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Journal articles on the topic 'Fractional Grunwald-Letnikov derivatives'

Author: Grafiati

Published: 5 June 2025

Last updated: 15 July 2025

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1

Khurshaid*, Adil, and Hajra Khurshaid. "Comparative Analysis and Definitions of Fractional Derivatives." Journal of Biomedical Research & Environmental Sciences 4, no. 12 (2023): 1684–88. http://dx.doi.org/10.37871/jbres1852.

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Fractional Calculus (FC) has emerged as a valuable tool in various fields. This study explores the historical development of (FC) and examines prominent definitions regarding Fractional Derivatives (FD), such as the Riemann-Liouville, Grunwald-Letnikov, Caputo Fractional Derivative, Katugampula derivatives, Caputo Fractional Derivative, Caputo-Fabrizio Fractional Derivative and as well as Atangana-Baleanu Fractional Derivative. It critically evaluates their strengths, weaknesses and implications on (FD) equations. The findings contribute to establishing a clearer understanding of Fractional De

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2

Menaria, Anil Kumar. "Existence and uniqueness of linear and non-linear fractional differential equations." Journal of Interdisciplinary Mathematics 27, no. 8 (2024): 1909–15. https://doi.org/10.47974/jim-2058.

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The existence and uniqueness of solutions for linear and nonlinear fractional differential equations (FDE) involving Miller-Ross sequential fractional derivatives (SFD) can be established through mathematical analysis and specific conditions, which allows direct applications with Riemann-Liouville fractional derivatives, the Caputo fractional derivatives and the Grunwald Letnikov derivatives, which is a particular case of the Miller-Ross SFD.

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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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4

Avci, Derya, Eroglu Iskender, and Necati Ozdemir. "Conformable heat equation on a radial symmetric plate." Thermal Science 21, no. 2 (2017): 819–26. http://dx.doi.org/10.2298/tsci160427302a.

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The conformable heat equation is defined in terms of a local and limit-based definition called conformable derivative which provides some basic properties of integer order derivative such that conventional fractional derivatives lose some of them due to their non-local structures. In this paper, we aim to find the fundamental solution of a conformable heat equation acting on a radial symmetric plate. Moreover, we give a comparison between the new conformable and the existing Grunwald-Letnikov solutions of heat equation. The computational results show that conformable formulation is quite succe

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Ciaschetti, Valentina, Isaac Elishakoff, and Alessandro Marzani. "Vibrations of fractional half- and single-degree of freedom systems." Vietnam Journal of Mechanics 39, no. 3 (2017): 259–73. http://dx.doi.org/10.15625/0866-7136/9772.

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In this paper we study vibrations of fractional oscillators by two methods: the triangular strip matrix approach, based on the Grunwald-Letnikov discretization of the fractional term, and the state variable analysis, which is suitable for systems with fractional derivatives of rational order. Some examples are solved in order to compare the two approaches and to conduct comparison with benchmark problems.

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6

Lopuh, Nazariy. "Numerical model of mass transfer processes using fractional derivatives." Physico-mathematical modelling and informational technologies, no. 28, 29 (December 27, 2019): 26–32. http://dx.doi.org/10.15407/fmmit2020.28.026.

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The article describes the scheme of construction and application of finite element method using Grunwald-Letnikov algorithm. Obtained results make it possible to estimate influence of fractional derivative order in terms of time and space on process of gas filtration in porous medium. Numerical ecperification and analysis performed

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7

Zhang, Xiang Mei, An Ping Xu, and Xian Zhou Guo. "Stability Analysis of Fractional Delay Differential Equations by Lagrange Polynomial." Advanced Materials Research 500 (April 2012): 591–95. http://dx.doi.org/10.4028/www.scientific.net/amr.500.591.

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The paper deals with the numerical stability analysis of fractional delay differential equations with non-smooth coefficients using the Lagrange collocation method. In this paper, based on the Grunwald-Letnikov fractional derivatives, we discuss the approximation of fractional differentiation by the Lagrange polynomial. Then we study the numerical stability of the fractional delay differential equations. Finally, the stability of the delayed Mathieu equation of fractional order is studied and examined by Lagrange collocation method.

