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Journal articles on the topic 'Fractional derivatives'

Author: Grafiati

Published: 4 June 2021

Last updated: 5 February 2022

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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

Ortigueira, Manuel Duarte, and Gabriel Bengochea. "Bilateral Tempered Fractional Derivatives." Symmetry 13, no. 5 (May 8, 2021): 823. http://dx.doi.org/10.3390/sym13050823.

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The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided tempered fractional derivatives and the two-sided fractional derivatives. An analysis of the tempered Riesz potential is done and shows that it cannot be considered as a derivative.

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3

Farayola, Musiliu Folarin, Sharidan Shafie, Fuaada Mohd Siam, Rozi Mahmud, and Suraju Olusegun Ajadi. "Mathematical modeling of cancer treatments with fractional derivatives: An Overview." Malaysian Journal of Fundamental and Applied Sciences 17, no. 4 (August 31, 2021): 389–401. http://dx.doi.org/10.11113/mjfas.v17n4.2062.

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This review article presents fractional derivative cancer treatment models to show the importance of fractional derivatives in modeling cancer treatments. Cancer treatment is a significant research area with many mathematical models developed by mathematicians to represent the cancer treatment processes like hyperthermia, immunotherapy, chemotherapy, and radiotherapy. However, many of these models were based on ordinary derivatives and the use of fractional derivatives is still new to many mathematicians. Therefore, it is imperative to review fractional cancer treatment models. The review was done by first presenting 22 various definitions of fractional derivative. Thereafter, 11 articles were selected from different online databases which included Scopus, EBSCOHost, ScienceDirect Journal, SpringerLink Journal, Wiley Online Library, and Google Scholar. These articles were summarized, and the used fractional derivative models were analyzed. Based on this analysis, the merit of modeling with fractional derivative, the most used fractional derivative definition, and the future direction for cancer treatment modeling were presented. From the results of the analysis, it was shown that fractional derivatives incorporated memory effects which gave it an advantage over ordinary derivative for cancer treatment modeling. Moreover, the fractional derivative is a general definition of all derivatives. Also, the fractional models can be applied to different cancer treatment procedures and the most used fractional derivative is the Caputo as well as its non-singular kernel versions. Finally, it was concluded that the future direction for cancer treatment modeling is the adoption of fractional derivative models corroborated with experimental or clinical data.

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4

Li, Changpin, Deliang Qian, and YangQuan Chen. "On Riemann-Liouville and Caputo Derivatives." Discrete Dynamics in Nature and Society 2011 (2011): 1–15. http://dx.doi.org/10.1155/2011/562494.

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Recently, many models are formulated in terms of fractional derivatives, such as in control processing, viscoelasticity, signal processing, and anomalous diffusion. In the present paper, we further study the important properties of the Riemann-Liouville (RL) derivative, one of mostly used fractional derivatives. Some important properties of the Caputo derivative which have not been discussed elsewhere are simultaneously mentioned. The partial fractional derivatives are also introduced. These discussions are beneficial in understanding fractional calculus and modeling fractional equations in science and engineering.

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5

TARASOV, VASILY E. "FRACTIONAL DERIVATIVE AS FRACTIONAL POWER OF DERIVATIVE." International Journal of Mathematics 18, no. 03 (March 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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6

Sene, Ndolane, and José Francisco Gómez Aguilar. "Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives." Fractal and Fractional 3, no. 3 (July 7, 2019): 39. http://dx.doi.org/10.3390/fractalfract3030039.

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This paper proposes novel analytical solutions of the mass-spring-damper systems described by certain generalized fractional derivatives. The Liouville–Caputo left generalized fractional derivative and the left generalized fractional derivative were used. The behaviors of the analytical solutions of the mass-spring-damper systems described by the left generalized fractional derivative and the Liouville–Caputo left generalized fractional derivative were represented graphically and the effect of the orders of the fractional derivatives analyzed. We finish by analyzing the global asymptotic stability and the converging-input-converging-state of the unforced mass-damper system, the unforced spring-damper, the spring-damper system, and the mass-damper system.

