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Journal articles on the topic 'First and second fractional derivative test'

Author: Grafiati

Published: 5 June 2025

Last updated: 15 July 2025

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1

Chii-Huei, Yu. "Study on Some Properties of Fractional Analytic Function." International Journal of Mechanical and Industrial Technology 10, no. 1 (2022): 31–35. https://doi.org/10.5281/zenodo.7016567.

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<strong>Abstract:</strong> In this paper, based on Jumarie’s modified Riemann-Liouville (R-L) fractional derivative, we study some properties of fractional analytic function, such as fractional Taylor’s theorem, first fractional derivative test, and second fractional derivative test. The major methods used in this paper are fractional Rolle’s theorem, fractional mean value theorem, product rule for fractional derivatives, and a new multiplication of fractional analytic functions. In fact, these new results are generalizations of those results in ordinary calculus. <strong>Key

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2

Fu, Cheng-Biao, Hei-Gang Xiong, and An-Hong Tian. "Study on the Effect of Fractional Derivative on the Hyperspectral Data of Soil Organic Matter Content in Arid Region." Journal of Spectroscopy 2019 (February 3, 2019): 1–11. http://dx.doi.org/10.1155/2019/7159317.

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Discussion on the application of fractional derivative algorithm in monitoring organic matter content in field soil is scarce. This study is aimed at improving the accuracy of soil organic matter (SOM) content estimation in arid region, and the undesirable model precision caused by the missing information associated with the larger discrepancy between conventional integer-order, i.e., first order and second order, derivative, and raw spectral data. We utilized fractional derivative (of zeroth order to second order in 0.2-order interval) processing on the field spectral reflectance (R) of the s

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3

Syam, Sondos M., Z. Siri, Sami H. Altoum, and R. Md. Kasmani. "Analytical and Numerical Methods for Solving Second-Order Two-Dimensional Symmetric Sequential Fractional Integro-Differential Equations." Symmetry 15, no. 6 (2023): 1263. http://dx.doi.org/10.3390/sym15061263.

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In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate the implementation of numerical and analytical approaches. Then, we propose a numerical approach based on the operational matrix method, which involves deriving operational matrices for the differential and integral terms of the equation and combining them to generate a sin

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Ma, Ruiqun, Min Ni, Quanlong Chen, Yinjia Zhou, and Jinglong Han. "Viscoelastic Fractional Model Based on Harmonic Excitation." Mathematical Problems in Engineering 2022 (July 18, 2022): 1–13. http://dx.doi.org/10.1155/2022/4799387.

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This paper analyzes the viscoelastic fractional constitutive model based on the analytical form of the fractional derivative of the sine (cosine) function. First, the polynomials of the linear model are combined into a fractional derivative term, then the complex modulus can be easily obtained, and the response characteristics of the model can be analyzed. Second, the fractional nonlinear model is expressed in harmonic form, which can easily identify nonlinear parameters. Finally, fractional-order nonlinear models can also be transformed into integer-order nonlinear forms. In this paper, the t

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Molavi-Arabshahi, Mahboubeh, and Zahra Saeidi. "Application of Compact Finite Difference Method for Solving Some Type of Fractional Derivative Equations." International Journal of Circuits, Systems and Signal Processing 15 (September 6, 2021): 1324–35. http://dx.doi.org/10.46300/9106.2021.15.143.

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In this paper, the compact finite difference scheme as unconditionally stable method is applied to some type of fractional derivative equation. We intend to solve with this scheme two kinds of a fractional derivative, first a fractional order system of Granwald-Letnikov type 1 for influenza and second fractional reaction sub diffusion equation. Also, we analyzed the stability of equilibrium points of this system. The convergence of the compact finite difference scheme in norm 2 are proved. Finally, various cases are used to test the numerical method. In comparison to other existing numerical m

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6

CAFAGNA, DONATO, and GIUSEPPE GRASSI. "FRACTIONAL-ORDER CHUA'S CIRCUIT: TIME-DOMAIN ANALYSIS, BIFURCATION, CHAOTIC BEHAVIOR AND TEST FOR CHAOS." International Journal of Bifurcation and Chaos 18, no. 03 (2008): 615–39. http://dx.doi.org/10.1142/s0218127408020550.

