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Journal articles on the topic 'Fractional derivative operators'

Author: Grafiati

Published: 2 June 2025

Last updated: 31 July 2025

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1

TARASOV, VASILY E. "FRACTIONAL DERIVATIVE AS FRACTIONAL POWER OF DERIVATIVE." International Journal of Mathematics 18, no. 03 (2007): 281–99. http://dx.doi.org/10.1142/s0129167x07004102.

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Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of selfadjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

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2

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

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The aim of this work is to introduce a novel concept, Riesz–Dunkl fractional derivatives, within the context of Dunkl-type operators. A particularly noteworthy revelation is that when a specific parameter κ equals zero, the Riesz–Dunkl fractional derivative smoothly reduces to both the well-known Riesz fractional derivative and the fractional second-order derivative. Furthermore, we introduce a new concept: the fractional Sobolev space. This space is defined and characterized using the versatile framework of the Dunkl transform.

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3

Pskhu, Arsen. "Nakhushev extremum principle for a class of integro-differential operators." Fractional Calculus and Applied Analysis 23, no. 6 (2020): 1712–22. http://dx.doi.org/10.1515/fca-2020-0085.

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Abstract We investigate extreme properties of a class of integro-differential operators. We prove an assertion that extends the Nakhushev extremum principle, known for fractional Riemann-Liouville derivatives, to integro-differential operators with kernels of a general form. We establish the weighted extremum principle for convolution operators and the Riemann-Liouville fractional derivative. In addition, as an application, we prove a uniqueness theorem for a boundary value problem in a non-cylindrical domain for the fractional diffusion equation with the Riemann-Lioville fractional derivative

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4

Inbaig, Abdlgader M., and Yasmina M. Bashon. "A comparative study on the behavior of Riemann-Liouville and Caputo fractional derivatives of some functions." Libyan Journal of Science &Technology 14, no. 2 (2025): 127–38. https://doi.org/10.37376/ljst.v14i2.7209.

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This paper presents an overview of fractional order derivative operators. Particular attention is devoted to the Riemann-Liouville and Caputo fractional derivative operators. A comparative study of these two frameworks to show how they behave geometrically. The computation results of some elementary function derivatives of fractional order are shown in graphic form and tabular for this purpose. The conclusion will include a few observations about derivatives of integer and fr Abdlgader M. Inbaig, Yasmina M. Bashon actional order.

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Mortari, Daniele, Roberto Garrappa, and Luigi Nicolò. "Theory of Functional Connections Extended to Fractional Operators." Mathematics 11, no. 7 (2023): 1721. http://dx.doi.org/10.3390/math11071721.

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The theory of functional connections, an analytical framework generalizing interpolation, was extended and applied in the context of fractional-order operators (integrals and derivatives). The extension was performed and presented for univariate functions, with the aim of determining the whole set of functions satisfying some constraints expressed in terms of integrals and derivatives of non-integer order. The objective of these expressions was to solve fractional differential equations or other problems subject to fractional constraints. Although this work focused on the Riemann–Liouville def

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6

Martínez-Fuentes, Oscar, Fidel Meléndez-Vázquez, Guillermo Fernández-Anaya, and José Francisco Gómez-Aguilar. "Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities." Mathematics 9, no. 17 (2021): 2084. http://dx.doi.org/10.3390/math9172084.

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In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we

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7

Tarasov, Vasily E. "Fractional Derivative Regularization in QFT." Advances in High Energy Physics 2018 (2018): 1–8. http://dx.doi.org/10.1155/2018/7612490.

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We propose in this paper a new regularization, where integer-order differential operators are replaced by fractional-order operators. Regularization for quantum field theories based on application of the Riesz fractional derivatives of noninteger orders is suggested. The regularized loop integrals depend on parameter that is the order α>0 of the fractional derivative. The regularization procedure is demonstrated for scalar massless fields in φ4-theory on n-dimensional pseudo-Euclidean space-time.