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7

Atangana, Abdon, and Aydin Secer. "A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/279681.

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The purpose of this note is to present the different fractional order derivatives definition that are commonly used in the literature on one hand and to present a table of fractional order derivatives of some functions in Riemann-Liouville sense On other the hand. We present some advantages and disadvantages of these fractional derivatives. And finally we propose alternative fractional derivative definition.

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8

İlhan, Esin, and İ. Onur Kıymaz. "A generalization of truncated M-fractional derivative and applications to fractional differential equations." Applied Mathematics and Nonlinear Sciences 5, no. 1 (March 31, 2020): 171–88. http://dx.doi.org/10.2478/amns.2020.1.00016.

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AbstractIn this paper, our aim is to generalize the truncated M-fractional derivative which was recently introduced [Sousa and de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Inter. of Jour. Analy. and Appl., 16 (1), 83–96, 2018]. To do that, we used generalized M-series, which has a more general form than Mittag-Leffler and hypergeometric functions. We called this generalization as truncated ℳ-series fractional derivative. This new derivative generalizes several fractional derivatives and satisfies important properties of the integer-order derivatives. Finally, we obtain the analytical solutions of some ℳ-series fractional differential equations.

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9

Hattaf, Khalid. "On Some Properties of the New Generalized Fractional Derivative with Non-Singular Kernel." Mathematical Problems in Engineering 2021 (May 27, 2021): 1–6. http://dx.doi.org/10.1155/2021/1580396.

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This paper presents some new formulas and properties of the generalized fractional derivative with non-singular kernel that covers various types of fractional derivatives such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. These new properties extend many recent results existing in the literature. Furthermore, the paper proposes some interesting inequalities that estimate the generalized fractional derivatives of some specific functions. These inequalities can be used to construct Lyapunov functions with the aim to study the global asymptotic stability of several fractional-order systems arising from diverse fields of science and engineering.

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10

Garrappa, Roberto, Eva Kaslik, and Marina Popolizio. "Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial." Mathematics 7, no. 5 (May 7, 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 some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo’s derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications.

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11

Trenčevski, Kostadin. "New approach to the fractional derivatives." International Journal of Mathematics and Mathematical Sciences 2003, no. 5 (2003): 315–25. http://dx.doi.org/10.1155/s0161171203206050.

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We introduce a new approach to the fractional derivatives of the analytical functions using the Taylor series of the functions. In order to calculate the fractional derivatives off, it is not sufficient to know the Taylor expansion off, but we should also know the constants of all consecutive integrations off. For example, any fractional derivative ofexisexonly if we assume that thenth consecutive integral ofexisexfor each positive integern. The method of calculating the fractional derivatives very often requires a summation of divergent series, and thus, in this note, we first introduce a method of such summation of series via analytical continuation of functions.

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12

Kaya, U. "CAUCHY FRACTIONAL DERIVATIVE." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 12, no. 4 (2020): 28–32. http://dx.doi.org/10.14529/mmph200403.

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In this paper, we introduce a new sort of fractional derivative. For this, we consider the Cauchy's integral formula for derivatives and modify it by using Laplace transform. So, we obtain the fractional derivative formula F(α)(s) = L{(–1)(α)L–1{F(s)}}. Also, we find a relation between Weyl's fractional derivative and the formula above. Finally, we give some examples for fractional derivative of some elementary functions.

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13

Odibat, Zaid, and Dumitru Baleanu. "On a New Modification of the Erdélyi–Kober Fractional Derivative." Fractal and Fractional 5, no. 3 (September 13, 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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14

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 (July 15, 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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15

Durur, Hülya, Ali Kurt, and Orkun Tasbozan. "New Travelling Wave Solutions for KdV6 Equation Using Sub Equation Method." Applied Mathematics and Nonlinear Sciences 5, no. 1 (April 10, 2020): 455–60. http://dx.doi.org/10.2478/amns.2020.1.00043.