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In this tutorial the chaotic behavior of the fractional-order Chua's circuit is investigated from the time-domain point of view. The objective is achieved using the Adomian decomposition method, which enables the solution of the fractional differential equations to be found in closed form. By exploiting the capabilities offered by the decomposition method, the paper presents two remarkable findings. The first result is that a novel bifurcation parameter is identified, that is, the fractional-order q of the derivative. The second result is that chaos exists in the fractional Chua's circuit with

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7

Bellabaci, Safia, and Bachir Dehda. "Laplacian edge detection method based on a new approximation of the second-order fractional derivative in the Caputo-Fabrizio sense." STUDIES IN ENGINEERING AND EXACT SCIENCES 5, no. 2 (2024): e8986. http://dx.doi.org/10.54021/seesv5n2-317.

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Edge detection is an important issue in image processing and computer vision problems. It includes all mathematical methods that aim to identify the discontinuous points in the image. This process leads to construct curves that indicate the boundaries of an object and therefore extract feature information. In recent years, this task has received great attention from researchers and scientists, as we find in the literature many methods with their different algorithms. Up to now, there are two methods, the gradient and Laplacian methods. The first depends on the first derivative, where pixels wi

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8

Tabak, Abdulsamed. "A novel fractional order PID plus derivative (PIλDµDµ2) controller for AVR system using equilibrium optimizer". COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 40, № 3 (2021): 722–43. http://dx.doi.org/10.1108/compel-02-2021-0044.

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Purpose The purpose of this paper is to improve transient response and dynamic performance of automatic voltage regulator (AVR). Design/methodology/approach This paper proposes a novel fractional order proportional–integral–derivative plus derivative (PIλDµDµ2) controller called FOPIDD for AVR system. The FOPIDD controller has seven optimization parameters and the equilibrium optimizer algorithm is used for tuning of controller parameters. The utilized objective function is widely preferred in AVR systems and consists of transient response characteristics. Findings In this study, results of AV

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9

Hayat, Afzaal Mubashir, Muhammad Bilal Riaz, Muhammad Abbas, Moataz Alosaimi, Adil Jhangeer, and Tahir Nazir. "Numerical Solution to the Time-Fractional Burgers–Huxley Equation Involving the Mittag-Leffler Function." Mathematics 12, no. 13 (2024): 2137. http://dx.doi.org/10.3390/math12132137.

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Fractional differential equations play a significant role in various scientific and engineering disciplines, offering a more sophisticated framework for modeling complex behaviors and phenomena that involve multiple independent variables and non-integer-order derivatives. In the current research, an effective cubic B-spline collocation method is used to obtain the numerical solution of the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. It is implemented with the help of a θ-weighted scheme to solve the proposed problem. The spatial derivative is interpolated using cubic B-spl

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Yee, Jennifer C., and Andrew P. Gould. "An Alternate Method for Minimizing X ²." Publications of the Astronomical Society of the Pacific 137, no. 5 (2025): 054501. https://doi.org/10.1088/1538-3873/adcb6c.

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Abstract In this paper, we describe an algorithm and associated software package (sfit_minimize) for maximizing the likelihood function of a set of parameters by minimizing χ 2. The algorithm estimates the second derivative of the χ 2 function using first derivatives of the function to be fitted and takes fractional, ϵ ≤ 1, steps at each iteration. The derivatives can also be used to calculate the uncertainties in each parameter. We test this algorithm against several standard minimization algorithms in SciPy.optimize.minimize() by fitting point lens models to light curves from the 2018 Korea

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Izadi, Mohammad. "Approximate Solutions for Solving Fractional-order Painlevé Equations." Contemporary Mathematics 1, no. 1 (2019): 12–24. http://dx.doi.org/10.37256/cm.11201947.12-24.

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In this work, Chebyshev orthogonal polynomials are employed as basis functions in the collocation scheme to solve the nonlinear Painlevé initial value problems known as the first and second Painlevé equations. Using the collocation points, representing the solution and its fractional derivative (in the Caputo sense) in matrix forms, and the matrix operations, the proposed technique transforms a solution of the initial-value problem for the Painlevé equations into a system of nonlinear algebraic equations. To get ride of nonlinearlity, the technique of quasi-linearization is also applied, which

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Behl, Ramandeep, M. Salimi, M. Ferrara, S. Sharifi, and Samaher Alharbi. "Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme." Symmetry 11, no. 2 (2019): 239. http://dx.doi.org/10.3390/sym11020239.