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8

Garrappa, Roberto, Eva Kaslik, and Marina Popolizio. "Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial." Mathematics 7, no. 5 (2019): 407. http://dx.doi.org/10.3390/math7050407.

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Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of so

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9

Azzouz, Noureddine, Bouharket Benaissa та Hüseyin Budak. "Ongeneralized ψ-conformable calculus: Properties and inequalities". Filomat 38, № 25 (2024): 8755–72. https://doi.org/10.2298/fil2425755a.

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In this paper, we first introduce a new fractional derivatives and integrals called generalized ?-conformable derivative and generalized ?-conformable integral operators, respectively. We also show that these operators generalize various well-known fractional integral operators. Then, we present several properties of these operators including semi-group property. Moreover, we apply these operators to obtain a new Hermite-Hadamard-type inequality for convex functions. Furthermore, we obtain corresponding midpoint and trapezoid type inequalities for functions whose derivatives in absolute value

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10

Odibat, Zaid, and Dumitru Baleanu. "On a New Modification of the Erdélyi–Kober Fractional Derivative." Fractal and Fractional 5, no. 3 (2021): 121. http://dx.doi.org/10.3390/fractalfract5030121.

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In this paper, we introduce a new Caputo-type modification of the Erdélyi–Kober fractional derivative. We pay attention to how to formulate representations of Erdélyi–Kober fractional integral and derivatives operators. Then, some properties of the new modification and relationships with other Erdélyi–Kober fractional derivatives are derived. In addition, a numerical method is presented to deal with fractional differential equations involving the proposed Caputo-type Erdélyi–Kober fractional derivative. We hope the presented method will be widely applied to simulate such fractional models.

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11

Wang, Haihua, та Jie Zhao. "Mild Solutions of the (ρ1,ρ2,k1,k2,φ)-Proportional Hilfer–Cauchy Problem". Symmetry 16, № 10 (2024): 1349. http://dx.doi.org/10.3390/sym16101349.

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Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the (ρ1,ρ2,k1,k2,φ)-proportional integral and the (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional derivative. Numerous previous studied fractional integrals and derivatives can be considered as particular instances of the novel operators introduced above. Some properties of the (ρ1,ρ2,k1,k2,φ)-proportional integral are discussed, including mapping properties, the generalized Laplace transform of the (ρ1,ρ2,k1,k2,φ)-proportional integral and (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fracti

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12

Zarin, Rahat, Abdur Raouf, Amir Khan, Aeshah A. Raezah, and Usa Wannasingha Humphries. "Computational modeling of financial crime population dynamics under different fractional operators." AIMS Mathematics 8, no. 9 (2023): 20755–89. http://dx.doi.org/10.3934/math.20231058.

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<abstract><p>This paper presents an analysis and numerical simulation of financial crime population dynamics using fractional order calculus and Newton's polynomial. The dynamics of financial crimes are modeled as a fractional-order system, which is then solved using numerical methods based on Newton's polynomial. The results of the simulation provide insights into the behavior of financial crime populations over time, including the stability and convergence of the systems. The study provides a new approach to understanding financial crime populations and has potential applications

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13

Saqib, Muhammad, Ilyas Khan, Yu-Ming Chu, Ahmad Qushairi, Sharidan Shafie, and Kottakkaran Sooppy Nisar. "Multiple Fractional Solutions for Magnetic Bio-Nanofluid Using Oldroyd-B Model in a Porous Medium with Ramped Wall Heating and Variable Velocity." Applied Sciences 10, no. 11 (2020): 3886. http://dx.doi.org/10.3390/app10113886.

