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Journal articles on the topic 'Caputo derivatives'

Author: Grafiati

Published: 28 June 2021

Last updated: 29 July 2025

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1

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 sc

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2

Agarwal, Ravi, Snezhana Hristova, and Donal O’Regan. "Generalized Proportional Caputo Fractional Differential Equations with Noninstantaneous Impulses: Concepts, Integral Representations, and Ulam-Type Stability." Mathematics 10, no. 13 (2022): 2315. http://dx.doi.org/10.3390/math10132315.

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The generalized proportional Caputo fractional derivative is a comparatively new type of derivative that is a generalization of the classical Caputo fractional derivative, and it gives more opportunities to adequately model complex phenomena in physics, chemistry, biology, etc. In this paper, the presence of noninstantaneous impulses in differential equations with generalized proportional Caputo fractional derivatives is discussed. Generalized proportional Caputo fractional derivatives with fixed lower limits at the initial time as well as generalized proportional Caputo fractional derivatives

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3

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

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

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4

Guswanto, Bambang Hendriya, Leony Rhesmafiski Andini, and Triyani Triyani. "On Conformable, Riemann-Liouville, and Caputo fractional derivatives." Bulletin of Applied Mathematics and Mathematics Education 2, no. 2 (2022): 59–64. http://dx.doi.org/10.12928/bamme.v2i2.7072.

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This article compares conformable fractional Derivative with Riemann-Liouville and Caputo fractional derivative by comparing solutions to fractional ordinary differential equations involving the three fractional derivatives via the numerical simulations of the solutions. The result shows that conformable fractional derivative can be used as an alternative to Riemann-Liouville and Caputo fractional derivative for order α with 1/2<α<1.

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5

Peng, Zhongqi, Yuan Li, Qi Zhang, and Yimin Xue. "Extremal Solutions for Caputo Conformable Differential Equations with p-Laplacian Operator and Integral Boundary Condition." Complexity 2021 (October 25, 2021): 1–14. http://dx.doi.org/10.1155/2021/1097505.

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The Caputo conformable derivative is a new Caputo-type fractional differential operator generated by conformable derivatives. In this paper, using Banach fixed point theorem, we obtain the uniqueness of the solution of nonlinear and linear Cauchy problem with the conformable derivatives in the Caputo setting, respectively. We also establish two comparison principles and prove the extremal solutions for nonlinear fractional p -Laplacian differential system with Caputo conformable derivatives by utilizing the monotone iterative technique. An example is given to verify the validity of the results

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6

Oliveira, Daniela S., and Edmundo Capelas de Oliveira. "On a Caputo-type fractional derivative." Advances in Pure and Applied Mathematics 10, no. 2 (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 an

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7

Tapé, Z. Grace Espérance Z., Mouhamadou Dosso, and Seydou Traoré. "On Numerical Schemes for Solving Fractional Advection-diffusion Equations in the Sense of Caputo." WSEAS TRANSACTIONS ON SYSTEMS AND CONTROL 20 (July 23, 2025): 305–28. https://doi.org/10.37394/23203.2025.20.34.

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This paper presents the application of concentration described by the fractional advection-diffusion equation, considering fractional derivatives in the sense of Caputo and a new modified fractional derivative also in the sense of Caputo. Specifically, approximations of fractional derivatives of order α (m−1≤α≤m, m∈N ∗ ) using the finite difference method in the two aforementioned cases have enabled us to develop numerical resolution schemes. The results of comparative numerical tests between the two schemes showed that, with the modified Caputo derivative, diffusion is slightly faster with a

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8

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 (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

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9

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

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

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10

Hattaf, Khalid. "A New Mixed Fractional Derivative with Applications in Computational Biology." Computation 12, no. 1 (2024): 7. http://dx.doi.org/10.3390/computation12010007.

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This study develops a new definition of a fractional derivative that mixes the definitions of fractional derivatives with singular and non-singular kernels. This developed definition encompasses many types of fractional derivatives, such as the Riemann–Liouville and Caputo fractional derivatives for singular kernel types, as well as the Caputo–Fabrizio, the Atangana–Baleanu, and the generalized Hattaf fractional derivatives for non-singular kernel types. The associate fractional integral of the new mixed fractional derivative is rigorously introduced. Furthermore, a novel numerical scheme is d

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11

Doungmo Goufo, Emile Franc, and Sunil Kumar. "Shallow Water Wave Models with and without Singular Kernel: Existence, Uniqueness, and Similarities." Mathematical Problems in Engineering 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/4609834.