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AbstractThis paper proposes obtaining the new wave solutions of time fractional sixth order nonlinear Equation (KdV6) using sub-equation method where the fractional derivatives are considered in conformable sense. Conformable derivative is an understandable and applicable type of fractional derivative that satisfies almost all the basic properties of Newtonian classical derivative such as Leibniz rule, chain rule and etc. Also conformable derivative has some superiority over other popular fractional derivatives such as Caputo and Riemann-Liouville. In this paper all the computations are carried out by computer software called Mathematica.

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16

Abdeljawad, Thabet, Dumitru Baleanu, Fahd Jarad, and Ravi P. Agarwal. "Fractional Sums and Differences with Binomial Coefficients." Discrete Dynamics in Nature and Society 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/104173.

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In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.

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17

Babenko, V. F., and M. S. Churilova. "On inequalities of Kolmogorov type for fractional derivatives of functions defined on the real domain." Researches in Mathematics 16 (February 7, 2021): 28. http://dx.doi.org/10.15421/240804.

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We obtain new inequalities that generalize known result of Geisberg, which was obtained for fractional Marchaud derivatives, to the case of higher derivatives, at that the fractional derivative is a Riesz one. The inequality with second higher derivative is sharp.

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18

Jiménez, Leonardo Martínez, J. Juan Rosales García, Abraham Ortega Contreras, and Dumitru Baleanu. "Analysis of Drude model using fractional derivatives without singular kernels." Open Physics 15, no. 1 (November 6, 2017): 627–36. http://dx.doi.org/10.1515/phys-2017-0073.

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AbstractWe report study exploring the fractional Drude model in the time domain, using fractional derivatives without singular kernels, Caputo-Fabrizio (CF), and fractional derivatives with a stretched Mittag-Leffler function. It is shown that the velocity and current density of electrons moving through a metal depend on both the time and the fractional order 0 <γ≤ 1. Due to non-singular fractional kernels, it is possible to consider complete memory effects in the model, which appear neither in the ordinary model, nor in the fractional Drude model with Caputo fractional derivative. A comparison is also made between these two representations of the fractional derivatives, resulting a considered difference whenγ< 0.8.

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19

Sene, Ndolane, Babacar Sène, Seydou Nourou Ndiaye, and Awa Traoré. "Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative." Discrete Dynamics in Nature and Society 2020 (July 25, 2020): 1–11. http://dx.doi.org/10.1155/2020/8047347.

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The value of an option plays an important role in finance. In this paper, we use the Black–Scholes equation, which is described by the nonsingular fractional-order derivative, to determine the value of an option. We propose both a numerical scheme and an analytical solution. Recent studies in fractional calculus have included new fractional derivatives with exponential kernels and Mittag-Leffler kernels. These derivatives have been found to be applicable in many real-world problems. As fractional derivatives without nonsingular kernels, we use a Caputo–Fabrizio fractional derivative and a Mittag-Leffler fractional derivative. Furthermore, we use the Adams–Bashforth numerical scheme and fractional integration to obtain the numerical scheme and the analytical solution, and we provide graphical representations to illustrate these methods. The graphical representations prove that the Adams–Bashforth approach is helpful in getting the approximate solution for the fractional Black–Scholes equation. Finally, we investigate the volatility of the proposed model and discuss the use of the model in finance. We mainly notice in our results that the fractional-order derivative plays a regulator role in the diffusion process of the Black–Scholes equation.

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Aljhani, Sami, Mohd Salmi Md Noorani, Khaled M. Saad, and A. K. Alomari. "Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott." Journal of Function Spaces 2021 (July 19, 2021): 1–12. http://dx.doi.org/10.1155/2021/2544688.

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A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transformation method to obtain approximate solutions for the time-fractional Gray-Scott model. This method gives a more realistic series of solutions that converge rapidly to the exact solution. We can ensure convergence by solving the series resultant. We study the convergence analysis of fractional homotopy analysis transformation method by determining the interval of convergence employing the ℏ u , v -curves and the average residual error. We also test the accuracy and the efficiency of this method by comparing our results numerically with the exact solution. Moreover, the effect of the fractionally obtained derivatives on the reaction-diffusion is analyzed. The fractional homotopy analysis transformation method algorithm can be easily applied for singular and nonsingular fractional derivative with partial differential equations, where a few terms of series solution are good enough to give an accurate solution.

