useful than the conventional model for describing the properties of viscoelastic fluid.
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Jin, Yongshun, YangQuan Chen, Chunyang Wang, and Ying Luo. "Fractional Order Proportional Derivative (FOPD) and FO[PD] Controller Design for Networked Position Servo Systems." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-87662.

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This paper considers the fractional order proportional derivative (FOPD) controller and fractional order [proportional derivative] (FO[PD]) controller for networked position servo systems. The systematic design schemes of the networked position servo system with a time delay are presented. It follows from the Bode plot of the FOPD system and the FO[PD] that the given gain crossover frequency and phase margin are fulfilled. Moreover, the phase derivative w.r.t. the frequency is zero, which means that the closed-loop system is robust to gain variations at the given gain crossover frequency. However, sometimes we can not get the controller parameters to meet our robustness requirement. In this paper, we have studied on this situation and presented the requirement of the gain cross frequency, and phase margin in the designing process. For the comparison of fractional order controllers with traditional integer order controller, the integer order proportional integral differential (IOPID) was also designed by using the same proposed method. The simulation results have verified that FOPD and FO[PD] are effective for networked position servo. The simulation results also reveal that both FOPD controller and FO[PD] controller outperform IO-PID controller for this type of system.
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Taub, Gordon N., Hyungoo Lee, S. Balachandar, and S. A. Sherif. "A Numerical Study of Swirling Buoyant Laminar Jets at Low Reynolds Numbers." In ASME 2009 International Mechanical Engineering Congress and Exposition. ASMEDC, 2009. http://dx.doi.org/10.1115/imece2009-13082.

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The effect of swirl on laminar buoyant jets with low Reynolds numbers is explored. Three dimensional direct numerical simulations are performed to solve the time-dependent, incompressible Navier-Stokes equations. We use a body fitted grid system and employ the finite volume method to discretize the governing equations. A second-order central difference scheme is employed for all spatial derivative terms. The numerical simulation is advanced in time by a fractional step method with the second-order Adams-Bashforth scheme for explicit-convection terms and the Crank-Nicholson scheme for implicit-diffusion terms. The amount of swirl and buoyancy is varied from zero to very large values and the effect on the velocity field, jet width, entrainment and vortex are examined. Comparisons with analytical and experimental models are discussed.
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Andersen, Pål Østebø. "Extended Fractional Flow Theory for Steady State Relative Permeability Experiments With Capillary End Effects – Transient Solutions and Time Scales." In 2022 SPWLA 63rd Annual Symposium. Society of Petrophysicists and Well Log Analysts, 2022. http://dx.doi.org/10.30632/spwla-2022-0031.

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Steady state core flooding experiments are regularly performed for estimation of relative permeability functions which are key inputs for simulation of multiphase flow in porous media. Such tests can be time demanding, thus it is valuable to find the relation between fluid flow properties, injection conditions and the time for the desired measurements of stabilized average saturation and pressure drop to be obtained. The phases we consider are oil and water but could be any immiscible phases. An injection condition is specified by the total injected volumetric rate Qt and the fraction F of the total rate due to water. At steady state at these injection conditions there is no change with time in pressures or saturations. We consider the transition from the steady state of one injection condition, when one or both of Qt and F are changed, until the steady state of the new injection conditions have developed. The spatial saturation distributions at steady state are known analytically from solutions derived by Andersen (2021a). From the method of characteristics, a solution is constructed to describe the velocity of each saturation traveling between the two steady state profiles. The velocity for each saturation is estimated based on an effective fractional flow function that combines contributions of advective and capillary forces to one saturation dependent function. The solution allows us to interpret experimental data, forecast the time scale of the tests at each condition and suggest a more optimal scheme to conduct the experiments as they proceed. Assuming an imbibition setup, the solution implies frontal displacement and short time scales when increasing the fraction F where the flow function is convex. This is the case at sufficiently low saturations. At higher saturations the flow function becomes concave and the saturations travel with individual speeds proportional to the flow function derivative. The time scale is limited by the slowest saturation. At high saturations the saturation speed approaches zero giving an infinite time scale. In a water-wet case, capillary end effects shift the saturation profiles up to include slower high saturations and increase the time scale. In an oil-wet case the saturation profiles are shifted down such that the slowest saturations are faster or travel a shorter distance, thus reducing the time scale. The model is validated by comparing the proposed analytical solution with numerical solutions.
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Maamri, N., and J. C. Trigeassou. "Integration of Fractional Differential Equations without Fractional Derivatives." In 2021 9th International Conference on Systems and Control (ICSC). IEEE, 2021. http://dx.doi.org/10.1109/icsc50472.2021.9666533.

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Pooseh, Shakoor, Helena Sofia Rodrigues, Delfim F. M. Torres, Theodore E. Simos, George Psihoyios, Ch Tsitouras, and Zacharias Anastassi. "Fractional Derivatives in Dengue Epidemics." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics. AIP, 2011. http://dx.doi.org/10.1063/1.3636838.

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