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After the recent introduction of the Caputo-Fabrizio derivative by authors of the same names, the question was raised about an eventual comparison with the old version, namely, the Caputo derivative. Unlike Caputo derivative, the newly introduced Caputo-Fabrizio derivative has no singular kernel and the concern was about the real impact of this nonsingularity on real life nonlinear phenomena like those found in shallow water waves. In this paper, a nonlinear Sawada-Kotera equation, suitable in describing the behavior of shallow water waves, is comprehensively analyzed with both types of deriva

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12

Feng, Xue, Baolin Feng, Ghulam Farid, Sidra Bibi, Qi Xiaoyan, and Ze Wu. "Caputo Fractional Derivative Hadamard Inequalities for Stronglym-Convex Functions." Journal of Function Spaces 2021 (April 21, 2021): 1–11. http://dx.doi.org/10.1155/2021/6642655.

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In this paper, two versions of the Hadamard inequality are obtained by using Caputo fractional derivatives and stronglym-convex functions. The established results will provide refinements of well-known Caputo fractional derivative Hadamard inequalities form-convex and convex functions. Also, error estimations of Caputo fractional derivative Hadamard inequalities are proved and show that these are better than error estimations already existing in literature.

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13

Agarwal, Ravi P., Snezhana Hristova, and Donal O’Regan. "Asymptotic Behavior of Delayed Reaction-Diffusion Neural Networks Modeled by Generalized Proportional Caputo Fractional Partial Differential Equations." Fractal and Fractional 7, no. 1 (2023): 80. http://dx.doi.org/10.3390/fractalfract7010080.

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In this paper, a delayed reaction-diffusion neural network model of fractional order and with several constant delays is considered. Generalized proportional Caputo fractional derivatives with respect to the time variable are applied, and this type of derivative generalizes several known types in the literature for fractional derivatives such as the Caputo fractional derivative. Thus, the obtained results additionally generalize some known models in the literature. The long term behavior of the solution of the model when the time is increasing without a bound is studied and sufficient conditio

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14

Ziada, Eman A. A., Salwa El-Morsy, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani, and Monica Botros. "Solution of the SIR epidemic model of arbitrary orders containing Caputo-Fabrizio, Atangana-Baleanu and Caputo derivatives." AIMS Mathematics 9, no. 7 (2024): 18324–55. http://dx.doi.org/10.3934/math.2024894.

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<abstract><p>The main aim of this study was to apply an analytical method to solve a nonlinear system of fractional differential equations (FDEs). This method is the Adomian decomposition method (ADM), and a comparison between its results was made by using a numerical method: Runge-Kutta 4 (RK4). It is proven that there is a unique solution to the system. The convergence of the series solution is given, and the error estimate is also proven. After that, the susceptible-infected-recovered (SIR) model was taken as an real phenomenon with such systems. This system is discussed with th

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15

Mahdi, N. K., and A. R. Khudair. "On Caputo Delta q−Fractional Dynamical Systems: Lyapunov Stability." Malaysian Journal of Mathematical Sciences 18, no. 4 (2024): 775–83. https://doi.org/10.47836/mjms.18.4.06.

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The investigation of dynamic systems that incorporate Caputo delta q−fractional derivatives has garnered significant interest due to their practicality in diverse scientific and engineering fields. This paper studies the stability of a dynamic system with the Caputo delta q−fractional derivative using Lyapunov's direct method. The motivation behind our work stems from the necessity to comprehend the dynamics and resilience of systems defined by Caputo delta q−fractional derivatives, which exemplify a category of operators that are both non-local and non-singular. This unique fractional derivat

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16

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

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

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17

Hasanah, Dahliatul. "On continuity properties of the improved conformable fractional derivatives." Jurnal Fourier 11, no. 2 (2022): 88–96. http://dx.doi.org/10.14421/fourier.2022.112.88-96.

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The conformable fractional derivative has been introduced to extend the familiar limit definition of the classical derivative. Despite having many advantages compared to other fractional derivatives such as satisfying nice properties as classical derivative and easy to solve numerically, it also has disadvantages as it gives large error compared to Riemann-Liouville and Caputo fractional derivatives. Modified types of conformable derivatives have been proposed to overcome the shortcoming. The improved conformal fractional derivatives are declared to be better approximations of Riemann-Liouvill

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18

Liaqat, Muhammad Imran. "A Hybrid Approach to Approximate and Exact Solutions for Linear and Nonlinear Fractional-Order Schrödinger Equations with Conformable Fractional Derivatives." Electronic Journal of Applied Mathematics 2, no. 3 (2024): 1–26. https://doi.org/10.61383/ejam.20242371.

