Relevant bibliographies by topics / Fractional calculus

Contents

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

Academic literature on the topic 'Fractional calculus'

Author: Grafiati
Published: 4 June 2021
Last updated: 27 July 2024

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Journal articles on the topic "Fractional calculus"

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Feng, Xiaobing, and Mitchell Sutton. "A new theory of fractional differential calculus." Analysis and Applications 19, no. 04 (February 20, 2021): 715–50. http://dx.doi.org/10.1142/s0219530521500019.

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This paper presents a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem calculus, product and chain rules, and integration by parts formulas are established for weak fractional derivatives. Additionally, relationships with classical fractional derivatives and detailed characterizations of weakly fractional differentiable functions are also established. Furthermore, the notion of weak fractional derivatives is also systematically extended to general distributions instead of only to some special distributions. This new theory lays down a solid theoretical foundation for systematically and rigorously developing new theories of fractional Sobolev spaces, fractional calculus of variations, and fractional PDEs as well as their numerical solutions in subsequent works.
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Baleanu, Dumitru. "About Fractional Calculus of Singular Lagrangians." Journal of Advanced Computational Intelligence and Intelligent Informatics 9, no. 4 (July 20, 2005): 395–98. http://dx.doi.org/10.20965/jaciii.2005.p0395.

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In this paper the solutions of the fractional Euler-Lagrange quations corresponding to singular fractional Lagrangians were examined. We observed that if a Lagrangian is singular in the classical sense, it remains singular after being fractionally generalized. The fractional Lagrangian is non-local but its gauge symmetry was preserved despite complexity of equations in fractional cases. We generalized four examples of singular Lagrangians admitting gauge symmetry in fractional case and found solutions to corresponding Euler-Lagrange equations.
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Zhao, Yan Chun. "Design and Application of Digital Filter Based on Calculus Computing Concept." Applied Mechanics and Materials 513-517 (February 2014): 3151–55. http://dx.doi.org/10.4028/www.scientific.net/amm.513-517.3151.

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Calculus has been widely applied in engineering fields. The development of Integer order calculus theory is more mature in the project which can obtain fractional calculus theory through the promotion of integration order. It extends the flexibility of calculation and achieves the engineering analysis of multi-degree of freedom. According to fractional calculus features and the characteristics of fractional calculus, this paper treats the frequency domain as the object of study and gives the fractional calculus definition of the frequency characteristics. It also designs the mathematical model of fractional calculus digital filters using Fourier transform and Laplace transform. At last, this paper stimulates and analyzes numerical filtering of fractional calculus digital filter circuit using matlab general numerical analysis software and FDATool filter toolbox provided by matlab. It obtains the one-dimensional and two-dimensional filter curves of fractional calculus method which achieves the fractional Calculus filter of complex digital filter.
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Sabzikar, Farzad, Mark M. Meerschaert, and Jinghua Chen. "Tempered fractional calculus." Journal of Computational Physics 293 (July 2015): 14–28. http://dx.doi.org/10.1016/j.jcp.2014.04.024.

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Lubich, Ch. "Discretized Fractional Calculus." SIAM Journal on Mathematical Analysis 17, no. 3 (May 1986): 704–19. http://dx.doi.org/10.1137/0517050.

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Tarasov, Vasily E. "Lattice fractional calculus." Applied Mathematics and Computation 257 (April 2015): 12–33. http://dx.doi.org/10.1016/j.amc.2014.11.033.

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Cont, Rama, and Ruhong Jin. "Fractional Ito calculus." Transactions of the American Mathematical Society, Series B 11, no. 22 (March 27, 2024): 727–61. http://dx.doi.org/10.1090/btran/185.

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We derive Itô–type change of variable formulas for smooth functionals of irregular paths with nonzero p p th variation along a sequence of partitions, where p ≥ 1 p \geq 1 is arbitrary, in terms of fractional derivative operators. Our results extend the results of the Föllmer–Itô calculus to the general case of paths with ‘fractional’ regularity. In the case where p p is not an integer, we show that the change of variable formula may sometimes contain a nonzero ‘fractional’ Itô remainder term and provide a representation for this remainder term. These results are then extended to functionals of paths with nonzero ϕ \phi -variation and multidimensional paths. Using these results, we derive an isometry property for the pathwise Föllmer integral in terms of ϕ \phi -variation.
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Mishura, Yuliya, Olha Hopkalo, and Hanna Zhelezniak. "Elements of fractional calculus. Fractional integrals." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 1 (2022): 11–19. http://dx.doi.org/10.17721/1812-5409.2022/1.1.