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21

Rossikhin, Yuriy A., and Marina V. Shitikova. "New Approach for the Analysis of Damped Vibrations of Fractional Oscillators." Shock and Vibration 16, no. 4 (2009): 365–87. http://dx.doi.org/10.1155/2009/387676.

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The dynamic behavior of linear and nonlinear mechanical oscillators with constitutive equations involving fractional derivatives defined as a fractional power of the operator of conventional time-derivative is considered. Such a definition of the fractional derivative enables one to analyse approximately vibratory regimes of the oscillator without considering the drift of its position of equilibrium. The assumption of small fractional derivative terms allows one to use the method of multiple time scales whereby a comparative analysis of the solutions obtained for different orders of low-level fractional derivatives and nonlinear elastic terms is possible to be carried out. The interrelationship of the fractional parameter (order of the fractional operator) and nonlinearity manifests itself in full measure when orders of the small fractional derivative term and of the cubic nonlinearity entering in the oscillator's constitutive equation coincide.

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Tenreiro Machado, J. A. "Fractional Coins and Fractional Derivatives." Abstract and Applied Analysis 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/205097.

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This paper discusses the fundamentals of negative probabilities and fractional calculus. The historical evolution and the main mathematical concepts are discussed, and several analogies between the two apparently unrelated topics are established. Based on the new conceptual perspective, some experiments are performed shading new light into possible future progress.

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Diethelm, Kai, Roberto Garrappa, Andrea Giusti, and Martin Stynes. "Why fractional derivatives with nonsingular kernels should not be used." Fractional Calculus and Applied Analysis 23, no. 3 (June 25, 2020): 610–34. http://dx.doi.org/10.1515/fca-2020-0032.

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AbstractIn recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.

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24

Hu, Shaoxiang, and Ping Liang. "Theory Analysis of Left-Handed Grünwald-Letnikov Formula with0<α<1to Detect and Locate Singularities." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/157542.

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We study fractional-order derivatives of left-handed Grünwald-Letnikov formula with0<α<1to detect and locate singularities in theory. The widely used four types of ideal singularities are analyzed by deducing their fractional derivative formula. The local extrema of fractional derivatives are used to locate the singularities. Theory analysis indicates that fractional-order derivatives of left-handed Grünwald-Letnikov formula with0<α<1can detect and locate four types of ideal singularities correctly, which shows better performance than classical 1-order derivatives in theory.

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25

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 system, considering all of the aforementioned fractional operators. Our analysis covers multiagent systems with linear and nonlinear dynamics, affected by bounded external disturbances and described by fixed directed graphs. Lastly, examples, which are solved analytically and numerically, are presented to validate our contributions.

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26

Tarasov, Vasily E., and Elias C. Aifantis. "Toward fractional gradient elasticity." Journal of the Mechanical Behavior of Materials 23, no. 1-2 (May 1, 2014): 41–46. http://dx.doi.org/10.1515/jmbm-2014-0006.

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AbstractThe use of an extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe the power law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one dimension. The second involves the Riesz fractional derivative in three dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case, stress equilibrium in a Caputo elastic bar requires the existence of a nonzero internal body force to equilibrate it. In the second case, in a Riesz-type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.

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27

Fedorov, Vladimir E., Marina V. Plekhanova, and Elizaveta M. Izhberdeeva. "Initial Value Problems of Linear Equations with the Dzhrbashyan–Nersesyan Derivative in Banach Spaces." Symmetry 13, no. 6 (June 11, 2021): 1058. http://dx.doi.org/10.3390/sym13061058.

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Among the many different definitions of the fractional derivative, the Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan–Nersesyan fractional derivative, which generalizes the Riemann–Liouville and the Gerasimov–Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. The issues of the unique solvability of initial value problems for some classes of linear inhomogeneous equations of general form with the fractional Dzhrbashyan–Nersesyan derivative in Banach spaces are investigated. An inhomogeneous equation containing a bounded operator at the fractional derivative is considered, and the solution is presented using the Mittag–Leffler functions. The result obtained made it possible to study the initial value problems for a linear inhomogeneous equation with a degenerate operator at the fractional Dzhrbashyan–Nersesyan derivative in the case of relative p-boundedness of the operator pair from the equation. Abstract results were used to study a class of initial boundary value problems for equations with the time-fractional Dzhrbashyan–Nersesyan derivative and with polynomials in a self-adjoint elliptic differential operator with respect to spatial variables.