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Fractional-order Schrödinger differential equations extend the classical Schrödinger equation by incorporating fractional calculus to describe more complex physical phenomena. The Schrödinger equations are solved using fractional derivatives expressed through the Caputo derivative. However, there is limited research on exact and approximate solutions involving conformable fractional derivatives. This study aims to address this gap by employing a hybrid approach that combines the Elzaki transform with the decomposition technique to solve the Schrödinger equation with conformable fractional deri

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19

Mamman, John Ojima. "Computational Algorithm for Approximating Fractional Derivatives of Functions." Journal of Modeling and Simulation of Materials 5, no. 1 (2022): 31–38. http://dx.doi.org/10.21467/jmsm.5.1.31-38.

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This paper presents an algorithmic approach for numerically solving Caputo fractional diﬀerentiation. The trapezoidal rule was modiﬁed, the new modiﬁcation was used to derive an algorithm to approximate fractional derivatives of order α > 0, the fractional derivative used was based on Caputo deﬁnition for a given function by a weighted sum of function and its ordinary derivatives values at speciﬁed points. The trapezoidal rule was used in conjunction with the finite difference scheme which is the forward, backward and central difference to derive the computational algorithm for the numerica

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20

Sami, Ahmed, and Sameer Qasim Hasan. "Stability of Composition Caputa– Katugampola Fractional Differential Nonlinear Control System with Delay Riemann −Katugampola." Mustansiriyah Journal of Pure and Applied Sciences 2, no. 4 (2024): 19–40. http://dx.doi.org/10.47831/mjpas.2024.2.4.19-40.

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work used the Composition Caputo-Katugampola Fractional Derivatives technique to tackle nonlinear problems and delay fractional differential equations. The fractional derivative is defined using the Caputo and Riemann-Katugampola Fractional Derivatives Method. Proposed method In comparison to other digital technologies, this one is simple, effective, and uncomplicated. Ensure authenticity and correctness proposed method Some exemplary problems have been solved

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21

Sami, Ahmed, and Sameer Qasim Hasan. "Stability of Composition Caputa– Katugampola Fractional Differential Nonlinear Control System with Delay Riemann −Katugampola." Mustansiriyah Journal of Pure and Applied Sciences 2, no. 4 (2024): 19–40. http://dx.doi.org/10.47831/mjpas.v2i4.131.

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work used the Composition Caputo-Katugampola Fractional Derivatives technique to tackle nonlinear problems and delay fractional differential equations. The fractional derivative is defined using the Caputo and Riemann-Katugampola Fractional Derivatives Method. Proposed method In comparison to other digital technologies, this one is simple, effective, and uncomplicated. Ensure authenticity and correctness proposed method Some exemplary problems have been solved

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22

Aitbrahim, Aabdessamad, J. El Ghordaf, A. El Hajaji, K. Hilal, and J. E. Napoles Valdes. "A Comparative Analysis of Conformable, Non-conformable, Riemann-Liouville, and Caputo Fractional Derivatives." European Journal of Pure and Applied Mathematics 17, no. 3 (2024): 1842–54. http://dx.doi.org/10.29020/nybg.ejpam.v17i3.5237.

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This study undertakes a comparative analysis of the non conformable and conformable fractional derivatives alongside the Riemann-Liouville and Caputo fractional derivatives. It examines their efficacy in solving fractional ordinary differential equations and explores their applications in physics through numerical simulations. The findings suggest that the conformable fractional derivative emerges as a promising substitute for the non conformable, Riemann-Liouville and Caputo fractional derivatives within the range of order $\alpha $ where $1/2 < \alpha < 1$.

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23

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

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

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24

Luo, D., J. R. Wang, and M. Fečkan. "Applying Fractional Calculus to Analyze Economic Growth Modelling." Journal of Applied Mathematics, Statistics and Informatics 14, no. 1 (2018): 25–36. http://dx.doi.org/10.2478/jamsi-2018-0003.

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Abstract In this work, we apply fractional calculus to analyze a class of economic growth modelling (EGM) of the Spanish economy. More precisely, the Grünwald-Letnnikov and Caputo derivatives are used to simulate GDP by replacing the previous integer order derivatives with the help of Matlab, SPSS and R software. As a result, we find that the data raised from the Caputo derivative are better than the data raised from the Grünwald-Letnnikov derivative. We improve the previous result in [12].

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25

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

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

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26

Liaqat, Muhammad Imran, and Hussam Aljarrah. "Conformable Sumudu Transform Based Adomian Decomposition Method for Linear and Nonlinear Fractional-Order Schrödinger Equations." European Journal of Pure and Applied Mathematics 17, no. 4 (2024): 3464–91. http://dx.doi.org/10.29020/nybg.ejpam.v17i4.5456.