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In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrate

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13

Zafar, Farhan, Suheel Abdullah Malik, Tayyab Ali, et al. "Stabilization and tracking control of underactuated ball and beam system using metaheuristic optimization based TID-F and PIDD2–PI control schemes." PLOS ONE 19, no. 2 (2024): e0298624. http://dx.doi.org/10.1371/journal.pone.0298624.

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In this paper, we propose two different control strategies for the position control of the ball of the ball and beam system (BBS). The first control strategy uses the proportional integral derivative-second derivative with a proportional integrator PIDD2-PI. The second control strategy uses the tilt integral derivative with filter (TID-F). The designed controllers employ two distinct metaheuristic computation techniques: grey wolf optimization (GWO) and whale optimization algorithm (WOA) for the parameter tuning. We evaluated the dynamic and steady-state performance of the proposed control str

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14

Voevoda, A., V. Shipagin, and K. Bobobekov. "Control device synthesis by polynomial matrix fractional descriptions method with limitation on controller structure." Journal of Physics: Conference Series 2131, no. 3 (2021): 032021. http://dx.doi.org/10.1088/1742-6596/2131/3/032021.

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Abstract Modification of the algorithm for the polynomial synthesis of a multi-channel controller was proposed to preserve all control channels in this article. In order to test the functionality of the proposed modification, an example of a linear model of an unstable multi-channel plant is considered. The choice of the plant was determined by the possibility of a visual algorithm demonstration for polynomial synthesis of the controller, taking into account the proposed modifications.The plant was represented as three series-connected standard links: an aperiodic link of the first order, an u

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15

Fathy, Ahmed, Hegazy Rezk, Seydali Ferahtia, Rania M. Ghoniem, Reem Alkanhel, and Mohamed M. Ghoniem. "A New Fractional-Order Load Frequency Control for Multi-Renewable Energy Interconnected Plants Using Skill Optimization Algorithm." Sustainability 14, no. 22 (2022): 14999. http://dx.doi.org/10.3390/su142214999.

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Connection between electric power networks is essential to cover any deficit in the generation of power from any of them. The exchange powers of the plants during load disturbance should not be violated beyond their specified values. This can be achieved by installing load frequency control (LFC); therefore, this paper proposes a new metaheuristic-based approach using a skill optimization algorithm (SOA) to design a fractional-order proportional integral derivative (FOPID)-LFC approach with multi-interconnected systems. The target is minimizing the integral time absolute error (ITAE) of freque

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16

Ali, Mushtaq Salh, Mostafa Shamsi, Hassan Khosravian-Arab, Delfim F. M. Torres, and Farid Bozorgnia. "A space–time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives." Journal of Vibration and Control 25, no. 5 (2018): 1080–95. http://dx.doi.org/10.1177/1077546318811194.

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We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi–Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer–order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are pr

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17

Rubin, R. H., I. A. McNabb, J. P. Simpson, et al. "Spitzer finds cosmic neon's and sulfur's sweet spot: part III, NGC 6822." Proceedings of the International Astronomical Union 5, S265 (2009): 249–50. http://dx.doi.org/10.1017/s1743921310000682.

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AbstractWe observed several H ii regions in the dwarf irregular galaxy NGC 6822 using the infrared spectrograph on the Spitzer Space Telescope. Our aim is twofold: first, to examine the neon to sulfur abundance ratio in order to determine how much it may vary and whether or not, it is fairly ‘universal’; second, to discriminate and test the predicted ionizing spectral energy distribution between various stellar atmosphere models by comparing with our derivation of the ratio of fractional ionizations involving neon and sulfur. This work extends our previous similar studies of H ii regions in M8

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18

Kamran, Shahzad Khan, Sharifah E. Alhazmi, Fahad M. Alotaibi, Massimiliano Ferrara, and Ali Ahmadian. "On the Numerical Approximation of Mobile-Immobile Advection-Dispersion Model of Fractional Order Arising from Solute Transport in Porous Media." Fractal and Fractional 6, no. 8 (2022): 445. http://dx.doi.org/10.3390/fractalfract6080445.