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8

Parmikanti, Kankan, and Endang Rusyaman. "Grundwald-Letnikov Operator and Its Role in Solving Fractional Differential Equations." EKSAKTA: Berkala Ilmiah Bidang MIPA 23, no. 03 (2022): 223–30. http://dx.doi.org/10.24036/eksakta/vol23-iss03/331.

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Leibnitz in 1663 introduced the derivative notation for the order of natural numbers, and then the idea of fractional derivatives appeared. Only a century later, this idea began to be realized with the discovery of the concepts of fractional derivatives by several mathematicians, including Riemann (1832), Grundwal, Fourier, and Caputo in 1969. The concepts in the definitions of fractional derivatives by Riemann-Liouville and Caputo are more frequently used than other definitions, this paper will discuss the Grunwald-Letnikov (GL) operator, which has been discovered in 1867. This concept is les

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9

Bereziuk, Volodymyr, and Yaroslav Sokolovskyi. "ENHANCEMENT OF MEDICAL MRI IMAGES BASED ON FRACTAL OPERATORS." Computer Design Systems. Theory and Practice 6, no. 2 (2024): 130–45. http://dx.doi.org/10.23939/cds2024.02.130.

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This article presents the research of texture enhancement algorithms on medical images. Medical MRI brain scans contain large areas with low level grey colors that carry useful information for doctors. Texture improvement allow to highlight large grey areas on images for future detailed recognition. Based on the study of existing texture enhancement methods, it was determined that fractal operators are effective for processing medical images. The mathematical framework of fractal operators is presented based on the approximation equation of the Grünwald-Letnikov fractional derivatives. The cre

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10

Zhang, Xiang Mei, Xian Zhou Guo, and Anping Xu. "Stability Analysis of Fractional Delay Differential Equations by Chebyshev Polynomial." Advanced Materials Research 500 (April 2012): 586–90. http://dx.doi.org/10.4028/www.scientific.net/amr.500.586.

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The paper is devoted to the numerical stability of fractional delay differential equations with non-smooth coefficients using the Chebyshev collocation method. In this paper, based on the Grunwald-Letnikov fractional derivatives, we discuss the approximation of fractional differentiation by the Chebyshev polynomial of the first kind. Then we solve the stability of the fractional delay differential equations. Finally, the stability of the delayed Mathieu equation of fractional order is examined for a set of case studies that contain the complexities of periodic coefficients, delays and disconti

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11

Mamatov, Mashrabzhan, Jalolkon Nuritdinov, and Egamberdi Esonov. "DIFFERENTIAL GAMES OF FRACTIONAL ORDER WITH DISTRIBUTED PARAMETERS." Journal of Automation and Information sciences 4 (July 1, 2021): 38–47. http://dx.doi.org/10.34229/1028-0979-2021-4-4.

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The article deals with the problem of pursuit in differential games of fractional order with distributed parameters. Partial fractional derivatives with respect to time and space variables are understood in the sense of Riemann - Liouville, and the Grunwald-Letnikov formula is used in the approximation. The problem of getting into some positive neighborhood of the terminal set is considered. To solve this problem, the finite difference method is used. The fractional Riemann-Liouville derivatives with respect to spatial variables on a segment are approximated using the Grunwald-Letnikov formula

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12

Baloochian, Hossein, Hamid Reza Ghaffary, and Saeed Balochian. "Enhancing Fingerprint Image Recognition Algorithm Using Fractional Derivative Filters." Open Computer Science 7, no. 1 (2017): 9–16. http://dx.doi.org/10.1515/comp-2017-0002.

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Abstract One of the most important steps in recognizing fingerprint is accurate feature extraction of the input image. To enhance the accuracy of fingerprint recognition, an algorithm using fractional derivatives is proposed in this paper. The proposed algorithm uses the definitions of fractional derivatives Riemann-Liouville (R-L) and Grunwald-Letnikov (G-L) in two sections of direction estimation and image enhancement for the first time. Based on it, new mask of fractional derivative Gabor filter is calculated. The proposed fractional derivative-based method enhances the image quality. This

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13

Pyanylo, Ya, M. Prytula, N. Prytula, and N. Lopuh. "Models of mass transfer in gas transmission systems." Mathematical Modeling and Computing 1, no. 1 (2014): 84–96. http://dx.doi.org/10.23939/mmc2014.01.084.