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Rehman, Hameed Ur, Maslina Darus, and Jamal Salah. "A Note on Caputo’s Derivative Operator Interpretation in Economy." Journal of Applied Mathematics 2018 (October 1, 2018): 1–7. http://dx.doi.org/10.1155/2018/1260240.

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We propound the economic idea in terms of fractional derivatives, which involves the modified Caputo’s fractional derivative operator. The suggested economic interpretation is based on a generalization of average count and marginal value of economic indicators. We use the concepts ofT-indicatorswhich analyses the economic performance with the presence of memory. The reaction of economic agents due to recurrence identical alteration is minimized by using the modified Caputo’s derivative operator of orderλinstead of integer order derivativen. The two sides of Caputo’s derivative are expressed by a brief time-line. The degree of attenuation is further depressed by involving the modified Caputo’s operator.

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Hu, Shaoxiang. "External Fractional-Order Gradient Vector Perona-Malik Diffusion for Sinogram Restoration of Low-Dosed X-Ray Computed Tomography." Advances in Mathematical Physics 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/516919.

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Existing fractional-order Perona-Malik Diffusion (FOPMD) algorithms are defined as fully spatial fractional-order derivatives (FSFODs). However, we argue that FSFOD is not the best way for diffusion since different parts of spatial derivative play different roles in Perona-Malik diffusion (PMD) and derivative orders should be decided according to their roles. Therefore, we adopt a novel fractional-order diffusion scheme, named external fractional-order gradient vector Perona-Malik diffusion (EFOGV-PMD), by only replacing integer-order derivatives of “external” gradient vector to their fractional-order counterparts while keeping integer-order derivatives of gradient vector for diffusion coefficients since the ability of edge indicator for 1-order derivative is demonstrated both in theory and applications. Here “external” indicates the spatial derivatives except for the derivatives used in diffusion coefficients. In order to demonstrate the power of the proposed scheme, some real sinograms of low-dosed computed tomography (LDCT) are used to compare the different performances. These schemes include PMD, regularized PMD (RPMD), and FOPMD. Experimental results show that the new scheme has good ability in edge preserving, is convergent quickly, has good stability for iteration number, and can avoid artifacts, dark resulting images, and speckle effect.

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Hilfer, Rudolf, and Yuri Luchko. "Desiderata for Fractional Derivatives and Integrals." Mathematics 7, no. 2 (February 4, 2019): 149. http://dx.doi.org/10.3390/math7020149.

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The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead they intend to stimulate the field by providing guidelines based on a small number of time honoured and well established criteria.

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Henriques, Manuel, Duarte Valério, Paulo Gordo, and Rui Melicio. "Fractional-Order Colour Image Processing." Mathematics 9, no. 5 (February 24, 2021): 457. http://dx.doi.org/10.3390/math9050457.

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Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative.

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Abboubakar, Hamadjam, Pushpendra Kumar, Vedat Suat Erturk, and Anoop Kumar. "A mathematical study of a tuberculosis model with fractional derivatives." International Journal of Modeling, Simulation, and Scientific Computing 12, no. 04 (March 26, 2021): 2150037. http://dx.doi.org/10.1142/s1793962321500379.

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In this work, we use a Predictor–Corrector method to implement and derive an iterative solution of an existing Tuberculosis (TB) model with two fractional derivatives, namely, Caputo–Fabrizio fractional derivative and the new generalized Caputo fractional derivative. We begin by recalling some existing results such as the basic reproduction number [Formula: see text] and the equilibrium points of the model. Then, we study the global asymptotic stability of disease-free equilibrium of the fractional models. We also prove, for each fractional model, the existence and uniqueness of solutions. An iterative solution of the two models is computed using the Predictor–Corrector method. Using realistic parameter values, we perform numerical simulations for different values of the fractional order. Simulation results permit to conclude that the new generalized Caputo fractional derivative provides a more realistic analysis than the Caputo–Fabrizio fractional derivative and the classical integer-order TB model.