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The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have been applied to many physical and engineering problems because of its simplicity in implementation and its superiority in solving different real-world problems easily. In this article, we propose an efficient local RBFs method coupled with Laplace transform (LT) for approximating the solution of fractional mobile/immobile solute transport model in the sense of Caputo derivative. In our meth

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Chilyabanyama, Obvious Nchimunya, Roma Chilengi, Natasha Makabilo Laban, et al. "Comparing growth velocity of HIV exposed and non-exposed infants: An observational study of infants enrolled in a randomized control trial in Zambia." PLOS ONE 16, no. 8 (2021): e0256443. http://dx.doi.org/10.1371/journal.pone.0256443.

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Background Impaired growth among infants remains one of the leading nutrition problems globally. In this study, we aimed to compare the growth trajectory rate and evaluate growth trajectory characteristics among children, who are HIV exposed uninfected (HEU) and HIV unexposed uninfected (HUU), under two years in Zambia. Method Our study used data from the ROVAS II study (PACTR201804003096919), an open-label randomized control trial of two verses three doses of live, attenuated, oral RotarixTM administered 6 &10 weeks or at 6 &10 weeks plus an additional dose at 9 months of age, conduct

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20

Kamal, Raheel, Kamran, Saleh Alzahrani, and Talal Alzahrani. "A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives." Fractal and Fractional 7, no. 5 (2023): 381. http://dx.doi.org/10.3390/fractalfract7050381.

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This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0<α<1, and 1<β<2. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to o

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Bokam, Jagadish Kumar, Naresh Patnana, Tarun Varshney, and Vinay Pratap Singh. "Sine Cosine Algorithm Assisted FOPID Controller Design for Interval Systems Using Reduced-Order Modeling Ensuring Stability." Algorithms 13, no. 12 (2020): 317. http://dx.doi.org/10.3390/a13120317.

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The focus of present research endeavor was to design a robust fractional-order proportional-integral-derivative (FOPID) controller with specified phase margin (PM) and gain cross over frequency (ωgc) through the reduced-order model for continuous interval systems. Currently, this investigation is two-fold: In the first part, a modified Routh approximation technique along with the matching Markov parameters (MPs) and time moments (TMs) are utilized to derive a stable reduced-order continuous interval plant (ROCIP) for a stable high-order continuous interval plant (HOCIP). Whereas in the second

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Qayyum, Mubashir, Farnaz Ismail, Muhammad Sohail, Naveed Imran, Sameh Askar, and Choonkil Park. "Numerical exploration of thin film flow of MHD pseudo-plastic fluid in fractional space: Utilization of fractional calculus approach." Open Physics 19, no. 1 (2021): 710–21. http://dx.doi.org/10.1515/phys-2021-0081.

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Abstract In this article, thin film flow of non-Newtonian pseudo-plastic fluid is investigated on a vertical wall through homotopy-based scheme along with fractional calculus. Three cases were examined after considering (i) partial fractional differential equation (PFDE) by altering first-order derivative to fractional derivative in the interval (0, 1), (ii) PFDE by altering second-order derivative to fractional derivative in the interval (1, 2), and (iii) fully FDE by altering first-order derivative to fractional derivative in (0, 1) and second-order derivative to fractional derivative in (1,

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Xu, Bo, Yufeng Zhang, and Sheng Zhang. "Fractional Rogue Waves with Translational Coordination, Steep Crest, and Modified Asymmetry." Complexity 2021 (April 20, 2021): 1–14. http://dx.doi.org/10.1155/2021/6669087.

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To construct fractional rogue waves, this paper first introduces a conformable fractional partial derivative. Based on the conformable fractional partial derivative and its properties, a fractional Schrödinger (NLS) equation with Lax integrability is then derived and first- and second-order fractional rogue wave solutions of which are finally obtained. The obtained fractional rogue wave solutions possess translational coordination, providing, to some extent, the degree of freedom to adjust the position of the rogue waves on the coordinate plane. It is shown that the obtained first- and second-

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Soni, R. C., and Deepika Singh. "The unified Riemann-Liouville fractional derivative formulae." Tamkang Journal of Mathematics 36, no. 3 (2005): 231–36. http://dx.doi.org/10.5556/j.tkjm.36.2005.115.