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The models of gas movement in pipelines and gas filtration processes in complex porous media are considered in entire and fractional derivatives. The method for linearization of equations, which are included in the mathematical model of mass transfer, is suggested as well as an iterative scheme for solving initial systems of nonlinear differential equations is constructed. The finite-element model of the problem with the use of the Petrov-Galerkin method and Grunwald-Letnikov scheme concerning derivatives of the fractional order are implemented. The research of the models is carried out as wel

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14

Prokhorov, Stanislav, and Anastasiya Ten. "Computer Simulations of the Heat Transfer Process in Heterogeneous Media Using Parallel Computing Technologies." Mathematical Physics and Computer Simulation, no. 3 (October 2023): 26–36. http://dx.doi.org/10.15688/mpcm.jvolsu.2023.3.3.

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This paper studied a two-dimensional linear equation of thermal conductivity with a fractional differentiation order and compiled a computational algorithm for its numerical solution. Fractional derivatives in space and time were represented using the Riemann—Liouville definition, and the Grunwald— Letnikov definition with a shift was used to approximate them. Based on the computational algorithm, a program was written for computer modeling of the heat exchange process in heterogeneous environments. A version of the program has been developed using OpenMP parallel computing technology. The com

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15

Ibrahim, M., and V. G. Pimenov. "Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay." Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta 57 (May 2021): 128–41. http://dx.doi.org/10.35634/2226-3594-2021-57-05.

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A two-dimensional in space fractional diffusion equation with functional delay of a general form is considered. For this problem, the Crank-Nicolson method is constructed, based on shifted Grunwald-Letnikov formulas for approximating fractional derivatives with respect to each spatial variable and using piecewise linear interpolation of discrete history with continuation extrapolation to take into account the delay effect. The Douglas scheme is used to reduce the emerging high-dimensional system to tridiagonal systems. The residual of the method is investigated. To obtain the order of the meth

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16

Nisardi, Muhammad Rifki, Kasbawati Kasbawati, and Khaeruddin Khaeruddin. "A Mathematical Model Analysis of COVID-19 Transmission with Vaccination in Caputo Fractional Derivatives: Case Study in Indonesia." JTAM (Jurnal Teori dan Aplikasi Matematika) 8, no. 4 (2024): 1183. http://dx.doi.org/10.31764/jtam.v8i4.24711.

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This study aims to investigate a fractional-order mathematical model of COVID-19 transmission using the Caputo derivative definition which suitable to epidemiological cases by its advantage to explain memory effects. The model incorporates compartments for asymptomatic infections and includes a vaccination strategy aimed at mitigating the spread of COVID-19. We derived the disease-free and endemic equilibrium points for the fractional model and computed the basic reproduction number (R_0 ) using the Next-generation Matrix method. Additionally, we conducted sensitivity analyses of parameters af

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17

Sweilam, Nasser, S. M. Al−Mekhlafi, Aya Ahmed, Doaa Mohamed, and Emad Abo-Eldahab. "Impact of Singular and Non-Singular Kernels on Crossover Monkeypox Mathematical Model." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 5913. https://doi.org/10.29020/nybg.ejpam.v18i2.5913.

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This study presents three crossover models describing monkeypox disease that includes Caputo, MittagLeffler, and Caputo Fabrizio definitions. To represent a monkeypox disease, three models of variable-order fractional, fractal-fractional, and stochastic, as well as their piecewise derivatives are provided at three different time periods. To approximate these models, we use the nonstandard Grunwald ¨ −Letnikov finite difference method to approximate the deterministic model with a singular kernel and a nonsingular Mittag-Leffler kernel to approximate the deterministic model using the Toufik-Atan

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18

Moroz, L. I., and A. G. Maslovskaya. "Fractional-Differential Model of Heat Conductivity Process in Ferroelectrics under the Intensive Heating Conditions." Mathematics and Mathematical Modeling, no. 2 (July 20, 2019): 29–47. http://dx.doi.org/10.24108/mathm.0219.0000185.

