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Journal articles on the topic 'Fractional calculus'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 July 2024

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Tarasov, Vasily E. "General Fractional Vector Calculus." Mathematics 9, no. 21 (November 5, 2021): 2816. http://dx.doi.org/10.3390/math9212816.

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A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus (GFC) in the Luchko approach, which was published in 2021. This paper proposed the following: (I) Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed. (II) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (III) The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves. All these three parts allow us to state that we proposed a calculus, which is a general fractional vector calculus (General FVC). The difficulties and problems of defining general fractional integral and differential vector operators are discussed to the nonlocal case, caused by the violation of standard product rule (Leibniz rule), chain rule (rule of differentiation of function composition) and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed.

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12

Sabatier, Jocelyn. "Design of Fractional Calculus Free Controllers with Fractional Behaviors." WSEAS TRANSACTIONS ON SYSTEMS AND CONTROL 18 (December 31, 2023): 602–11. http://dx.doi.org/10.37394/23203.2023.18.62.

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Faced with the complexity and drawbacks of fractional calculus highlighted in the literature, this paper proposes simple solutions to avoid its use in the field of feedback control and especially to define fractional PID- and CRONE-like controllers. It shows that it is possible to generate fractional behaviors, which are known since the work of Bode to be useful in the field of control, without invoking fractional calculus and fractional models. Fractional calculus based models and fractional behaviors are indeed two different concepts: one denotes a particular class of models and the other a class of dynamical behaviors that can be generated and modelled by a wide variety of mathematical tools other than fractional calculus. Solutions to tune the fractional PID- and Crone-like controllers defined in this paper are proposed.

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13

Zine, Houssine, and Delfim F. M. Torres. "A Stochastic Fractional Calculus with Applications to Variational Principles." Fractal and Fractional 4, no. 3 (August 1, 2020): 38. http://dx.doi.org/10.3390/fractalfract4030038.

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We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.

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14

Gohar, Abdelrahman, Mayada Younes, and Salah Doma. "On Gohar Fractional Calculus." Journal of Fractional Calculus and Nonlinear Systems 5, no. 1 (June 22, 2024): 32–51. http://dx.doi.org/10.48185/jfcns.v5i1.1048.

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Recently, Gohar et al. introduced a novel, local, and well-behaved fractional calculus. It possesses all the classical properties, and Its locality imposes simplicity and accuracy in modeling fractional order systems. In this article, we further develop the definitions and extend the classical properties of Gohar fractional calculus to address some of the open problems in Calculus. The fractional Gronwall's integral inequality, Taylor power series expansion, and Laplace transform are defined and applied to overcome some of the limitations in the classical integer-order calculus. The fractional Laplace transform is applied to solve Bernoulli-type logistic and Bertalanffy nonlinear fractional differential equations, and the criteria under which it can be applied to solve linear differential equations are investigated.

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15

Tarasov, Vasily E. "General Fractional Calculus in Multi-Dimensional Space: Riesz Form." Mathematics 11, no. 7 (March 29, 2023): 1651. http://dx.doi.org/10.3390/math11071651.

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An extension of the general fractional calculus (GFC) is proposed as a generalization of the Riesz fractional calculus, which was suggested by Marsel Riesz in 1949. The proposed Riesz form of GFC can be considered as an extension GFC from the positive real line and the Laplace convolution to the m-dimensional Euclidean space and the Fourier convolution. To formulate the general fractional calculus in the Riesz form, the Luchko approach to construction of the GFC, which was suggested by Yuri Luchko in 2021, is used. The general fractional integrals and derivatives are defined as convolution-type operators. In these definitions the Fourier convolution on m-dimensional Euclidean space is used instead of the Laplace convolution on positive semi-axis. Some properties of these general fractional operators are described. The general fractional analogs of first and second fundamental theorems of fractional calculus are proved. The fractional calculus of the Riesz potential and the fractional Laplacian of the Riesz form are special cases of proposed general fractional calculus of the Riesz form.