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In this paper, we obtain two unified fractional derivative formulae. The first involves the product of two general class of polynomials and the multivariable $H$-function. The second fractional derivative formula also involves the product of two general class of polynomials and the multivariable $H$-function and has been obtained by the application of the first fractional derivative formula twice and it has two independent variables instead of one. The polynomials and the functions involved in both the fractional derivative formulae as well as their arguments are quite general in nature and so

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Tarasov, Vasily E., and Elias C. Aifantis. "Toward fractional gradient elasticity." Journal of the Mechanical Behavior of Materials 23, no. 1-2 (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

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

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

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Alipour, Mohsen, and Dumitru Baleanu. "Approximate Analytical Solution for Nonlinear System of Fractional Differential Equations by BPs Operational Matrices." Advances in Mathematical Physics 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/954015.

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We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs). In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD), and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI). The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the soluti

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LING, LEEVAN, and MASAHIRO YAMAMOTO. "NUMERICAL SIMULATIONS FOR SPACE–TIME FRACTIONAL DIFFUSION EQUATIONS." International Journal of Computational Methods 10, no. 02 (2013): 1341001. http://dx.doi.org/10.1142/s0219876213410016.

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We consider the solutions of a space–time fractional diffusion equation on the interval [-1, 1]. The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann–Liouville fractional derivative of order between one and two, and the first-order time derivative by a Caputo fractional derivative of order between zero and one. As the fundamental solution of this fractional equation is unknown (if exists), an eigenfunction approach is applied to obtain approximate fundamental solutions which are then used to solve the space–time fractional di

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Luchko, Yuri. "On Complete Monotonicity of Solution to the Fractional Relaxation Equation with the nth Level Fractional Derivative." Mathematics 8, no. 9 (2020): 1561. http://dx.doi.org/10.3390/math8091561.

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In this paper, we first deduce the explicit formulas for the projector of the nth level fractional derivative and for its Laplace transform. Afterwards, the fractional relaxation equation with the nth level fractional derivative is discussed. It turns out that, under some conditions, the solutions to the initial-value problems for this equation are completely monotone functions that can be represented in form of the linear combinations of the Mittag–Leffler functions with some power law weights. Special attention is given to the case of the relaxation equation with the second level derivative.

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Volosova, Natalya K., Konstantin A. Volosov, Aleksandra K. Volosova, Mikhail I. Karlov, Dmitriy F. Pastukhov та Yuriy F. Pastukhov. "Модифицированная формула Герасимова–Капуто". Вестник Пермского университета. Математика. Механика. Информатика, № 1 (64) (2024): 5–14. http://dx.doi.org/10.17072/1993-0550-2024-1-5-14.

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In this work, modified Gerasimov–Caputo formulas were obtained for the first time. The modified formulas take into account the value of the derivative of the function at zero with an order of one less than the order of the derivative under the sign of the Gerasimov–Caputo integral. Without taking into account the new term in the Gerasimov–Caputo formulas, it is not always possible to calculate the fractional derivative on any order interval and for any function. The paper also describes a simple numerical algorithm with the Gaussian quadrature formula, which allows one to calculate the fractio

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Alsharari, Fahad, Raouf Fakhfakh, and Abdelghani Lakhdari. "On Fractal–Fractional Simpson-Type Inequalities: New Insights and Refinements of Classical Results." Mathematics 12, no. 24 (2024): 3886. https://doi.org/10.3390/math12243886.

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In this paper, we introduce a novel fractal–fractional identity, from which we derive new Simpson-type inequalities for functions whose first-order local fractional derivative exhibits generalized s-convexity in the second sense. This work introduces an approach that uses the first-order local fractional derivative, enabling the treatment of functions with lower regularity requirements compared to earlier studies. Additionally, we present two distinct methodological frameworks, one of which achieves greater precision by refining classical outcomes in the existing literature. The paper conclude

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Sagar Hassan, Muhammad Amin, and Saima Mushtaq. "APPLICATION OF A HYBRID B-SPLINE METHOD FOR THE NUMERICAL ANALYSIS OF TIME-FRACTIONAL DIFFUSION WAVE EQUATION." Kashf Journal of Multidisciplinary Research 2, no. 06 (2025): 104–26. https://doi.org/10.71146/kjmr515.