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Three different fractional models of Oldroyd-B fluid are considered in this work. Blood is taken as a special example of Oldroyd-B fluid (base fluid) with the suspension of gold nanoparticles, making the solution a biomagnetic non-Newtonian nanofluid. Based on three different definitions of fractional operators, three different models of the resulting nanofluid are developed. These three operators are based on the definitions of Caputo (C), Caputo–Fabrizio (CF), and Atnagana–Baleanu in the Caputo sense (ABC). Nanofluid is taken over an upright plate with ramped wall heating and time-dependent

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14

Solís-Pérez, Jesús Emmanuel, and José Francisco Gómez-Aguilar. "Novel Fractional Operators with Three Orders and Power-Law, Exponential Decay and Mittag–Leffler Memories Involving the Truncated M-Derivative." Symmetry 12, no. 4 (2020): 626. http://dx.doi.org/10.3390/sym12040626.

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In this research, novel M-truncated fractional derivatives with three orders have been proposed. These operators involve truncated Mittag–Leffler function to generalize the Khalil conformable derivative as well as the M-derivative. The new operators proposed are the convolution of truncated M-derivative with a power law, exponential decay and the complete Mittag–Leffler function. Numerical schemes based on Lagrange interpolation to predict chaotic behaviors of Rucklidge, Shimizu–Morioka and a hybrid strange attractors were considered. Additionally, numerical analysis based on 0–1 test and sens

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15

Hattaf, Khalid. "A new generalized class of fractional operators with weight and respect to another function." Journal of Fractional Calculus and Nonlinear Systems 5, no. 2 (2024): 53–68. https://doi.org/10.48185/jfcns.v5i2.1269.

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This paper introduces and investigates the properties of a new generalized class of fractional differential and integral operators. Such newly class covers various definitions of fractional derivatives with singular and non-singular kernels, weighted fractional derivatives with respect to another function, as well as the new mixed fractional derivative in the sense of Caputo and Riemann-Liouville. Furthermore, the newly introduced class includes all existing forms of fractional integrals, weighted fractionalintegrals and also the weighted fractional integrals with respect to another function i

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16

Sitnik, Sergey M., Vladimir E. Fedorov, Nikolay V. Filin, and Viktor A. Polunin. "On the Solvability of Equations with a Distributed Fractional Derivative Given by the Stieltjes Integral." Mathematics 10, no. 16 (2022): 2979. http://dx.doi.org/10.3390/math10162979.

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Linear equations in Banach spaces with a distributed fractional derivative given by the Stieltjes integral and with a closed operator A in the right-hand side are considered. Unlike the previously studied classes of equations with distributed derivatives, such kinds of equations may contain a continuous and a discrete part of the integral, i.e., a standard integral of the fractional derivative with respect to its order and a linear combination of fractional derivatives with different orders. Resolving families of operators for such equations are introduced into consideration, and their propert

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17

Nava-Antonio, G., G. Fernández-Anaya, E. G. Hernández-Martínez, J. J. Flores-Godoy, and E. D. Ferreira-Vázquez. "Consensus of Multiagent Systems Described by Various Noninteger Derivatives." Complexity 2019 (February 26, 2019): 1–14. http://dx.doi.org/10.1155/2019/3297410.

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In this paper, we unify and extend recent developments in Lyapunov stability theory to present techniques to determine the asymptotic stability of six types of fractional dynamical systems. These differ by being modeled with one of the following fractional derivatives: the Caputo derivative, the Caputo distributed order derivative, the variable order derivative, the conformable derivative, the local fractional derivative, or the distributed order conformable derivative (the latter defined in this work). Additionally, we apply these results to study the consensus of a fractional multiagent syst

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18

Rosales García, J. Juan, J. David Filoteo, and Andrés González. "A comparative analysis of the RC circuit with local and non-local fractional derivatives." Revista Mexicana de Física 64, no. 6 (2018): 647. http://dx.doi.org/10.31349/revmexfis.64.647.

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This work is devoted to investigate solutions to RC circuits using four different types of time fractional diferential operators of order 0 < γ ≤ 1. The fractional derivatives considered are, Caputo, Caputo-Fabrizio, Atangana-Baleanu and the conformable derivative. It is shown that Atangana-Baleanu fractional derivative (non-local), and the conformable (local) derivative could describe a wider class of physical processes then the Caputo and Caputo-Fabrizio. The solutions are exactly equal for all four erivatives only for the case γ=1.