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The paper is devoted to the basic properties of fractional integrals. It is a survey of the well-known properties of fractional integrals, however, the authors tried to present the known information about fractional integrals as short and transparently as possible. We introduce fractional integrals on the compact interval and on the semi-axes, consider the famous Hardy-Littlewood theorem and other properties of integrability of fractional integrals. Among other basic properties, we consider Holder continuity and establish to what extent fractional integration increases the smoothness of the integrand. Also, we establish continuity of fractional integrals according to the index of fractional integration, both at strictly positive value and at zero. Then we consider properties of restrictions of fractional integrals from semi-axes on the compact interval. Generalized Minkowsky inequality is applied as one of the important tools. Some examples of calculating fractional integrals are provided.
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Medina, Gustavo D., Nelson R. Ojeda, Jose H. Pereira, and Luis G. Romero. "Fractional Laplace transform and fractional calculus." International Mathematical Forum 12, no. 20 (2017): 991–1000. http://dx.doi.org/10.12988/imf.2017.71194.

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Hanif, Usama, Ammara Nosheen, Rabia Bibi, Khuram Ali Khan, and Hamid Reza Moradi. "Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals." Mathematical Problems in Engineering 2021 (May 28, 2021): 1–14. http://dx.doi.org/10.1155/2021/9939468.

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In this paper, Jensen and Hardy inequalities, including Pólya–Knopp type inequalities for superquadratic functions, are extended using Riemann–Liouville delta fractional integrals. Furthermore, some inequalities are proved by using special kernels. Particular cases of obtained inequalities give us the results on time scales calculus, fractional calculus, discrete fractional calculus, and quantum fractional calculus.
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Dissertations / Theses on the topic "Fractional calculus"

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Tavares, Dina dos Santos. "Fractional calculus of variations." Doctoral thesis, Universidade de Aveiro, 2017. http://hdl.handle.net/10773/22184.

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Doutoramento em Matemática e Aplicações
O cálculo de ordem não inteira, mais conhecido por cálculo fracionário, consiste numa generalização do cálculo integral e diferencial de ordem inteira. Esta tese é dedicada ao estudo de operadores fracionários com ordem variável e problemas variacionais específicos, envolvendo também operadores de ordem variável. Apresentamos uma nova ferramenta numérica para resolver equações diferenciais envolvendo derivadas de Caputo de ordem fracionária variável. Consideram- -se três operadores fracionários do tipo Caputo, e para cada um deles é apresentada uma aproximação dependendo apenas de derivadas de ordem inteira. São ainda apresentadas estimativas para os erros de cada aproximação. Além disso, consideramos alguns problemas variacionais, sujeitos ou não a uma ou mais restrições, onde o funcional depende da derivada combinada de Caputo de ordem fracionária variável. Em particular, obtemos condições de otimalidade necessárias de Euler–Lagrange e sendo o ponto terminal do integral, bem como o seu correspondente valor, livres, foram ainda obtidas as condições de transversalidade para o problema fracionário.
The calculus of non–integer order, usual known as fractional calculus, consists in a generalization of integral and differential integer-order calculus. This thesis is devoted to the study of fractional operators with variable order and specific variational problems involving also variable order operators. We present a new numerical tool to solve differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them, an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. Furthermore, we consider variational problems subject or not to one or more constraints, where the functional depends on a combined Caputo derivative of variable fractional order. In particular, we establish necessary optimality conditions of Euler–Lagrange. As the terminal point in the cost integral, as well the terminal state, are free, thus transversality conditions are obtained.
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Kimeu, Joseph M. "Fractional Calculus: Definitions and Applications." TopSCHOLAR®, 2009. http://digitalcommons.wku.edu/theses/115.

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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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Ferreira, Rui Alexandre Cardoso. "Calculus of variations on time scales and discrete fractional calculus." Doctoral thesis, Universidade de Aveiro, 2010. http://hdl.handle.net/10773/2921.