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16

Peng, Gang, Zhanqing Chen, and Jiarui Chen. "Research on Rock Creep Characteristics Based on the Fractional Calculus Meshless Method." Advances in Civil Engineering 2018 (September 26, 2018): 1–6. http://dx.doi.org/10.1155/2018/1472840.

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The application of fractional calculus in the rheological problems has been widely accepted. In this study, the constitutive relationship of the generalized Kelvin model based on fractional calculus was studied, and the meshless method was introduced so as to derive a new meshless algorithm formula based on the fractional calculus of the generalized Kelvin model. By using the MTS815.02 hydraulic servo rock mechanics test system, the creep test of mudstones is carried out, and the related data of the creep process were obtained. Based on the generalized Kelvin model of fractional calculus, the related creep parameters of the argillaceous sandstone under compression were fitted. The results showed that the solution of the generalized Kelvin model based on fractional calculus was greatly consistent with the numerical method solution. Meanwhile, the meshless algorithm based on fractional calculus had a favorable stability and accuracy.

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17

Huang, Guo, Hong-ying Qin, Qingli Chen, Zhanzhan Shi, Shan Jiang, and Chenying Huang. "Research on Application of Fractional Calculus Operator in Image Underlying Processing." Fractal and Fractional 8, no. 1 (January 5, 2024): 37. http://dx.doi.org/10.3390/fractalfract8010037.

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Fractional calculus extends traditional, integer-based calculus to include non-integer orders, offering a powerful tool for a range of engineering applications, including image processing. This work delves into the utility of fractional calculus in two crucial aspects of image processing: image enhancement and denoising. We explore the foundational theories of fractional calculus together with its amplitude–frequency characteristics. Our focus is on the effectiveness of fractional differential operators in enhancing image features and reducing noise. Experimental results reveal that fractional calculus offers unique benefits for image enhancement and denoising. Specifically, fractional-order differential operators outperform their integer-order counterparts in accentuating details such as weak edges and strong textures in images. Moreover, fractional integral operators excel in denoising images, not only improving the signal-to-noise ratio but also better preserving essential features such as edges and textures when compared to traditional denoising techniques. Our empirical results affirm the effectiveness of the fractional-order calculus-based image-processing approach in yielding optimal results for low-level image processing.

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18

Magin, Richard, Xu Feng, and Dumitru Baleanu. "Fractional Calculus in NMR." IFAC Proceedings Volumes 41, no. 2 (2008): 9613–18. http://dx.doi.org/10.3182/20080706-5-kr-1001.01626.

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19

Anastassiou, George A. "Fuzzy fractional calculus revisited." New Trends in Mathematical Science 1 Proceeding, no. 9 (July 26, 2021): 173–90. http://dx.doi.org/10.20852/ntmsci.2021.447.

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20

MA, LI, and CHANGPIN LI. "ON HADAMARD FRACTIONAL CALCULUS." Fractals 25, no. 03 (May 11, 2017): 1750033. http://dx.doi.org/10.1142/s0218348x17500335.

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This paper is devoted to the investigation of the Hadamard fractional calculus in three aspects. First, we study the semigroup and reciprocal properties of the Hadamard-type fractional operators. Then, the definite conditions of certain class of Hadamard-type fractional differential equations (HTFDEs) are proposed through the Banach contraction mapping principle. Finally, we prove a novel Gronwall inequality with weak singularity and analyze the dependence of solutions of HTFDEs on the derivative order and the perturbation terms along with the proposed initial value conditions. The illustrative examples are presented as well.

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21

Banerji, P. K., and Deshna Loonker. "Marvels of fractional calculus." Tbilisi Mathematical Journal 10, no. 1 (January 2017): 295–314. http://dx.doi.org/10.1515/tmj-2017-0019.

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22

Eloe, Paul, and Jeffrey Neugebauer. "Concavity in fractional calculus." Filomat 32, no. 9 (2018): 3123–28. http://dx.doi.org/10.2298/fil1809123e.