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Ferroelectrics, due a number of characteristics, behave as hereditary materials with fractal structure. To model mathematically the systems with so-called memory effects one can use the fractional time-derivatives. The pyro-electric properties of ferroelectrics arouse interest in developing the fractional-differential approach to simulating heat conductivity process.The present study deals with development and numerical implementation of fractal heat conductivity model for hereditary materials using the concepts of fractional-differential calculus applied to the simulation of intensive heating

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19

Sokolovskyy, Yaroslav, Volodymyr Yarkun, Maryana Levkovych, and Dmytro Ratynchuk. "Software and algorithmic provision of parallel calculation of non-isothermal moisture transfer based on the apparatus of fractional derivatives." Computer Design Systems. Theory and Practice 4, no. 1 (2022): 95–106. http://dx.doi.org/10.23939/cds2022.01.095.

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A new mathematical model of the nonstationary process of heat and moisture transfer in the two-dimensional region is constructed on the basis of the use of Caputo and Grunwald- Letnikov derivatives. An implicit finite-difference scheme for approximation of a mathematical model of noisothermal moisture transfer taking into account the fractional integro-differential apparatus is developed. The given algorithm of numerical realization of model allows to receive values of function of temperature and humidity for all points of area of partition. The method of fractional steps is adapted for numeri

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20

Tarasov, Vasily. "On History of Mathematical Economics: Application of Fractional Calculus." Mathematics 7, no. 6 (2019): 509. http://dx.doi.org/10.3390/math7060509.

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Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At the present moment the new revolution, which can be called “Memory revolution”, is actually taking place in modern economics. This revolution is intended to “cure amnesia” of modern economic theory, which is caused by the use of differential and integral operators of integer orders. In economics, the des

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21

Sene, Ndolane. "Fractional Model for a Class of Diffusion-Reaction Equation Represented by the Fractional-Order Derivative." Fractal and Fractional 4, no. 2 (2020): 15. http://dx.doi.org/10.3390/fractalfract4020015.

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This paper proposes the analytical solution for a class of the fractional diffusion equation represented by the fractional-order derivative. We mainly use the Grunwald–Letnikov derivative in this paper. We are particularly interested in the application of the Laplace transform proposed for this fractional operator. We offer the analytical solution of the fractional model as the diffusion equation with a reaction term expressed by the Grunwald–Letnikov derivative by using a double integration method. To illustrate our findings in this paper, we represent the analytical solutions for different v

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22

STANKUNAS, GEDIMINAS. "FRACTAL MODEL OF FISSION PRODUCT RELEASE IN NUCLEAR FUEL." International Journal of Modern Physics C 23, no. 09 (2012): 1250057. http://dx.doi.org/10.1142/s012918311250057x.

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A model of fission gas migration in nuclear fuel pellet is proposed. Diffusion process of fission gas in granular structure of nuclear fuel with presence of inter-granular bubbles in the fuel matrix is simulated by fractional diffusion model. The Grunwald–Letnikov derivative parameter characterizes the influence of porous fuel matrix on the diffusion process of fission gas. A finite-difference method for solving fractional diffusion equations is considered. Numerical solution of diffusion equation shows correlation of fission gas release and Grunwald–Letnikov derivative parameter. Calculated p

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23

Mukherjee, Deep, Palash Kumar Kundu, and Apurba Ghosh. "A performance analysis of fractional order based MARC controller over optimal fractional order PID controller on inverted pendulum." International Journal of Engineering & Technology 7, no. 2.21 (2018): 29. http://dx.doi.org/10.14419/ijet.v7i2.21.11830.

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This paper presents a new way to design MIT rule as an advanced technique of MARC (Model Adaptive Reference Controller) for an integer order inverted pendulum system. Here, our work aims to study the performance characteristics of fractional order MIT rule of MARC controller followed by optimal fractional order PID controller in MATLAB SIMULINK environment with respect to time domain specifications. Here, to design fractional order MIT rule Grunwald-Letnikov fractional derivative calculus method has been considered and based on Grunwald-Letnikov fractional calculus rule fractional MIT rule has

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24

Gao, Feng, and Chunmei Chi. "Pade Method for Construction of Numerical Algorithms for Fractional Initial Value Problem." Journal of Mathematics 2020 (July 8, 2020): 1–7. http://dx.doi.org/10.1155/2020/8964759.