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In this article, approximate solutions of Time-Fractional Diffusion Wave Equation has been investigated using a hybrid cubic B-spline technique with finite difference scheme. For the discretization of time fractional derivative Caputo-Fabrizo formula is employed. To get the numerical out comes the Caputo-Fabrizo fractional derivative and a hybrid cubic B-spline strategy is delved. The presented method is proved to be unconditionally stable with a second order convergence. The proposed scheme is validated using some test problems, demonstrating its feasibility and reasonable accuracy. Numerical

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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 di

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Yang, Yong-Ge, Wei Xu, YangQuan Chen, and Bingchang Zhou. "Bifurcation Analysis of a Vibro-Impact Viscoelastic Oscillator with Fractional Derivative Element." International Journal of Bifurcation and Chaos 28, no. 14 (2018): 1850170. http://dx.doi.org/10.1142/s0218127418501705.

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To the best of authors’ knowledge, little work has been focused on the noisy vibro-impact systems with fractional derivative element. In this paper, stochastic bifurcation of a vibro-impact oscillator with fractional derivative element and a viscoelastic term under Gaussian white noise excitation is investigated. First, the viscoelastic force is approximately replaced by damping force and stiffness force. Thus the original oscillator is converted to an equivalent oscillator without a viscoelastic term. Second, the nonsmooth transformation is introduced to remove the discontinuity of the vibro-

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Ji, Cuicui, and Zhizhong Sun. "An Unconditionally Stable and High-Order Convergent Difference Scheme for Stokes’ First Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative." Numerical Mathematics: Theory, Methods and Applications 10, no. 3 (2017): 597–613. http://dx.doi.org/10.4208/nmtma.2017.m1605.

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AbstractThis article is intended to fill in the blank of the numerical schemes with second-order convergence accuracy in time for nonlinear Stokes’ first problem for a heated generalized second grade fluid with fractional derivative. A linearized difference scheme is proposed. The time fractional-order derivative is discretized by second-order shifted and weighted Gr¨unwald-Letnikov difference operator. The convergence accuracy in space is improved by performing the average operator. The presented numerical method is unconditionally stable with the global convergence order of in maximum norm,

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Atanacković, Teodor M., Stevan Pilipović, and Dušan Zorica. "Properties of the Caputo-Fabrizio fractional derivative and its distributional settings." Fractional Calculus and Applied Analysis 21, no. 1 (2018): 29–44. http://dx.doi.org/10.1515/fca-2018-0003.

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Abstract The Caputo-Fabrizio fractional derivative is analyzed in classical and distributional settings. The integral inequalities needed for application in linear viscoelasticity are presented. They are obtained from the entropy inequality in a weak form. Moreover, integration by parts, an expansion formula, approximation formula and generalized variational principles of Hamilton’s type are given. Hamilton’s action integral in the first principle, do not coincide with the lower bound in the fractional integral, while in the second principle the minimization is performed with respect to a func

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Awadalla, Muath, Yves Yameni Noupoue Yannick, and Kinda Abu Asbeh. "Modeling the Dependence of Barometric Pressure with Altitude Using Caputo and Caputo–Fabrizio Fractional Derivatives." Journal of Mathematics 2020 (November 24, 2020): 1–9. http://dx.doi.org/10.1155/2020/2417681.

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This work is dedicated to the study of the relationship between altitude and barometric atmospheric pressure. There is a consistent literature on this relationship, out of which an ordinary differential equation with initial value problems is often used for modeling. Here, we proposed a new modeling technique of the relationship using Caputo and Caputo–Fabrizio fractional differential equations. First, the proposed model is proven well-defined through existence and uniqueness of its solution. Caputo–Fabrizio fractional derivative is the main tool used throughout the proof. Then, follow experim

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Albalawi, Kholoud Saad, and Badr Saad T. Alkahtani. "Piecewise Fractional Analysis of Omicron Type Covid-19 Infection." Pakistan Journal of Medical and Health Sciences 16, no. 7 (2022): 851–54. http://dx.doi.org/10.53350/pjmhs22167851.