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We discuss a concavity like property for functions u satisfying D?0+u ? C[0, b] with u(0) = 0 and -D?0+u(t) ? 0 for all t ? [0,b]. We develop the property for ? ? (1,2], where D?0+ is the standard Riemann-Liouville fractional derivative. We observe the property is also valid in the case ? = 1. Finally, we show that under certain conditions, -D?0+u(t) ? 0 implies u is concave in the classical sense.

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23

Abdeljawad, Thabet. "On conformable fractional calculus." Journal of Computational and Applied Mathematics 279 (May 2015): 57–66. http://dx.doi.org/10.1016/j.cam.2014.10.016.

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24

West, Bruce J. "Fractional Calculus in Bioengineering." Journal of Statistical Physics 126, no. 6 (February 13, 2007): 1285–86. http://dx.doi.org/10.1007/s10955-007-9294-0.

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25

Meilanov, R. P., and R. A. Magomedov. "Thermodynamics in Fractional Calculus." Journal of Engineering Physics and Thermophysics 87, no. 6 (November 2014): 1521–31. http://dx.doi.org/10.1007/s10891-014-1158-2.

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26

Anastassiou, George A. "On right fractional calculus." Chaos, Solitons & Fractals 42, no. 1 (October 2009): 365–76. http://dx.doi.org/10.1016/j.chaos.2008.12.013.

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27

Sopasakis, Pantelis, Haralambos Sarimveis, Panos Macheras, and Aristides Dokoumetzidis. "Fractional calculus in pharmacokinetics." Journal of Pharmacokinetics and Pharmacodynamics 45, no. 1 (October 3, 2017): 107–25. http://dx.doi.org/10.1007/s10928-017-9547-8.

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28

Raja, Valarmathi, and Arulprakash Gowrisankar. "On the Variable Order Fractional Calculus Characterization for the Hidden Variable Fractal Interpolation Function." Fractal and Fractional 7, no. 1 (December 28, 2022): 34. http://dx.doi.org/10.3390/fractalfract7010034.

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In this study, the variable order fractional calculus of the hidden variable fractal interpolation function is explored. It extends the constant order fractional calculus to the case of variable order. The Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus are investigated for hidden variable fractal interpolation function. Moreover, the conditions for the variable fractional order μ on a specified range are also derived. It is observed that, under certain conditions, the Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus of the hidden variable fractal interpolation function are again the hidden variable fractal interpolation functions interpolating the new data set.

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29

Rani, Noosheza, and Arran Fernandez. "An operational calculus formulation of fractional calculus with general analytic kernels." Electronic Research Archive 30, no. 12 (2022): 4238–55. http://dx.doi.org/10.3934/era.2022216.

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<abstract><p>Fractional calculus with analytic kernels provides a general setting of integral and derivative operators that can be connected to Riemann–Liouville fractional calculus via convergent infinite series. We interpret these operators from an algebraic viewpoint, using Mikusiński's operational calculus, and utilise this algebraic formalism to solve some fractional differential equations.</p></abstract>

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30

Failla, Giuseppe, and Massimiliano Zingales. "Advanced materials modelling via fractional calculus: challenges and perspectives." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2172 (May 11, 2020): 20200050. http://dx.doi.org/10.1098/rsta.2020.0050.

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Fractional calculus is now a well-established tool in engineering science, with very promising applications in materials modelling. Indeed, several studies have shown that fractional operators can successfully describe complex long-memory and multiscale phenomena in materials, which can hardly be captured by standard mathematical approaches as, for instance, classical differential calculus. Furthermore, fractional calculus has recently proved to be an excellent framework for modelling non-conventional fractal and non-local media, opening valuable prospects on future engineered materials. The theme issue gathers cutting-edge theoretical, computational and experimental studies on advanced materials modelling via fractional calculus, with a focus on complex phenomena and non-conventional media. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

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31

Lazorenko, O., and L. Chernogor. "FRACTAL RADIOPHYSICS. PART 3. FRACTIONAL CALCULUS IN ELECTRODYNAMICS." Radio physics and radio astronomy 29, no. 1 (2024): 046–67. http://dx.doi.org/10.15407/rpra29.01.046.