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19

EL-NABULSI, AHMAD RAMI. "FRACTIONAL LAGRANGIAN FORMULATION OF GENERAL RELATIVITY AND EMERGENCE OF COMPLEX, SPINORIAL AND NONCOMMUTATIVE GRAVITY." International Journal of Geometric Methods in Modern Physics 06, no. 01 (2009): 25–76. http://dx.doi.org/10.1142/s021988780900345x.

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Fractional calculus of variations and in particular the Fractional Action-Like Variational Approach has recently gained importance in studying nonconservative and weak decaying dynamical systems. Until now, in high-energy physics including cosmology and quantum field theories the derivative and integral operators have only been used in integer steps. In this work, we develop the fractional Lagrangian formulation of General Relativity based on the Cresson's fractional differential operators that generalize the differential operators of conventional Einstein's General Relativity but that reduces

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20

Chen, Minghua, and Weihua Deng. "Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators." Communications in Computational Physics 16, no. 2 (2014): 516–40. http://dx.doi.org/10.4208/cicp.120713.280214a.

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AbstractHigh order discretization schemes play more important role in fractional operators than classical ones. This is because usually for classical derivatives the stencil for high order discretization schemes is wider than low order ones; but for fractional operators the stencils for high order schemes and low order ones are the same. Then using high order schemes to solve fractional equations leads to almost the same computational cost with first order schemes but the accuracy is greatly improved. Using the fractional linear multistep methods, Lubich obtains thev-th order (v <6) approxi

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21

Cichoń, Mieczysław, Hussein A. H. Salem, and Wafa Shammakh. "Regularity Results for Hybrid Proportional Operators on Hölder Spaces." Fractal and Fractional 9, no. 2 (2025): 58. https://doi.org/10.3390/fractalfract9020058.

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Recently, a new type of derivative has been introduced, known as Caputo proportional derivatives. These are motivated by the applications of such derivatives (which are a generalization of Caputo’s standard fractional derivative) and the need to incorporate such calculus into the research on operators. The investigation therefore focuses on the equivalence of differential and integral problems for proportional calculus problems. The operators are always studied in the appropriate function spaces. Furthermore, the investigation extends these results to encompass the more general notion of Hilfe

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22

Samadi, Ayub, Sotiris K. Ntouyas, and Jessada Tariboon. "Fractional Sequential Coupled Systems of Hilfer and Caputo Integro-Differential Equations with Non-Separated Boundary Conditions." Axioms 13, no. 7 (2024): 484. http://dx.doi.org/10.3390/axioms13070484.

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In studying boundary value problems and coupled systems of fractional order in (1,2], involving Hilfer fractional derivative operators, a zero initial condition is necessary. The consequence of this fact is that boundary value problems and coupled systems of fractional order with non-zero initial conditions cannot be studied. For example, such boundary value problems and coupled systems of fractional order are those including separated, non-separated, or periodic boundary conditions. In this paper, we propose a method for studying a coupled system of fractional order in (1,2], involving fracti

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23

Jebril, Iqbal H., Mohammed S. El-Khatib, Ahmad A. Abubaker, Suha B. Al-Shaikh, and Iqbal M. Batiha. "Results on Katugampola Fractional Derivatives and Integrals." International Journal of Analysis and Applications 21 (October 12, 2023): 113. http://dx.doi.org/10.28924/2291-8639-21-2023-113.

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In this paper, we introduce and develop a new definitions for Katugampola derivative and Katugampola integral. In particular, we defined a (left) fractional derivative starting from a of a function f of order α∈(m-1, m] and a (right) fractional derivative terminating at b, where m ∈ N. Then, we give some proprieties in relation to these operators such as linearity, product rule, quotient rule, power rule, chain rule, and vanishing derivatives for constant functions.

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24

Xu, Qinwu, and Zhoushun Zheng. "Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator." International Journal of Differential Equations 2019 (January 1, 2019): 1–14. http://dx.doi.org/10.1155/2019/3734617.