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In this article, we investigate the new type of COVID-19 caused by the Omicron virus in the sense of piece-wise fractional derivative. The total interval is divided into two sub-intervals under fractional Caputo and Atangana Baleanu operators respectively. The whole model is divided into six compartments in which the agent of infection from Omicron is included. The proposed piecewise fractional model is tested for fixed points using the different theorems of fixed point theory. The approximate solution is carried out by the technique of the piecewise fractional Adams-Bashforth method. All the

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Wu, Cailian, Congcong Wei, Zhe Yin, and Ailing Zhu. "A Crank–Nicolson Compact Difference Method for Time-Fractional Damped Plate Vibration Equations." Axioms 11, no. 10 (2022): 535. http://dx.doi.org/10.3390/axioms11100535.

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This paper discusses the Crank–Nicolson compact difference method for the time-fractional damped plate vibration problems. For the time-fractional damped plate vibration equations, we introduce the second-order space derivative and the first-order time derivative to convert fourth-order differential equations into second-order differential equation systems. We discretize the space derivative via compact difference and approximate the time-integer-order derivative and fraction-order derivative via central difference and L1 interpolation, respectively, to obtain the compact difference formats wi

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Khan, Zareen A., Muhammad Imran Liaqat, Ali Akgül, and J. Alberto Conejero. "Qualitative Analysis of Stochastic Caputo–Katugampola Fractional Differential Equations." Axioms 13, no. 11 (2024): 808. http://dx.doi.org/10.3390/axioms13110808.

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Stochastic pantograph fractional differential equations (SPFDEs) combine three intricate components: stochastic processes, fractional calculus, and pantograph terms. These equations are important because they allow us to model and analyze systems with complex behaviors that traditional differential equations cannot capture. In this study, we achieve significant results for these equations within the context of Caputo–Katugampola derivatives. First, we establish the existence and uniqueness of solutions by employing the contraction mapping principle with a suitably weighted norm and demonstrate

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Albidah, Abdulrahman B. "Application of Riemann–Liouville Derivatives on Second-Order Fractional Differential Equations: The Exact Solution." Fractal and Fractional 7, no. 12 (2023): 843. http://dx.doi.org/10.3390/fractalfract7120843.

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This paper applies two different types of Riemann–Liouville derivatives to solve fractional differential equations of second order. Basically, the properties of the Riemann–Liouville fractional derivative depend mainly on the lower bound of the integral involved in the Riemann–Liouville fractional definition. The Riemann–Liouville fractional derivative of first type considers the lower bound as a zero while the second type applies negative infinity as a lower bound. Due to the differences in properties of the two operators, two different solutions are obtained for the present two classes of fr

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He, Haiyan, Kaijie Liang, and Baoli Yin. "A numerical method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation." International Journal of Modeling, Simulation, and Scientific Computing 10, no. 01 (2019): 1941005. http://dx.doi.org/10.1142/s1793962319410058.

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In this paper, we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation. In order to avoid using higher order elements, we introduce an intermediate variable [Formula: see text] and translate the fourth-order derivative of the original problem into a second-order coupled system. We discretize the fractional time derivative terms by using the [Formula: see text]-approximation and discretize the first-order time derivative term by using the second-order backward differentiation formula. In the fully discrete scheme, we implement

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Rossikhin, Yury A., Marina V. Shitikova, and Basem Ajarmah. "Numerical analysis of non-linear vibrations of a fractionally damped cylindrical shell under the conditions of combinational internal resonance." MATEC Web of Conferences 148 (2018): 03006. http://dx.doi.org/10.1051/matecconf/201814803006.

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Non-linear damped vibrations of a cylindrical shell embedded into a fractional derivative medium are investigated for the case of the combinational internal resonance, resulting in modal interaction, using two different numerical methods with further comparison of the results obtained. The damping properties of the surrounding medium are described by the fractional derivative Kelvin-Voigt model utilizing the Riemann-Liouville fractional derivatives. Within the first method, the generalized displacements of a coupled set of nonlinear ordinary differential equations of the second order are estim

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Atangana, Abdon, and Ali Akgül. "Analysis of a derivative with two variable orders." AIMS Mathematics 7, no. 5 (2022): 7274–93. http://dx.doi.org/10.3934/math.2022406.