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Doutoramento em Matemática
Estudamos problemas do cálculo das variações e controlo óptimo no contexto das escalas temporais. Especificamente, obtemos condições necessárias de optimalidade do tipo de Euler–Lagrange tanto para lagrangianos dependendo de derivadas delta de ordem superior como para problemas isoperimétricos. Desenvolvemos também alguns métodos directos que permitem resolver determinadas classes de problemas variacionais através de desigualdades em escalas temporais. No último capítulo apresentamos operadores de diferença fraccionários e propomos um novo cálculo das variações fraccionário em tempo discreto. Obtemos as correspondentes condições necessárias de Euler– Lagrange e Legendre, ilustrando depois a teoria com alguns exemplos.
We study problems of the calculus of variations and optimal control within the framework of time scales. Specifically, we obtain Euler–Lagrange type equations for both Lagrangians depending on higher order delta derivatives and isoperimetric problems. We also develop some direct methods to solve certain classes of variational problems via dynamic inequalities. In the last chapter we introduce fractional difference operators and propose a new discrete-time fractional calculus of variations. Corresponding Euler–Lagrange and Legendre necessary optimality conditions are derived and some illustrative examples provided.
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Ito, Yu. "Rough path theory via fractional calculus." 京都大学 (Kyoto University), 2015. http://hdl.handle.net/2433/199445.

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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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Abdelsheed, Ismail Gad Ameen. "Fractional calculus: numerical methods and SIR models." Doctoral thesis, Università degli studi di Padova, 2016. http://hdl.handle.net/11577/3422267.

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Fractional calculus is ”the theory of integrals and derivatives of arbitrary order, which unify and generalize the notions of integer-order differentiation and n-fold integration”. The idea of generalizing differential operators to a non-integer order, in particular to the order 1/2, first appears in the correspondence of Leibniz with L’Hopital (1695), Johann Bernoulli (1695), and John Wallis (1697) as a mere question or maybe even play of thoughts. In the following three hundred years a lot of mathematicians contributed to the fractional calculus: Laplace (1812), Lacroix (1812), Fourier (1822), Abel (1823-1826), Liouville (1832-1837), Riemann (1847), Grunwald (1867-1872), Letnikov (1868-1872), Sonin (1869), Laurent (1884), Heaviside (1892-1912), Weyl (1917), Davis (1936), Erde`lyi (1939-1965), Gelfand and Shilov (1959-1964), Dzherbashian (1966), Caputo (1969), and many others. Yet, it is only after the First Conference on Fractional Calculus and its applications that the fractional calculus becomes one of the most intensively developing areas of mathematical analysis. Recently, many mathematicians and applied researchers have tried to model real processes using the fractional calculus. This is because of the fact that the realistic modeling of a physical phenomenon does not depend only on the instant time, but also on the history of the previous time which can be successfully achieved by using fractional calculus. In other words, the nature of the definition of the fractional derivatives have provided an excellent instrument for the modeling of memory and hereditary properties of various materials and processes.
Il calcolo frazionario e` ”the theory of integrals and derivatives of arbitrary order, which unify and generalize the notions of integer-order differentiation and n-fold integration”. L’ idea di generalizzare operatori differenziali ad un ordine non intero, in particolare di ordine 1/2, compare per la prima volta in una corrispondenza di Leibniz con L’Hopital (1695), Johann Bernoulli (1695), e John Wallis (1697), come una semplice domanda o forse un gioco di pensieri. Nei successive trecento anni molti matematici hanno contribuito al calcolo frazionario: Laplace (1812), Lacroix (1812), di Fourier (1822), Abel (1823-1826), Liouville (1832-1837), Riemann (1847), Grunwald (1867-1872), Letnikov (1868-1872), Sonin (1869), Laurent (1884), Heaviside (1892-1912), Weyl (1917), Davis (1936), Erde`lyi (1939-1965), Gelfand e Shilov (1959-1964), Dzherbashian (1966), Caputo (1969), e molti altri. Eppure, è solo dopo la prima conferenza sul calcolo frazionario e le sue applicazioni che questo tema diventa una delle le aree più intensamente studiate dell’analisi matematica. Recentemente, molti matematici e ingegneri hanno cercato di modellare i processi reali utilizzando il calcolo frazionario. Questo a causa del fatto che spesso, la modellazione realistica di un fenomeno fisico non è locale nel tempo, ma dipende anche dalla storia, e questo comportamento può essere ben rappresentato attraverso modelli basati sul calcolo frazionario. In altre parole, la definizione dei derivata frazionaria fornisce un eccellente strumento per la modellazione della memoria e delle proprietà ereditarie di vari materiali e processi.
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Shen, Xin. "Applications of Fractional Calculus In Chemical Engineering." Thesis, Université d'Ottawa / University of Ottawa, 2018. http://hdl.handle.net/10393/37577.