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Generalized fractional operators are generalization of the Riemann-Liouville and Caputo fractional derivatives, which include Erdélyi-Kober and Hadamard operators as their special cases. Due to the complicated form of the kernel and weight function in the convolution, it is even harder to design high order numerical methods for differential equations with generalized fractional operators. In this paper, we first derive analytical formulas for α-th (α>0) order fractional derivative of Jacobi polynomials. Spectral approximation method is proposed for generalized fractional operators through a

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25

Torres-Hernandez, Anthony, Fernando Brambila-Paz, and Rafael Ramirez-Melendez. "Proposal for Use of the Fractional Derivative of Radial Functions in Interpolation Problems." Fractal and Fractional 8, no. 1 (2023): 16. http://dx.doi.org/10.3390/fractalfract8010016.

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This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions for use in interpolation problems. Furthermore, a technique is employed to precondition the matrices generated in the presented problems through QR decomposition. Similarly, a method is introduced to define two different types of abelian groups for any fractional operator defined in the interval [0,1), among which the

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26

Zhang, Zhiqiang, Ghulam Farid, Sajid Mehmood, Kamsing Nonlaopon, and Tao Yan. "Generalized k-Fractional Integral Operators Associated with Pólya-Szegö and Chebyshev Types Inequalities." Fractal and Fractional 6, no. 2 (2022): 90. http://dx.doi.org/10.3390/fractalfract6020090.

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Inequalities related to derivatives and integrals are generalized and extended via fractional order integral and derivative operators. The present paper aims to define an operator containing Mittag-Leffler function in its kernel that leads to deduce many already existing well-known operators. By using this generalized operator, some well-known inequalities are studied. The results of this paper reproduce Chebyshev and Pólya-Szegö type inequalities for Riemann-Liouville and many other fractional integral operators.

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27

Baeumer, Boris, Mark M. Meerschaert, and Jeff Mortensen. "Space-time fractional derivative operators." Proceedings of the American Mathematical Society 133, no. 8 (2005): 2273–82. http://dx.doi.org/10.1090/s0002-9939-05-07949-9.

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28

Turov, M. M., V. E. Fedorov, and B. T. Kien. "Linear Inverse Problems for Multi-term Equations with Riemann — Liouville Derivatives." Bulletin of Irkutsk State University. Series Mathematics 38 (2021): 36–53. http://dx.doi.org/10.26516/1997-7670.2021.38.36.

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The issues of well-posedness of linear inverse coefficient problems for multi-term equations in Banach spaces with fractional Riemann – Liouville derivatives and with bounded operators at them are considered. Well-posedness criteria are obtained both for the equation resolved with respect to the highest fractional derivative, and in the case of a degenerate operator at the highest derivative in the equation. Two essentially different cases are investigated in the degenerate problem: when the fractional part of the order of the second-oldest derivative is equal to or different from the fraction

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29

Yépez-Martínez, H., and J. F. Gómez-Aguilar. "Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel." Mathematical Modelling of Natural Phenomena 13, no. 1 (2018): 13. http://dx.doi.org/10.1051/mmnp/2018002.

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Analytical and numerical simulations of nonlinear fractional differential equations are obtained with the application of the homotopy perturbation transform method and the fractional Adams-Bashforth-Moulton method. Fractional derivatives with non singular Mittag-Leffler function in Liouville-Caputo sense and the fractional derivative of Liouville-Caputo type are considered. Some examples have been presented in order to compare the results obtained, classical behaviors are recovered when the derivative order is 1.

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Momenzadeh, Mohammad, and Sajedeh Norozpou. "Alternative fractional derivative operator on non-newtonian calculus and its approaches." Nexo Revista Científica 34, no. 02 (2021): 906–15. http://dx.doi.org/10.5377/nexo.v34i02.11616.