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In this paper, we propose an efficient method for constructing numerical algorithms for solving the fractional initial value problem by using the Pade approximation of fractional derivative operators. We regard the Grunwald–Letnikov fractional derivative as a kind of Taylor series and get the approximation equation of the Taylor series by Pade approximation. Based on the approximation equation, we construct the corresponding numerical algorithms for the fractional initial value problem. Finally, we use some examples to illustrate the applicability and efficiency of the proposed technique.

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25

Nasir, Haniffa Mohamed, and Khadija Al Hasani. "Analysis of Fractional Linear Multi-Step Methods of Order Four from Super-Convergence." Sultan Qaboos University Journal for Science [SQUJS] 28, no. 2 (2023): 44–55. http://dx.doi.org/10.53539/squjs.vol28iss2pp44-55.

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We analyze two implicit fractional linear multi-step methods of order four for solving fractional initial value problems. The methods are derived from the Grunwald-Letnikov approximation of fractional derivative at a non-integer shift point with super-convergence. The weight coefficients of the methods are computed from fundamental G unwald weights, making them computationally efficient when compared with other known methods of order four. We also show that the stability regions are larger than that of the fractional Adams-Moulton and fractional backward difference formula methods. We present

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26

Kim, Valentine, and Roman Parovik. "Mathematical Model of Fractional Duffing Oscillator with Variable Memory." Mathematics 8, no. 11 (2020): 2063. http://dx.doi.org/10.3390/math8112063.

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The article investigates a mathematical model of the Duffing oscillator with a variable fractional order derivative of the Riemann–Liouville type. The study of the model is carried out using a numerical scheme based on the approximation of the fractional derivative of the Riemann–Liouville type by a discrete analog—the fractional derivative of Grunwald–Letnikov. The adequacy of the numerical scheme is verified using specific examples. Using a numerical algorithm, oscillograms and phase trajectories are constructed depending on the values of the model parameters. Chaotic regimes of the Duffing

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27

Pavlenko, Ivan, Marek Ochowiak, Praveen Agarwal, Radosław Olszewski, Bernard Michałek, and Andżelika Krupińska. "Improvement of Mathematical Model for Sedimentation Process." Energies 14, no. 15 (2021): 4561. http://dx.doi.org/10.3390/en14154561.

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In this article, the fractional-order differential equation of particle sedimentation was obtained. It considers the Basset force’s fractional origin and contains the Riemann–Liouville fractional integral rewritten as a Grunwald–Letnikov derivative. As a result, the general solution of the proposed fractional-order differential equation was found analytically. The belonging of this solution to the real range of values was strictly theoretically proven. The obtained solution was validated on a particular analytical case study. In addition, it was proven numerically with the approach based on th

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28

Pratima Devi, Rajkumari, and Md Indraman Khan. "Comparative Analysis of Fractional Derivative Operators: Stability, Asymptotic Behavior, and Computational Challenges." International Journal of Research in Science and Technology 14, no. 2 (2024): 13–23. http://dx.doi.org/10.37648/ijrst.v14i02.002.

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This research paper explores novel theorems related to fractional differential operators, including Grunwald-Letnikov, Riemann-Liouville, Caputo, and Weyl. Each operator is rigorously defined, and their mathematical properties are investigated. The paper presents a detailed analysis of the asymptotic behavior of solutions to fractional differential equations governed by these operators. The advantages and disadvantages of each operator in capturing non-local behaviors, power-law decay, and handling initial conditions are discussed. Special emphasis is given to the stability characteristics of

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29

Boutiba, Malika, Selma Baghli-Bendimerad, and Abbès Benaïssa. "Three Approximations of Numerical Solution's by Finite Element Method for Resolving Space-Time Partial Differential Equations Involving Fractional Derivative's Order." Mathematical Modelling of Engineering Problems 9, no. 5 (2022): 1179–86. http://dx.doi.org/10.18280/mmep.090503.

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In this paper, we apply to a class of partial differential equation the finite element method when the problem is involving the Riemann-Liouville fractional derivative for time and space variables on a bounded domain with bounded conditions. The studied equation is obtained from the standard time diffusion equation by replacing the first order time derivative by  for 0<<1 and for the second standard order space derivative by  for 1<<2 respectively. The existence of the unique solution is proved by the Lax-Milgram Lemma. We present here three schemes to approximate numerically t

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30

Issakhov, A. A., A. B. Abylkassymova, R. E. Zhailybaev, and S. L. Yun. "STABILITY CONDITION OF FINITE DIFFERENCE SCHEMES FOR PARABOLIC AND HYPERBOLIC EQUATIONS: A COMPARISON WITH FINITE VOLUME METHODS FOR FRACTIONAL-ORDER DIFFUSION." Herald of the Kazakh-British technical university 22, no. 1 (2025): 184–96. https://doi.org/10.55452/1998-6688-2025-22-1-184-196.