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Subject and Purpose. At the beginning of the 21st century, a fundamentally new scientific direction was formed, currently known as fractal radiophysics. The present work is an overview of the principal theoretical and practical ideas concerning "fractalization" in radio physics. The purpose is a systematic presentation of the main practical results suitable for application of the fractional calculus in modern theoretical radiophysics. Methods and Methodology. The basic theoretical principles of fractional calculus are outlined in a structured form. Results of applying fractional calculus methods in electrodynamics are systematized. Essential features, advantages and disadvantages of the technique are demonstrated and the problems still remaining discussed. Results. The basics of fractional (or fractal) calculus have been considered with emphasis on practical application to problems of radiophysics. A variety of approaches to constructing fractional integrals and Riemann–Liouville, etc. fractional derivatives have been presented. Using the Newton-Leibnitz formula and fundamental theorems of fractional calculus, principles of generalization of the classic vector calculus to fractal problems have been discussed, suggesting the examples of fractional vector-differential and vector-integral operators, Green’s and Stokes’ fractional formulas, etc. With the use of Gauss’s fractional formula the basics of fractal electrodynamics are expounded. Some different types of fractal Maxwellian equations has been induced and analyzed. Also, the main approaches to solving radio wave propagation problems in fractal media are discussed. Conclusions. As a practical example of applying fractals in modern theoretical radiophysics, results have been presented of the use of fractional calculus in electrodynamics. These results signify appearance of a fundamentally new direction in radiophysics, namely fractal electrodynamics.

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32

Lazopoulos, Konstantinos A., and Anastasios A. Lazopoulos. "Fractional Vector Calculus and Fractional Continuum Mechanics." Progress in Fractional Differentiation and Applications 2, no. 2 (April 1, 2016): 85–104. http://dx.doi.org/10.18576/pfda/020202.

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33

Mu’lla, Mariam Almadi Mohammed. "Fractional Calculus, Fractional Differential Equations and Applications." OALib 07, no. 06 (2020): 1–9. http://dx.doi.org/10.4236/oalib.1106244.

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34

Meerschaert, Mark M., Jeff Mortensen, and Stephen W. Wheatcraft. "Fractional vector calculus for fractional advection–dispersion." Physica A: Statistical Mechanics and its Applications 367 (July 2006): 181–90. http://dx.doi.org/10.1016/j.physa.2005.11.015.

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35

Schiavone, S. E., and W. Lamb. "A fractional power approach to fractional calculus." Journal of Mathematical Analysis and Applications 149, no. 2 (July 1990): 377–401. http://dx.doi.org/10.1016/0022-247x(90)90049-l.

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36

Tarasov, Vasily E. "Fractional vector calculus and fractional Maxwell’s equations." Annals of Physics 323, no. 11 (November 2008): 2756–78. http://dx.doi.org/10.1016/j.aop.2008.04.005.

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37

David, S. A., J. L. Linares, and E. M. J. A. Pallone. "Fractional order calculus: historical apologia, basic concepts and some applications." Revista Brasileira de Ensino de Física 33, no. 4 (December 2011): 4302. http://dx.doi.org/10.1590/s1806-11172011000400002.

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Fractional order calculus (FOC) deals with integrals and derivatives of arbitrary (i.e., non-integer) order, and shares its origins with classical integral and differential calculus. However, until recently, it has been investigated mainly from a mathematical point of view. Advances in the field of fractals have revealed its subtle relationships with fractional calculus. Nonetheless, fractional calculus is generally excluded from standard courses in mathematics, partly because many mathematicians are unfamiliar with its nature and its applications. This area has emerged as a useful tool among researchers. One of the objectives of this paper is to discuss the usefulness of fractional calculus in applied sciences and engineering. In view of the increasing interest in the development of the new paradigm, another objective is to encourage the use of this mathematical idea in various scientific areas by means of a historical apologia for the development of fractional calculus.

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38

Wei, Yiheng, Linlin Zhao, Xuan Zhao, and Jinde Cao. "Enhancing the Mathematical Theory of Nabla Tempered Fractional Calculus: Several Useful Equations." Fractal and Fractional 7, no. 4 (April 14, 2023): 330. http://dx.doi.org/10.3390/fractalfract7040330.