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Fractional calculus, which is a generalization of classical calculus, has been the subject of numerous applications in physics and engineering during the last decade. In this thesis, fractional calculus has been implemented for chemical engineering applications, namely in process control and in the modeling mass transfer in adsorption. With respect to process control, some researchers have proposed fractional PIλDμ controllers based on fractional calculus to replace classical PI and PID controllers. The closed-loop control of different benchmark dynamic systems using optimally-tuned fractional PIλDμ controllers were investigated to determine for which dynamic systems this more computationally-intensive controller would be beneficial. Four benchmark systems were used: first order plus dead time system, high order system, nonlinear system, and first order plus integrator system. The optimal tuning of the fractional PIλDμ controller for each system was performed using multi-objective optimization minimizing three performance criteria, namely the ITAE, OZ, and ISDU. Conspicuous advantages of using PIλDμ controllers were confirmed and compared with other types of controllers for these systems. In some cases, a PIλ controller was also a good alternative to the PIλDμ controller with the advantage of being less computationally intensive. For the optimal tuning of fractional controllers for each benchmark dynamic system, a new version of the non-dominated sorting genetic algorithm (NSGA-III) was used to circumscribe the Pareto domain. However, it was found that for the tuning of PIλDμ controllers, it was difficult to circumscribe the complete Pareto domain using NSGA-III. Indeed, the Pareto domain obtained was sometimes fragmentary, unstable and/or susceptible to user-defined parameters and operators of NSGA-III. To properly use NSGA-III and determine a reliable Pareto domain, an investigation on the effect of these user-defined operators and parameters of this algorithm was performed. It was determined that a reliable Pareto domain was obtained with a crossover operator with a significant extrapolation component, a Gaussian mutation operator, and a large population. The findings on the proper use of NSGA-III can also be used for the optimization of other systems. Fractional calculus was also implemented in the modeling of breakthrough curves in packed adsorption columns using finite differences. In this investigation, five models based on different assumptions were proposed for the adsorption of butanol on activated carbon. The first four models are based on integer order partial differential equations accounting for the convective mass transfer through the packed bed and the diffusion and adsorption of an adsorbate within adsorbent particles. The fifth model assumes that the diffusion inside adsorbent particles is potentially anomalous diffusion and expressed by a fractional partial differential equation. For all these models, the best model parameters were determined by nonlinear regression for different sets of experimental data for the adsorption of butanol on activated carbon. The recommended model to represent the breakthrough curves for the two different adsorbents is the model that includes diffusion within the adsorbent particles. For the breakthrough experiments for the adsorption of butanol on activated carbon F-400, it is recommended using a model which accounts for the inner diffusion within the adsorbent particles. It was found that instantaneous or non-instantaneous adsorption models can be used. Best predictions were obtained with fractional order diffusion with instantaneous adsorption. For the adsorption of butanol on activated carbon Norit ROW 0.8, it is recommended using an integer diffusion model with instantaneous adsorption. The gain of using fractional order diffusion equation, given the intensity in computation, was not sufficient to recommend its use.
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Sikaneta, Ishuwa Christopher. "From fractional calculus to split dimensional regularization." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0012/MQ31867.pdf.

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Beig, Mirza Tanweer Ahmad. "Fractional Calculus and Dynamic Approach to Complexity." Thesis, University of North Texas, 2015. https://digital.library.unt.edu/ark:/67531/metadc822832/.

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Fractional calculus enables the possibility of using real number powers or complex number powers of the differentiation operator. The fundamental connection between fractional calculus and subordination processes is explored and affords a physical interpretation for a fractional trajectory, that being an average over an ensemble of stochastic trajectories. With an ensemble average perspective, the explanation of the behavior of fractional chaotic systems changes dramatically. Before now what has been interpreted as intrinsic friction is actually a form of non-Markovian dissipation that automatically arises from adopting the fractional calculus, is shown to be a manifestation of decorrelations between trajectories. Nonlinear Langevin equation describes the mean field of a finite size complex network at criticality. Critical phenomena and temporal complexity are two very important issues of modern nonlinear dynamics and the link between them found by the author can significantly improve the understanding behavior of dynamical systems at criticality. The subject of temporal complexity addresses the challenging and especially helpful in addressing fundamental physical science issues beyond the limits of reductionism.
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Books on the topic "Fractional calculus"

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Agarwal, Praveen, Dumitru Baleanu, YangQuan Chen, Shaher Momani, and José António Tenreiro Machado, eds. Fractional Calculus. Singapore: Springer Singapore, 2019. http://dx.doi.org/10.1007/978-981-15-0430-3.