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<abstract><p>In this paper, we investigate a derivative with the two variable orders. The first one shows the variable order fractal dimension and the second one presents the fractional order. We consider these derivatives with the power law kernel, exponential decay kernel and Mittag-Leffler kernel. We give the theory of this derivative in details. We also present the numerical approximation. The results we obtained in this work are very useful for researchers to improve many things for fractal fractional derivative with two variable orders.</p></abstract>

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Gou, Haide, and Yongxiang Li. "A Study on Impulsive Hilfer Fractional Evolution Equations with Nonlocal Conditions." International Journal of Nonlinear Sciences and Numerical Simulation 21, no. 2 (2020): 205–18. http://dx.doi.org/10.1515/ijnsns-2019-0015.

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AbstractIn this paper, we concern with the existence of mild solution to nonlocal initial value problem for nonlinear Sobolev-type impulsive evolution equations with Hilfer fractional derivative which generalized the Riemann–Liouville fractional derivative. At first, we establish an equivalent integral equation for our main problem. Second, by means of the properties of Hilfer fractional calculus, combining measure of noncompactness with the fixed-point methods, we obtain the existence results of mild solutions with two new characteristic solution operators. The results we obtained are new and

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Luc, DJERAYOM, Djibet Mbainguesse, Bakari Abbo, and Youssouf Paré. "Numerical Solution of Fractional Diffusion Equation by Shifted Legendre Operational Matrix Method and Fractional Linear Multi-step Methods." Journal of Advances in Mathematics and Computer Science 39, no. 12 (2024): 110–25. https://doi.org/10.9734/jamcs/2024/v39i121953.

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The paper deals with an efficient scheme to solve fractional diffusion equation including both time and spatial fractional derivative in Caputo sense. In the first time, the so-called operational matrice was obtained by computating fractional derivative of shifted Legendre polynomial followed by applying the spectral Tau method that convert the original equation in the system of fractionnal ordinary differential equation (FODE). The fractionnal linear multi-step metthods (FLMMS) can be used in the second time to give the approximate solution. To acces the accuracy and validity of the method, t

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Srivastava, Hari M., Abedel-Karrem N. Alomari, Khaled M. Saad, and Waleed M. Hamanah. "Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method." Fractal and Fractional 5, no. 3 (2021): 131. http://dx.doi.org/10.3390/fractalfract5030131.

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Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations,

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Wang, C. L. "Fractional kinetics of photocatalytic degradation." Journal of Advanced Dielectrics 08, no. 05 (2018): 1850034. http://dx.doi.org/10.1142/s2010135x18500340.

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In this paper, photocatalytic degradation processes of different materials are fitted to the first-order kinetic model, second-order kinetic model and fractional first-order kinetic model. Deterministic coefficients are calculated for the evaluation of the validity of these models. The fitting results show clearly that the degradation process can fit the fractional first-order kinetic model in a very good manner. In this way, two material parameters can be well defined. One is the degradation time, which can be used to describe the photocatalytic degradation process quantitatively. Another is

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Wang, Zhen, Luhan Sun, and Jianxiong Cao. "Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation." Fractal and Fractional 6, no. 7 (2022): 349. http://dx.doi.org/10.3390/fractalfract6070349.

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This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order α∈(0,1). Considering the weak singularity of the solution u(x,t) at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as t→0+ albeit the function itself being right continuous at t=0, two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2-1σ formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Gale

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Sugimoto, N. "Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves." Journal of Fluid Mechanics 225 (April 1991): 631–53. http://dx.doi.org/10.1017/s0022112091002203.

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This paper deals with initial-value problems for the Burgers equation with the inclusion of a hereditary integral known as the fractional derivative of order ½. Emphasis is placed on the difference between the local and global dissipation due to the second-order and the half-order derivatives, respectively. Exploiting the smallness of the coefficient of the second-order derivative, an asymptotic analysis is first developed. When a discontinuity appears, the matched-asymptotic expansion method is employed to derive a uniformly valid solution. If the coefficient of the half-order derivative is a

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