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Although many applications of fractional calculus have been reported in literature, modeling the physical world using this technique is still a challenge. One of the main difficulties in solving this problem is that the long memory property is necessary, whereas the infinite memory is undesirable. To address this challenge, a new type of nabla fractional calculus with a weight function is formulated, which combines the benefits of nabla fractional calculus and its tempered counterpart, making it highly valuable for modeling practical systems. However, many properties of this calculus are still unclear and need to be discovered. Therefore, this paper gives particular emphasis to the topic, developing some remarkable properties, i.e., the equivalence relation, the nabla Taylor formula, and the nabla Laplace transform of such nabla tempered fractional calculus. All the developed properties greatly enrich the mathematical theory of nabla tempered fractional calculus and provide high value and potential for further applications.

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39

Sikora, Ryszard, and Stanislaw Pawłowski. "Fractional derivatives and the laws of electrical engineering." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 37, no. 4 (July 2, 2018): 1384–91. http://dx.doi.org/10.1108/compel-08-2017-0347.

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Purpose This paper aims to evaluate the possibilities of fractional calculus application in electrical circuits and magnetic field theories. Design/methodology/approach The analysis of mathematical notation is used for physical phenomena description. The analysis aims to challenge or prove the correctness of applied notation. Findings Fractional calculus is sometimes applied correctly and sometimes erroneously in electrical engineering. Originality/value This paper provides guidelines regarding correct application of fractional calculus in description of electrical circuits’ phenomena. It can also inspire researchers to find new applications for fractional calculus in the future.

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40

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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41

KA, Lazopoulos, and Lazopoulos AK. "On Λ-fractional variational calculus." Annals of Mathematics and Physics 6, no. 1 (March 1, 2023): 036–40. http://dx.doi.org/10.17352/amp.000074.

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Pointing out that Λ-fractional analysis is the unique fractional calculus theory including mathematically acceptable fractional derivatives, variational calculus for Λ-fractional analysis is established. Since Λ-fractional analysis is a non-local procedure, global extremals are only accepted. That means the extremals should satisfy not only the Euler–Lagrange equation but also the additional Weierstrass-Erdmann corner conditions. Hence non-local stability criteria are introduced. The proposed variational procedure is applied to any branch of physics, mechanics, biomechanics, etc. The present analysis is applied to the Λ-fractional refraction of light.

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42

Bas, Erdal, Resat Yilmazer, and Etibar Panakhov. "Fractional Solutions of Bessel Equation with -Method." Scientific World Journal 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/685695.

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This paper deals with the design fractional solution of Bessel equation. We obtain explicit solutions of the equation with the help of fractional calculus techniques. Using the -fractional calculus operator method, we derive the fractional solutions of the equation.

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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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Fang, C.-Q., H. Y. Sun, and J. P. Gu. "Application of Fractional Calculus Methods to Viscoelastic Response of Amorphous Shape Memory Polymers." Journal of Mechanics 31, no. 4 (August 2015): 427–32. http://dx.doi.org/10.1017/jmech.2014.98.

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AbstractConstitutive models based on fractional calculus are utilized to investigate the viscoelastic response of thermally activated shape memory polymers (SMPs). Fractional calculus-based viscoelastic equations are fitted to experimental data existing in literature compared with traditional viscoelastic models. In addition, a fractional rheology model is applied to simulate the isothermal recovery of an amorphous SMP. The fit results show a significant improvement in the description of the strain recovery response of SMP by the fractional calculus method.

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Liang, Guishu, and Yulan Yang. "Fractional calculus-based analysis of soil electrical properties." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 39, no. 2 (November 28, 2019): 279–95. http://dx.doi.org/10.1108/compel-05-2019-0179.