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C, McBride A., and Roach G. F, eds. Fractional calculus. Boston: Pitman Advanced Pub. Program, 1985.

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Xue, Dingyü, and Lu Bai. Fractional Calculus. Singapore: Springer Nature Singapore, 2024. http://dx.doi.org/10.1007/978-981-99-2070-9.

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Anastassiou, George A. Generalized Fractional Calculus. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-56962-4.

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Das, Shantanu. Functional Fractional Calculus. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-20545-3.

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Goodrich, Christopher, and Allan C. Peterson. Discrete Fractional Calculus. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-25562-0.

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Lorenzo, Carl F. Initialized fractional calculus. Cleveland, Ohio: National Aeronautics and Space Administration, Glenn Research Center, 2000.

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Das, Shantanu. Functional Fractional Calculus. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg, 2011.

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Daftardar-Gejji, Varsha, ed. Fractional Calculus and Fractional Differential Equations. Singapore: Springer Singapore, 2019. http://dx.doi.org/10.1007/978-981-13-9227-6.

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Rubin, Boris. Fractional integrals and potentials. Harlow: Longman, 1996.

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Book chapters on the topic "Fractional calculus"

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Malinowska, Agnieszka B., Tatiana Odzijewicz, and Delfim F. M. Torres. "Fractional Calculus." In Advanced Methods in the Fractional Calculus of Variations, 7–21. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-14756-7_2.

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Mathai, A. M., Ram Kishore Saxena, and Hans J. Haubold. "Fractional Calculus." In The H-Function, 75–117. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/978-1-4419-0916-9_3.

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Petráš, Ivo. "Fractional Calculus." In Fractional-Order Nonlinear Systems, 7–42. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-18101-6_2.

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Vyawahare, Vishwesh, and Paluri S. V. Nataraj. "Fractional Calculus." In Fractional-order Modeling of Nuclear Reactor: From Subdiffusive Neutron Transport to Control-oriented Models, 1–10. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-10-7587-2_1.

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Gorenflo, R., and F. Mainardi. "Fractional Calculus." In Fractals and Fractional Calculus in Continuum Mechanics, 223–76. Vienna: Springer Vienna, 1997. http://dx.doi.org/10.1007/978-3-7091-2664-6_5.

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Gorenflo, R. "Fractional Calculus." In Fractals and Fractional Calculus in Continuum Mechanics, 277–90. Vienna: Springer Vienna, 1997. http://dx.doi.org/10.1007/978-3-7091-2664-6_6.

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Mainardi, F. "Fractional Calculus." In Fractals and Fractional Calculus in Continuum Mechanics, 291–348. Vienna: Springer Vienna, 1997. http://dx.doi.org/10.1007/978-3-7091-2664-6_7.

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Lepik, Ülo, and Helle Hein. "Fractional Calculus." In Mathematical Engineering, 107–22. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-04295-4_8.

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Balachandran, K. "Fractional Calculus." In Industrial and Applied Mathematics, 23–49. Singapore: Springer Nature Singapore, 2023. http://dx.doi.org/10.1007/978-981-99-6080-4_2.

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Jin, Bangti. "Fractional Calculus." In Fractional Differential Equations, 19–58. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-76043-4_2.

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Conference papers on the topic "Fractional calculus"

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Alahmad, Rami, and Ahmad Abdelhadi. "Introduction to Fractional Calculus." In 2019 Advances in Science and Engineering Technology International Conferences (ASET). IEEE, 2019. http://dx.doi.org/10.1109/icaset.2019.8714417.

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"Applied fractional order calculus." In 2016 IEEE International Conference on Automation, Quality and Testing, Robotics (AQTR). IEEE, 2016. http://dx.doi.org/10.1109/aqtr.2016.7501362.