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KUMAR, SACHIN, PRASHANT PANDEY, J. F. GÓMEZ-AGUILAR, and D. BALEANU. "DOUBLE-QUASI-WAVELET NUMERICAL METHOD FOR THE VARIABLE-ORDER TIME FRACTIONAL AND RIESZ SPACE FRACTIONAL REACTION–DIFFUSION EQUATION INVOLVING DERIVATIVES IN CAPUTO–FABRIZIO SENSE." Fractals 28, no. 08 (September 18, 2020): 2040047. http://dx.doi.org/10.1142/s0218348x20400472.

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Our motive in this scientific contribution is to deal with nonlinear reaction–diffusion equation having both space and time variable order. The fractional derivatives which are used are non-singular having exponential kernel. These derivatives are also known as Caputo–Fabrizio derivatives. In our model, time fractional derivative is Caputo type while spatial derivative is variable-order Riesz fractional type. To approximate the variable-order time fractional derivative, we used a difference scheme based upon the Taylor series formula. While approximating the variable order spatial derivatives, we apply the quasi-wavelet-based numerical method. Here, double-quasi-wavelet numerical method is used to investigate this type of model. The discretization of boundary conditions with the help of quasi-wavelet is discussed. We have depicted the efficiency and accuracy of this method by solving the some particular cases of our model. The error tables and graphs clearly show that our method has desired accuracy.

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NABER, MARK. "DISTRIBUTED ORDER FRACTIONAL SUB-DIFFUSION." Fractals 12, no. 01 (March 2004): 23–32. http://dx.doi.org/10.1142/s0218348x04002410.

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A distributed order fractional diffusion equation is considered. Distributed order derivatives are fractional derivatives that have been integrated over the order of the derivative within a given range. In this paper, sub-diffusive cases are considered. That is, the order of the time derivative ranges from zero to one. The equation is solved for Dirichlet, Neumann and Cauchy boundary conditions. The time dependence for each of the three cases is found to be a functional of the diffusion parameter. This functional is shown to have decay properties. Upper and lower bounds are computed for the functional. Examples are also worked out for comparative decay rates.

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Golmankhaneh, Alireza Khalili, and Renat Timergalievich Sibatov. "Fractal Stochastic Processes on Thin Cantor-Like Sets." Mathematics 9, no. 6 (March 15, 2021): 613. http://dx.doi.org/10.3390/math9060613.

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We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.

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TARASOV, VASILY E. "COMMENTS ON "RIEMANN–CHRISTOFFEL TENSOR IN DIFFERENTIAL GEOMETRY OF FRACTIONAL ORDER APPLICATION TO FRACTAL SPACE-TIME", [FRACTALS 21 (2013) 1350004]." Fractals 23, no. 02 (May 28, 2015): 1575001. http://dx.doi.org/10.1142/s0218348x15750018.

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We prove that main properties represented by Eq. (4.2) for fractional derivative of power function and the non-fractional Leibniz rule in the form (4.3) of the considered paper, cannot hold together for derivatives of non-integer order. As a result, we prove that the usual Leibniz rule (4.3) cannot hold for fractional derivatives.

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Oliveira, Daniela S., and Edmundo Capelas de Oliveira. "On a Caputo-type fractional derivative." Advances in Pure and Applied Mathematics 10, no. 2 (April 1, 2019): 81–91. http://dx.doi.org/10.1515/apam-2017-0068.

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Abstract In this paper, we present a new differential operator of arbitrary order defined by means of a Caputo-type modification of the generalized fractional derivative recently proposed by Katugampola. The generalized fractional derivative, when convenient limits are considered, recovers the Riemann–Liouville and the Hadamard derivatives of arbitrary order. Our differential operator recovers as limiting cases the arbitrary order derivatives proposed by Caputo and by Caputo–Hadamard. Some properties are presented as well as the relation between this differential operator of arbitrary order and the Katugampola generalized fractional operator. As an application we prove the fundamental theorem of fractional calculus associated with our operator.