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Purpose This paper aims to analyze soil electrical properties based on fractional calculus theory due to the fact that the frequency dependence of soil electrical parameters at high frequencies exhibits a fractional effect. In addition, for the fractional-order formulation, this paper aims to provide a more accurate numerical algorithm for solving the fractional differential equations. Design/methodology/approach This paper analyzes the frequency-dependence of soil electrical properties based on fractional calculus theory. A collocation method based on the Puiseux series is proposed to solve fractional differential equations. Findings The algorithm proposed in this paper can be used to solve fractional differential equations of arbitrary order, especially for 0.5th-order equations, obtaining accurate numerical solutions. Calculating the impact response of the grounding electrode based on the fractional calculus theory can obtain a more accurate result. Originality/value This paper proposes an algorithm for solving fractional differential equations of arbitrary order, especially for 0.5th-order equations. Using fractional calculus theory to analyze the frequency-dependent effect of soil electrical properties, provides a new idea for ground-related transient calculation.

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Alarifi, Najla M., and Rabha W. Ibrahim. "Specific Classes of Analytic Functions Communicated with a Q-Differential Operator Including a Generalized Hypergeometic Function." Fractal and Fractional 6, no. 10 (September 27, 2022): 545. http://dx.doi.org/10.3390/fractalfract6100545.

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A special function is a function that is typically entitled after an early scientist who studied its features and has a specific application in mathematical physics or another area of mathematics. There are a few significant examples, including the hypergeometric function and its unique species. These types of special functions are generalized by fractional calculus, fractal, q-calculus, (q,p)-calculus and k-calculus. By engaging the notion of q-fractional calculus (QFC), we investigate the geometric properties of the generalized Prabhakar fractional differential operator in the open unit disk ∇:={ξ∈C:|ξ|<1}. Consequently, we insert the generalized operator in a special class of analytic functions. Our methodology is indicated by the usage of differential subordination and superordination theory. Accordingly, numerous fractional differential inequalities are organized. Additionally, as an application, we study the solution of special kinds of q–fractional differential equation.

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Persechino, A. "An introduction to fractional calculus." Advanced Electromagnetics 9, no. 1 (February 19, 2020): 19–30. http://dx.doi.org/10.7716/aem.v9i1.1192.

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The aim of this work is to introduce the main concepts of Fractional Calculus, followed by one of its application to classical electrodynamics, illustrating how non-locality can be interpreted naturally in a fractional scenario. In particular, a result relating fractional dynamics to high frequency dielectric response is used as motivation. In addition to the theoretical discussion, a comprehensive review of two numerical procedures for fractional integration is carried out, allowing one immediately to build numerical models applied to high frequency electromagnetics and correlated fields.

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Pu, Yi-Fei, Ji-Liu Zhou, Patrick Siarry, Ni Zhang, and Yi-Guang Liu. "Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach-Based Multiscale Denoising Model for Texture Image." Abstract and Applied Analysis 2013 (2013): 1–19. http://dx.doi.org/10.1155/2013/483791.

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The traditional integer-order partial differential equation-based image denoising approaches often blur the edge and complex texture detail; thus, their denoising effects for texture image are not very good. To solve the problem, a fractional partial differential equation-based denoising model for texture image is proposed, which applies a novel mathematical method—fractional calculus to image processing from the view of system evolution. We know from previous studies that fractional-order calculus has some unique properties comparing to integer-order differential calculus that it can nonlinearly enhance complex texture detail during the digital image processing. The goal of the proposed model is to overcome the problems mentioned above by using the properties of fractional differential calculus. It extended traditional integer-order equation to a fractional order and proposed the fractional Green’s formula and the fractional Euler-Lagrange formula for two-dimensional image processing, and then a fractional partial differential equation based denoising model was proposed. The experimental results prove that the abilities of the proposed denoising model to preserve the high-frequency edge and complex texture information are obviously superior to those of traditional integral based algorithms, especially for texture detail rich images.

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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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Wei, Zhang, Huang Fan, and Nobuyuki Shimizu. "58075 CHAOS OF NONLINEAR FRACTIONAL-CALCULUS OSCILLATOR(Miscellaneous Applications)." Proceedings of the Asian Conference on Multibody Dynamics 2010.5 (2010): _58075–1_—_58075–6_. http://dx.doi.org/10.1299/jsmeacmd.2010.5._58075-1_.

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