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Nowadays, study on fractional derivative and integral operators is one of the hot topics of mathematics and lots of investigations and studies make their attentions in this field. Most of these concerns raised from the vast application of these operators in study of phenomena’s models. These operators interpreted by Newtonian calculus, however different types of calculi are existed and we introduce the fractional derivative operators focused on Bi-geometric calculus and also their fractional differential equations are studied.

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Cruz, Fátima, Ricardo Almeida, and Natália Martins. "Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels." Mathematics 9, no. 14 (2021): 1665. http://dx.doi.org/10.3390/math9141665.

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In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

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Rahman, Gauhar, Muhammad Samraiz, Cetin Yildiz, Thabet Abdeljawad, and Manar A. Alqudah. "New Generalized Results for Modified Atangana-Baleanu Fractional Derivatives and Integral Operators." European Journal of Pure and Applied Mathematics 18, no. 1 (2025): 5697. https://doi.org/10.29020/nybg.ejpam.v18i1.5697.

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In this current study, first we establish the modified power Atangana-Baleanu fractional derivative operators (MPC) in both the Caputo and Riemann-Liouville (MPRL) senses. Using the convolution approach and Laplace transformation, the so-called modified power fractional Caputo and R-L derivative operators with non-singular kernels are introduced. We establish theboundedness of the modified Caputo fractional derivative operator in this study. The fractional differential equations are solved with the generalised Laplace transform (GLT). In addition, the corresponding form of the fractional integ

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Luchko, Yuri. "On a Generic Fractional Derivative Associated with the Riemann–Liouville Fractional Integral." Axioms 13, no. 9 (2024): 604. http://dx.doi.org/10.3390/axioms13090604.

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In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its domain, null-space, and projector operator; establish the interrelations between its different realizations; and present a generalized fractional Taylor formula involving the generic fractional derivative. Then, we consider the fractional relaxation equation containing the generic fractional de

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34

Mallah, Ishfaq, Idris Ahmed, Ali Akgul, Fahd Jarad, and Subhash Alha. "On $ \psi $-Hilfer generalized proportional fractional operators." AIMS Mathematics 7, no. 1 (2022): 82–103. http://dx.doi.org/10.3934/math.2022005.

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<abstract><p>In this paper, we introduce a generalized fractional operator in the setting of Hilfer fractional derivatives, the $ \psi $-Hilfer generalized proportional fractional derivative of a function with respect to another function. The proposed operator can be viewed as an interpolator between the Riemann-Liouville and Caputo generalized proportional fractional operators. The properties of the proposed operator are established under some classical and standard assumptions. As an application, we formulate a nonlinear fractional differential equation with a nonlocal initial co

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35

Fedorov, Vladimir E., Mikhail M. Turov, and Bui Trong Kien. "A Class of Quasilinear Equations with Riemann–Liouville Derivatives and Bounded Operators." Axioms 11, no. 3 (2022): 96. http://dx.doi.org/10.3390/axioms11030096.

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The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Nonlinearity in the equation is assumed to be Lipschitz continuous and dependent on lower order fractional derivatives, which orders have the same fractional part as the order of the highest fractional derivative. The obtained abstract result is applied to study a class of initial-boundary value problems to time-fractional order equations with polynomials o

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Kochubei, Anatoly N., Yuri Kondratiev, and José Luís da Silva. "On fractional heat equation." Fractional Calculus and Applied Analysis 24, no. 1 (2021): 73–87. http://dx.doi.org/10.1515/fca-2021-0004.

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Abstract In this paper, the long-time behavior of the Cesaro mean of the fundamental solution for fractional Heat equation corresponding to random time changes in the Brownian motion is studied. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.

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Fahad, Hafiz Muhammad, and Arran Fernandez. "Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications." Fractional Calculus and Applied Analysis 24, no. 2 (2021): 518–40. http://dx.doi.org/10.1515/fca-2021-0023.