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Liu, Yin-Ping, and Zhi-Bin Li. "Homotopy Analysis Method for Nonlinear Differential Equations with Fractional Orders." Zeitschrift für Naturforschung A 63, no. 5-6 (June 1, 2008): 241–47. http://dx.doi.org/10.1515/zna-2008-5-602.

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The aim of this paper is to solve nonlinear differential equations with fractional derivatives by the homotopy analysis method. The fractional derivative is described in Caputo’s sense. It shows that the homotopy analysis method not only is efficient for classical differential equations, but also is a powerful tool for dealing with nonlinear differential equations with fractional derivatives.

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Salem, Hussein A. H. "On the Theory of Fractional Calculus in the Pettis-Function Spaces." Journal of Function Spaces 2018 (2018): 1–13. http://dx.doi.org/10.1155/2018/8746148.

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Throughout this paper, we outline some aspects of fractional calculus in Banach spaces. Some examples are demonstrated. In our investigations, the integrals and the derivatives are understood as Pettis integrals and the corresponding derivatives. Our results here extended all previous contributions in this context and therefore are new. To encompass the full scope of our paper, we show that a weakly continuous solution of a fractional order integral equation, which is modeled off some fractional order boundary value problem (where the derivatives are taken in the usual definition of the Caputo fractional weak derivative), may not solve the problem.

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Włodarczyk, M., and A. Zawadzki. "The application of hypergeometric functions to computing fractional order derivatives of sinusoidal functions." Bulletin of the Polish Academy of Sciences Technical Sciences 64, no. 1 (March 1, 2016): 243–48. http://dx.doi.org/10.1515/bpasts-2016-0026.

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Abstract In the paper, the analytical forms of fractional order derivatives of sinusoidal function according to definitions of Riemann - Liouville and Caputo are presented. To determine the analytical form of the integrals appearing in definitions of derivatives of fractional order the Lommel functions from hypergeometrical functions family were applied. With the use of properties of the derivatives of fractional order - differential-integral there were presented the conception of generalized element of a single equation, which depending on the value of the derivative order, can be inductor, resistor, capacitor, or a hypothetical element of a fractional order differential equation.

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Chyzhykov, I., and Yu Kosaniak. "Asymptotic Behavior of Fractional Derivatives of Bounded Analytic Functions." Zurnal matematiceskoj fiziki, analiza, geometrii 13, no. 2 (June 25, 2017): 107–18. http://dx.doi.org/10.15407/mag13.02.107.

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Aguilar, J. F. Gómez, T. Córdova-Fraga, J. Tórres-Jiménez, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, and G. V. Guerrero-Ramírez. "Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation." Mathematical Problems in Engineering 2016 (2016): 1–15. http://dx.doi.org/10.1155/2016/7845874.

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The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range(0,2]. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to1.

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Popović, Jovan K., Stevan Pilipović, and Teodor M. Atanacković. "Two compartmental fractional derivative model with fractional derivatives of different order." Communications in Nonlinear Science and Numerical Simulation 18, no. 9 (September 2013): 2507–14. http://dx.doi.org/10.1016/j.cnsns.2013.01.004.

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Hu, Langhua, Duan Chen, and Guo-Wei Wei. "High-order fractional partial differential equation transform for molecular surface construction." Computational and Mathematical Biophysics 1 (December 20, 2012): 1–25. http://dx.doi.org/10.2478/mlbmb-2012-0001.

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AbstractFractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

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Elbeleze, Asma Ali, Adem Kılıçman, and Bachok M. Taib. "Fractional Variational Iteration Method and Its Application to Fractional Partial Differential Equation." Mathematical Problems in Engineering 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/543848.

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We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations in fluid mechanics and in financial models. The fractional derivatives are described in Riemann-Liouville sense. To show the efficiency of the considered method, some examples that include the fractional Klein-Gordon equation, fractional Burgers equation, and fractional Black-Scholes equation are investigated.