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This paper compares the finite difference and finite volume methods for solving time-fractional diffusion equations. These methods are widely known for diffusion equations with integer order, but their effectiveness for time-fractional diffusion equations has not been sufficiently studied. The definition of the Grunwald-Letnikov fractional derivative is used to approximate the equation. An explicit difference scheme for the finite difference method is obtained and a stability condition for the fractional time order difference scheme is derived, which is also a generalisation for parabolic and

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31

El Bhih, A., Y. Benfatah, A. Ghazaoui, and M. Rachik. "On the maximal output set of fractional-order discrete-time linear systems." Mathematical Modeling and Computing 9, no. 2 (2022): 262–77. http://dx.doi.org/10.23939/mmc2022.02.262.

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In this paper, we consider a linear discrete-time fractional-order system defined by \[\Delta ^{\alpha }x_ {k+1}=Ax_k+B u_k, \quad k \geq 0, \quad x_{0} \in \mathbb{R}^{n};\] \[y_{k}=Cx_k, \quad k \geq 0,\] where $A$, $B$ and $C$ are appropriate matrices, $x_{0}$ is the initial state, $\alpha$ is the order of the derivative, $y_k$ is the signal output and $u_k=K x_k$ is feedback control. By defining the fractional derivative in the Grunwald–Letnikov sense, we investigate the characterization of the maximal output set, $\Gamma(\Omega)=\lbrace x_{0} \in \mathbb{R}^{n}/y_{i} \in \Omega,\forall i

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Issakhov, A. A., A. B. Abylkassymova, S. Yun, and R. Zhailybaev. "OPTIMIZATION ALGORITHM FOR NUMERICAL IMPLEMENTATION OF THE FRACTIONAL GRUNWALD-LETNIKOV DERIVATIVE BASED ON THE MEMORIZATION PRINCIPLE FOR ORDINARY DIFFERENTIAL EQUATIONS." Herald of the Kazakh-British Technical University 22, no. 2 (2025): 242–59. https://doi.org/10.55452/1998-6688-2025-22-2-242-259.

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Matuszak, Zenon. "Fractional Bloch's Equations Approach to Magnetic Relaxation." Current Topics in Biophysics 37, no. 1 (2015): 9–22. http://dx.doi.org/10.2478/ctb-2014-0069.

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It is the goal of this paper to present general strategy for using fractional operators to model the magnetic relaxation in complex environments revealing time and spacial disorder. Such systems have anomalous temporal and spacial response (non-local interactions and long memory) compared to systems without disorder. The systems having no memory can be modeled by linear differential equations with constant coefficients (exponential relaxation); the differential equations governing the systems with memory are known as Fractional Order Differential Equations (FODE). The relaxation of the spin sy

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34

Effati, Sohrab, Seyed Ali Rakhshan, and Samane Saqi. "Formulation of Euler–Lagrange Equations for Multidelay Fractional Optimal Control Problems." Journal of Computational and Nonlinear Dynamics 13, no. 6 (2018). http://dx.doi.org/10.1115/1.4039900.

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In this paper, a new numerical scheme is proposed for multidelay fractional order optimal control problems where its derivative is considered in the Grunwald–Letnikov sense. We develop generalized Euler–Lagrange equations that results from multidelay fractional optimal control problems (FOCP) with final terminal. These equations are created by using the calculus of variations and the formula for fractional integration by parts. The derived equations are then reduced into system of algebraic equations by using a Grunwald–Letnikov approximation for the fractional derivatives. Finally, for confir

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Palsaniya, Vandana, Ekta Mittal, Sunil Joshi, and D. L. Suthar. "Gronwall type inequality on generalized fractional conformable integral operators." Analysis, November 30, 2023. http://dx.doi.org/10.1515/anly-2022-1105.