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Cruz-Duarte, Jorge M., and Porfirio Toledo-Hernández. "Fractional Calculus in Mexico: The 5th Mexican Workshop on Fractional Calculus (MWFC)." In MWFC 2022. Basel Switzerland: MDPI, 2023. http://dx.doi.org/10.3390/cmsf2022004007.

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Zeng, Caibin, and YangQuan Chen. "Optimal Random Search, Fractional Dynamics and Fractional Calculus." 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/detc2013-12734.

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What is the most efficient search strategy for the random located target sites subject to the physical and biological constraints? Previous results suggested the Levy flight is the best option to characterize this optimal problem, however, which ignores the understanding and learning abilities of the searcher agents. In the paper we propose the Continuous Time Random Walk (C-TRW) optimal search framework and find the optimum for both of search length’s and waiting time’s distributions. Based on fractional calculus technique, we further derive its master equation to show the mechanism of such complex fractional dynamics. Numerous simulations are provided to illustrate the non-destructive and destructive cases.
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Zhang, Yanshan, Feng Zhang, and Mingfeng Lu. "Relationship between fractional calculus and fractional Fourier transform." In SPIE Optical Engineering + Applications, edited by Oliver E. Drummond and Richard D. Teichgraeber. SPIE, 2015. http://dx.doi.org/10.1117/12.2187649.

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Schirmer, Pascal A., and Iosif Mporas. "Energy Disaggregation Using Fractional Calculus." In ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2020. http://dx.doi.org/10.1109/icassp40776.2020.9054713.

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Liu, Kai, Xi Zhang, and YangQuan Chen. "Energy Informatics and Fractional Calculus." 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/detc2017-67485.

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Energy informatics (EI) is relatively a blossoming and dynamic research area especially in today’s green manufacturing world. Using renewable energy and clean technology are the keys to a revitalization of the world manufacturing and job creation. Green manufacturing, which reduces resource use, waste and emissions and saving the energy, has become the priority for the manufacturers. Therefore, EI has come into a desirable solution. The fractional calculus (FC) is a mighty tool which can characterize the complex properties of the natural and social phenomena. In this paper, we have provided an overview on the current EI by describing current research topics and methods and then pointed out an outlook of how this new field might be evolved with FC in the coming future.
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Yangquan Chen. "Applied Fractional Calculus in controls." In 2009 American Control Conference. IEEE, 2009. http://dx.doi.org/10.1109/acc.2009.5159794.

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Ma, Qingxia, Xianguang Lin, and Huijuan Li. "A Comparative Teaching of Fractional Calculus and Integer Calculus." In 2018 International Conference on Social Science and Education Reform (ICSSER 2018). Paris, France: Atlantis Press, 2018. http://dx.doi.org/10.2991/icsser-18.2018.22.

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Machado, J. A. Tenreiro, Isabel S. Jesus, and Alexandra Galhano. "A Fractional Calculus Perspective in Electromagnetics." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84862.

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Some experimentation with magnets was beginning in the late 19th century. By then reliable batteries had been developed and the electric current was recognized as a stream of charge particles. Maxwell developed a set of equations expressing the basic laws of electricity and magnetism, and demonstrated that these two phenomena are complementary aspects of electromagnetism. He showed that electric and magnetic fields travel through space, in the form of waves, at a constant velocity. Maxwell is generally regarded as the nineteenth century scientist who had the greatest influence on twentieth century physics, making contributions to the fundamental models of nature. Bearing these ideas in mind, in this study we apply the concept of fractional calculus and some aspects of electromagnetism, to the static electric potential, and we develop a new fractional order approximation method to the electrical potential.
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Reports on the topic "Fractional calculus"

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D'Elia, Marta, Mamikon Gulian, George Karniadakis, and Hayley Olson. A Unified Theory of Fractional Nonlocal and Weighted Nonlocal Vector Calculus. Office of Scientific and Technical Information (OSTI), May 2020. http://dx.doi.org/10.2172/1618398.

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Altınkaya, Şahsene, Shigeyoshi Owa, and Sibel Yalçın. On a New Class of Analytic Functions Related to Fractional Calculus. "Prof. Marin Drinov" Publishing House of Bulgarian Academy of Sciences, January 2020. http://dx.doi.org/10.7546/crabs.2020.01.02.

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D'Elia, Marta, Mamikon Gulian, Tadele Mengesha, and James Scott. Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition. Office of Scientific and Technical Information (OSTI), December 2021. http://dx.doi.org/10.2172/1855046.

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