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Fu, Jing-Li, Lijun Zhang, Chaudry Khalique, and Ma-Li Guo. "Motion equations and non-Noether symmetries of Lagrangian systems with conformable fractional derivative." Thermal Science 25, no. 2 Part B (2021): 1365–72. http://dx.doi.org/10.2298/tsci200520035f.

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In this paper, we present the fractional motion equations and fractional non-Noether symmetries of Lagrangian systems with the conformable fractional derivatives. The exchanging relationship between isochronous variation and fractional derivative, and the fractional Hamilton?s principle of the holonomic conservative and non-conservative systems under the conformable fractional derivative are proposed. Then the fractional motion equations of these systems based on the Hamil?ton?s principle are established. The fractional Euler operator, the definition of fractional non-Noether symmetries, non-Noether theorem, and Hojman?s conserved quantities for the Lagrangian systems are obtained with conformable fractional derivative. An example is given to illustrate the results.

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Cai, Min, and Changpin Li. "On Riesz derivative." Fractional Calculus and Applied Analysis 22, no. 2 (April 24, 2019): 287–301. http://dx.doi.org/10.1515/fca-2019-0019.

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Abstract This paper focuses on studying Riesz derivative. An interesting investigation on properties of Riesz derivative in one dimension indicates that it is distinct from other fractional derivatives such as Riemann-Liouville derivative and Caputo derivative. In the existing literatures, Riesz derivative is commonly considered as a proxy for fractional Laplacian on ℝ. We show the equivalence between Riesz derivative and fractional Laplacian on ℝn with n ≥ 1 in details.

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Mittal, A. K. "Spectrally accurate approximate solutions and convergence analysis of fractional Burgers’ equation." Arabian Journal of Mathematics 9, no. 3 (June 24, 2020): 633–44. http://dx.doi.org/10.1007/s40065-020-00286-x.

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Abstract In this paper, a new numerical technique implements on the time-space pseudospectral method to approximate the numerical solutions of nonlinear time- and space-fractional coupled Burgers’ equation. This technique is based on orthogonal Chebyshev polynomial function and discretizes using Chebyshev–Gauss–Lobbato (CGL) points. Caputo–Riemann–Liouville fractional derivative formula is used to illustrate the fractional derivatives matrix at CGL points. Using the derivatives matrices, the given problem is reduced to a system of nonlinear algebraic equations. These equations can be solved using Newton–Raphson method. Two model examples of time- and space-fractional coupled Burgers’ equation are tested for a set of fractional space and time derivative order. The figures and tables show the significant features, effectiveness, and good accuracy of the proposed method.

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Trencevski, Kostadin, and Zivorad Tomovski. "On some fractional derivatives of functions of exponential type." Publikacije Elektrotehnickog fakulteta - serija: matematika, no. 13 (2002): 77–84. http://dx.doi.org/10.2298/petf0213077t.

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In this paper a new proof of the well known fact that the derivative of e " of order ? ? R is equal to ??e?x is given. It enables to conclude that sin(?)(x) = sin(x + ??/2) and cos(?)(x) = cos (x + ??/2) which is initial assumption (axiom) for the classical theory of fractional derivatives. Namely we use a new method for calculation of fractional derivatives of functions of exponential type.

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Gómez-Aguilar, José Francisco, and Baleanu Dumitru. "Fractional Transmission Line with Losses." Zeitschrift für Naturforschung A 69, no. 10-11 (November 1, 2014): 539–46. http://dx.doi.org/10.5560/zna.2014-0049.

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AbstractIn this manuscript, the fractional transmission line with losses is presented. The order of the Caputo derivative is considered as 0 < β ≤ 1 and 0 < γ ≤ 1 for the fractional equation in space and time domain, respectively. Two cases are solved, with fractional spatial and fractional temporal derivatives, and also numerical simulations were carried out, where there are taken both derivatives in simultaneous form. Two parameters σx and σt are introduced and a physical relation between these parameters is reported. Solutions in space and time are given in terms of the Mittag-Leffler functions. The classic cases are recovered when β and γ are equal to 1.

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