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Abstract In 2015, Abdeljawad defined the conformable fractional derivative (Grunwald–Letnikov technique) to iterate the conformable fractional integral of order 0 < α ≤ 1 {0<\alpha\leq{1}} (Riemann approach), yielding Hadamard fractional integrals when α = 0 {\alpha=0} . The Gronwall type inequality for generalized operators unifying Riemann–Liouville and Hadamard fractional operators is obtained in this study. We use this inequality to show how the order and initial conditions affect the solution of differential equations with generalized fractional derivatives. More features for genera

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Bueno, Atila Madureira, Daniel Celso Daltin, Paulo J. A. Serni, Jose Manoel Balthazar, and Angelo Marcelo Tusset. "Sub-Optimal State Tracking Control Applied to a Nonlinear Fractional-Order Slewing Motion Flexible Structure." Journal of Computational and Nonlinear Dynamics, May 17, 2022. http://dx.doi.org/10.1115/1.4054570.

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Abstract The control of slewing motion flexible structures is important to a number of systems found in Engineering and Physical Sciences applications, such as Aerospace, Automotive, Robotics, Atomic Force Microscopy and many others. In this kind of system the controller must provide a stable and well damped behavior for the flexible structure vibrations, with admissible control signal amplitudes. Recently, many works have used fractional-order derivatives to model complex and nonlinear dynamical behavior present in the mentioned systems. In order to perform digital computer based control of f

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-, Konthoujam Ibochouba Singh, Md Indraman Khan -, and Prof Irom Tomba Singh -. "Methods of Fractional Differential Equations in Algebra and Calculus." International Journal on Science and Technology 16, no. 2 (2025). https://doi.org/10.71097/ijsat.v16.i2.6558.

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This paper introduces a novel approach, the Inverse Fractional Shehu Transform Method, for solving both homogeneous and non-homogeneous linear fractional differential equations. The fractional derivatives are considered in the Riemann-Liouville and Caputo senses. By applying the Laplace transform and the convolution product to the Riemann-Liouville fractional of matrices, we obtain accurate solutions for systems of matrix fractional differential equations. The method’s effectiveness is demonstrated through examples, and its accuracy is verified by comparing the results with existing solutions

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Saeed, Muhammad Mudassir, Malik Sajjad Mehmood, and Muddassar Muhammad. "Spectroscopic analysis of residual radicals of gamma sterilized UHMWPE with fractional order differential operators." Revista Mexicana de Física 70, no. 1 Jan-Feb (2024). http://dx.doi.org/10.31349/revmexfis.70.011001.

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The purpose of this research was to evaluate the Grunwald-Letnikov definition of fractional order derivatives for determining the fractional derivative of ESR signals from UHMWPE free radicals using the fitted Gaussian distribution method. Specifically, the study focused on analyzing two long-lasting oxygen-induced residual radicals di- or tri-enyls with a carbon center radical (R1) and the oxygen-containing radical (R2). The impact of the derivative order on ESR spectral parameters, such as the Landé g-factor and peak-to-peak separation, was analyzed, and new spectral parameters were establis

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Barrera, O. "A unified modelling and simulation for coupled anomalous transport in porous media and its finite element implementation." Computational Mechanics, August 5, 2021. http://dx.doi.org/10.1007/s00466-021-02067-5.

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AbstractThis paper presents an unified mathematical and computational framework for mechanics-coupled “anomalous” transport phenomena in porous media. The anomalous diffusion is mainly due to variable fluid flow rates caused by spatially and temporally varying permeability. This type of behaviour is described by a fractional pore pressure diffusion equation. The diffusion transient phenomena is significantly affected by the order of the fractional operators. In order to solve 3D consolidation problems of large scale structures, the fractional pore pressure diffusion equation is implemented in

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Zhang, Yabing, Tongjun Chen, Hejun Zhu, Yang Liu, Tao Xing, and Xin Zhang. "Approximating Constant-Q Seismic Wave Propagations in Acoustic and Elastic Media Using a Cole–Cole Model." Bulletin of the Seismological Society of America, December 2, 2022. http://dx.doi.org/10.1785/0120220143.