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Abstract Mikusiński’s operational calculus is a formalism for understanding integral and derivative operators and solving differential equations, which has been applied to several types of fractional-calculus operators by Y. Luchko and collaborators, such as for example [26], etc. In this paper, we consider the operators of Riemann–Liouville fractional differentiation of a function with respect to another function, and discover that the approach of Luchko can be followed, with small modifications, in this more general setting too. The Mikusiński’s operational calculus approach is used to obtai

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Ntouyas, Sotiris K., Bashir Ahmad, and Jessada Tariboon. "A Survey on Recent Results on Lyapunov-Type Inequalities for Fractional Differential Equations." Fractal and Fractional 6, no. 5 (2022): 273. http://dx.doi.org/10.3390/fractalfract6050273.

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This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville, ψ-Caputo, ψ-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, and proportional fractional derivative operators.

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Holm, Sverre. "Dispersion analysis for wave equations with fractional Laplacian loss operators." Fractional Calculus and Applied Analysis 22, no. 6 (2019): 1596–606. http://dx.doi.org/10.1515/fca-2019-0082.

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Abstract Several wave equations for power-law attenuation have a spatial fractional derivative in the loss term. Both one-sided and two-sided spatial fractional derivatives can give causal solutions and a phase velocity dispersion which satisfies the Kramers–Kronig relation. The Chen–Holm and the Treeby–Cox equations both have the two-sided fractional Laplacian derivative, but only the latter satisfies this relation. There also exists several seemingly different expressions for the phase velocity for these equations and it is shown here that they are approximately equivalent. Causality of the

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Yang, Ai-Min, Zeng-Shun Chen, H. M. Srivastava, and Xiao-Jun Yang. "Application of the Local Fractional Series Expansion Method and the Variational Iteration Method to the Helmholtz Equation Involving Local Fractional Derivative Operators." Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/259125.

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We investigate solutions of the Helmholtz equation involving local fractional derivative operators. We make use of the series expansion method and the variational iteration method, which are based upon the local fractional derivative operators. The nondifferentiable solution of the problem is obtained by using these methods.

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Piotrowska, Ewa, and Krzysztof Rogowski. "Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements." Electronics 10, no. 4 (2021): 475. http://dx.doi.org/10.3390/electronics10040475.

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The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Con

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Liu, Zelin, Xiaobin Yu, and Yajun Yin. "On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel." Fractal and Fractional 8, no. 11 (2024): 653. http://dx.doi.org/10.3390/fractalfract8110653.

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Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional calculus operators using the generalized fractional calculus with the Sonine kernel. In this framework, we prove that the α-th power of a differential operator corresponds precisely to the α-th fractional derivative, ensuring both accuracy and consistency. Th

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Bourafa, Safa, Mohammed Salah Abdelouahab, and René Lozi. "On periodic solutions of fractional-order differential systems with a fixed length of sliding memory." Journal of Innovative Applied Mathematics and Computational Sciences 1, no. 1 (2021): 64–78. http://dx.doi.org/10.58205/jiamcs.v1i1.6.

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The fractional-order derivative of a non-constant periodic function is not periodic with the same period. Consequently, any time-invariant fractional-order systems do not have a non-constant periodic solution. This property limits the applicability of fractional derivatives and makes it unfavorable to model periodic real phenomena.This article introduces a modification to the Caputo and Rieman-Liouville fractional-order operators by fixing their memory length and varying the lower terminal. It is shown that this modified definition of fractional derivative preserves the periodicity. Therefore,

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Almeida, Ricardo. "Optimizing Variational Problems through Weighted Fractional Derivatives." Fractal and Fractional 8, no. 5 (2024): 272. http://dx.doi.org/10.3390/fractalfract8050272.