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ABSTRACT Seismic wave propagation in the Earth’s interior inevitably encounters attenuation and dispersion effects, which usually can be represented by a constant-Q model. However, solving the constant-Q wave equations formulated by fractional Laplacians is computationally intensive. Alternatively, the Cole–Cole model provides an optimal description of seismic attenuation. Because of the fractional time derivatives of both stress and strain in the expression, this method exhibits good adaptability and flexibility. In this article, we investigate the performance of the Cole–Cole model to approx

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Kaveh, Amin, Mohammad Vahedi, and Majid Gandomkar. "Improving the performance of a chaotic nonlinear system of fractional-order brushless direct current electric motor using fractional-order sliding mode control." An International Journal of Optimization and Control: Theories & Applications (IJOCTA), April 22, 2025, 8407. https://doi.org/10.36922/ijocta.8407.

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Brushless direct current (BLDC) motors are widely used in industrial applications due to their high efficiency and reliability. However, these motors exhibit inherent nonlinear and chaotic behavior, which can degrade performance and cause instability under certain operating conditions. This paper proposes a fractional-order sliding mode controller (FO-SMC) for robust chaos suppression and improved stability in BLDC motor systems to address this issue. The proposed controller leverages fractional-order calculus to enhance robustness, mitigate chattering, and provide better disturbance rejection

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Saeed, Muhammad, M. Muddassar, M. Sajjad Mehmood, and N. Siddiqui. "Radiation-responsive polymers: a novel spectral approach to investigate ultrahigh molecular weight polyethylene modifications using Grunwald-Letnikov and Caputo fractional order derivatives." Revista Mexicana de Física 71, no. 1 Jan-Feb (2025). https://doi.org/10.31349/revmexfis.71.011005.

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The impact of gamma radiation on Ultrahigh Molecular Weight Polyethylene (UHMWPE) using fractional differential transformations of FTIR spectra has been probed during the current study. The gamma irradiated samples of UHMWPE for doses of 0, 25, and 50 kGy have been tested with FTIR spectroscopy, and subsequent to testing each spectrum has been analyzed with fraction order differentiation ranging from 0.5 to 1. The exhibited significant shifts in absorbance due to radiation-induced physical and chemical changes, including C=C unsaturation, C=O carbonyl absorption, and variations in CH2 bending

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Shukurova, Mubashiraxon Furqatovna Mohinur Xaydar qizi Raupova. "ON THE HISTORY OF THE ORIGIN OF FRACTIONAL INTEGRALS." September 7, 2022. https://doi.org/10.5281/zenodo.7058010.

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<em>The article provides brief information about the history of the origin of fractional order integrals. Concepts of fractional integro-differential calculus have been given. The contribution of each scientist has been mentioned separately. General properties of fractional operators, definition of fractional Riemann-Liouville integrals, brief history of integro-differential calculus theory have been described.</em>

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G. Omarova, Asiyat. "A numerical method of solving the Cauchy problem for one differential equation with the Riemann – Liouville fractional derivative." University proceedings. Volga region. Physical and mathematical sciences, no. 3 (2024). https://doi.org/10.21685/2072-3040-2024-3-2.

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Background. The object of study in this work is the Cauchy problem for an ordinary differential equation with the Riemann – Liouville fractional derivative on an interval [0,Т ]. A distinctive feature of the problem is that the order is a variable function α = α(t), that depends on time and satisfies the condition 0 < α(t) < 1. The purpose of the study is to construct a numerical method for solving the designated Cauchy problem. Materials and methods. For a numerical solution, the finite difference method is used, with the help of which the transition from a continuous region to a discre

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Han, Yu-Xing, Jia-Xin Zhang, and Yu-Lan Wang. "Dynamic behavior of a two-mass nonlinear fractional-order vibration system." Frontiers in Physics 12 (September 4, 2024). http://dx.doi.org/10.3389/fphy.2024.1452138.

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The two-mass nonlinear vocal cord vibration system (VCVS) serves as a mechanical representation of the fundamental vocalization process. Traditional models of the VCVS, which are based on integer-order dynamics, often overlook the impact of memory effects. To address this limitation and enhance the accuracy of simulations, this study incorporates the memory effects of vocal cord vibrations by integrating the Grunwald–Letnikov fractional derivative into the two-mass nonlinear VCVS framework. Initially, a high-precision computational scheme is formulated for the two-mass nonlinear fractional-ord

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