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In this article, we explore a variety of problems within the domain of calculus of variations, specifically in the context of fractional calculus. The fractional derivative we consider incorporates the notion of weighted fractional derivatives along with derivatives with respect to another function. Besides the fractional operators, the Lagrange function depends on extremal points. We examine the fundamental problem, providing the fractional Euler–Lagrange equation and the associated transversality conditions. Both the isoperimetric and Herglotz problems are also explored. Finally, we conclude

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Alzaid, Sara Salem, Badr Saad T. Alkahtani, Shivani Sharma, and Ravi Shanker Dubey. "Numerical Solution of Fractional Model of HIV-1 Infection in Framework of Different Fractional Derivatives." Journal of Function Spaces 2021 (March 19, 2021): 1–10. http://dx.doi.org/10.1155/2021/6642957.

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In this paper, we have extended the model of HIV-1 infection to the fractional mathematical model using Caputo-Fabrizio and Atangana-Baleanu fractional derivative operators. A detailed proof for the existence and the uniqueness of the solution of fractional mathematical model of HIV-1 infection in Atangana-Baleanu sense is presented. Numerical approach is used to find and study the behavior of the solution of the stated model using different derivative operators, and the graphical comparison between the solutions obtained for the Caputo-Fabrizio and the Atangana-Baleanu operator is presented t

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Mamatov, Tulkin, Nemat Mustafoev, Dilshod Barakaev, and Rano Sabirova. "Hardy-Littlewood-Type Theorem for Mixed Fractional Integrals in Hölder Spaces." Indian Journal of Advanced Mathematics 1, no. 2 (2021): 15–19. http://dx.doi.org/10.54105/ijam.b1105.101221.

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We study mixed Riemann-Liouville fractional integration operators and mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The obtained are results generalized to the case of Hölder spaces with power weight. Keywords: functions of two variables, fractional derivative of Marchaud form, mixed fractional derivative, weight, mixed fractional integral, Hölder space.

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Tong, Ce-Zhong, Cheng Yuan, and Ze-Hua Zhou. "Topological Structures of Derivative Weighted Composition Operators on the Bergman Space." Journal of Function Spaces 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/672385.

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We characterize the difference of derivative weighted composition operators on the Bergman space in the unit disk and determine when linear-fractional derivative weighted composition operators belong to the same component of the space of derivative weighted composition operators on the Bergman space under the operator norm topology.

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Aljahdaly, Noufe H., Ravi P. Agarwal, Rasool Shah, and Thongchai Botmart. "Analysis of the Time Fractional-Order Coupled Burgers Equations with Non-Singular Kernel Operators." Mathematics 9, no. 18 (2021): 2326. http://dx.doi.org/10.3390/math9182326.

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In this article, we have investigated the fractional-order Burgers equation via Natural decomposition method with nonsingular kernel derivatives. The two types of fractional derivatives are used in the article of Caputo–Fabrizio and Atangana–Baleanu derivative. We employed Natural transform on fractional-order Burgers equation followed by inverse Natural transform, to achieve the result of the equations. To validate the method, we have considered a two examples and compared with the exact results.

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Deogan, Aneesh S., Roeland Dilz, and Diego Caratelli. "On the Application of Fractional Derivative Operator Theory to the Electromagnetic Modeling of Frequency Dispersive Media." Mathematics 12, no. 7 (2024): 932. http://dx.doi.org/10.3390/math12070932.

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Fractional derivative operators are finding applications in a wide variety of fields with their ability to better model certain phenomena exhibiting spatial and temporal nonlocality. One area in which these operators are applicable is in the field of electromagnetism, thereby modelling transient wave propagation in complex media. To apply fractional derivative operators to electromagnetic problems, the operator must adhere to certain principles, like the trigonometric functions invariance property. The Grünwald–Letnikov and Marchaud fractional derivative operators comply with these principles

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Ortigueira, Manuel Duarte. "Riesz potential operators and inverses via fractional centred derivatives." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–12. http://dx.doi.org/10.1155/ijmms/2006/48391.

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Fractional centred differences and derivatives definitions are proposed, generalizing to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalizations of the Cauchy integral derivative. To compute this integral, a special two-straight-line path was used. With this the referred integrals lead to the well-known Riesz potential operators and their inverses that emerge as true fractional centred derivatives, but that can be computed through summations si

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