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Fractional-order Schrödinger differential equations extend the classical Schrödinger equation by incorporating fractional calculus to describe more complex physical phenomena. In the literature, the Schr ̈odinger equation is mostly solved using fractional derivatives expressed through the Caputo derivative. However, there is limited research on exact and approximate solutionsinvolving conformable fractional derivatives. This study aims to fill this gap by employing a hybrid approach that combines the Sumudu transform with the decomposition technique to solve the Schrödinger equation with confo

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27

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 (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

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28

Sene, Ndolane, and José Francisco Gómez Aguilar. "Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives." Fractal and Fractional 3, no. 3 (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 stabi

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29

Baleanu, Dumitru, Bahram Agheli, and Maysaa Mohamed Al Qurashi. "Fractional advection differential equation within Caputo and Caputo–Fabrizio derivatives." Advances in Mechanical Engineering 8, no. 12 (2016): 168781401668330. http://dx.doi.org/10.1177/1687814016683305.

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In this research, we applied the variational homotopic perturbation method and q-homotopic analysis method to find a solution of the advection partial differential equation featuring time-fractional Caputo derivative and time-fractional Caputo–Fabrizio derivative. A detailed comparison of the obtained results was reported. All computations were done using Mathematica.

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30

Raza, Nauman, and Muhammad Asad Ullah. "A comparative study of heat transfer analysis of fractional Maxwell fluid by using Caputo and Caputo–Fabrizio derivatives." Canadian Journal of Physics 98, no. 1 (2020): 89–101. http://dx.doi.org/10.1139/cjp-2018-0602.

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A comparative analysis is carried out to study the unsteady flow of a Maxwell fluid in the presence of Newtonian heating near a vertical flat plate. The fractional derivatives presented by Caputo and Caputo–Fabrizio are applied to make a physical model for a Maxwell fluid. Exact solutions of the non-dimensional temperature and velocity fields for Caputo and Caputo–Fabrizio time-fractional derivatives are determined via the Laplace transform technique. Numerical solutions of partial differential equations are obtained by employing Tzou’s and Stehfest’s algorithms to compare the results of both

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31

Baloch, Imran Abbas, Thabet Abdeljawad, Sidra Bibi, Aiman Mukheimer, Ghulam Farid, and Absar Ul Haq. "Some new Caputo fractional derivative inequalities for exponentially $ (\theta, h-m) $–convex functions." AIMS Mathematics 7, no. 2 (2022): 3006–26. http://dx.doi.org/10.3934/math.2022166.

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<abstract><p>Firstly, we obtain some inequalities of Hadamard type for exponentially $ (\theta, h-m) $–convex functions via Caputo $ k $–fractional derivatives. Secondly, using integral identity including the $ (n+1) $–order derivative of a given function via Caputo $ k $-fractional derivatives we prove some of its related results. Some new results are given and known results are recaptured as special cases from our results.</p></abstract>

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32

WU, CONG. "A GENERAL COMPARISON PRINCIPLE FOR CAPUTO FRACTIONAL-ORDER ORDINARY DIFFERENTIAL EQUATIONS." Fractals 28, no. 04 (2020): 2050070. http://dx.doi.org/10.1142/s0218348x2050070x.

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In this paper, we work on a general comparison principle for Caputo fractional-order ordinary differential equations. A full result on maximal solutions to Caputo fractional-order systems is given by using continuation of solutions and a newly proven formula of Caputo fractional derivatives. Based on this result and the formula, we prove a general fractional comparison principle under very weak conditions, in which only the Caputo fractional derivative is involved. This work makes up deficiencies of existing results.

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33

Khalighi, Moein, Leila Eftekhari, Soleiman Hosseinpour, and Leo Lahti. "Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators." Symmetry 13, no. 3 (2021): 368. http://dx.doi.org/10.3390/sym13030368.

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In this paper, we apply the concept of fractional calculus to study three-dimensional Lotka-Volterra differential equations. We incorporate the Caputo-Fabrizio fractional derivative into this model and investigate the existence of a solution. We discuss the uniqueness of the solution and determine under what conditions the model offers a unique solution. We prove the stability of the nonlinear model and analyse the properties, considering the non-singular kernel of the Caputo-Fabrizio operator. We compare the stability conditions of this system with respect to the Caputo-Fabrizio operator and

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34

Sweilam, Nasser H., Seham M. Al-Mekhlafi, Waleed S. Abdel Kareem та Ghader Alqurishi. "Comparative Study of Crossover Mathematical Model of Breast Cancer Based on Ψ-Caputo Derivative and Mittag-Leffler Laws: Numerical Treatments". Symmetry 16, № 9 (2024): 1172. http://dx.doi.org/10.3390/sym16091172.

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Two novel crossover models for breast cancer that incorporate Ψ-Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion and the crossover model for breast cancer that incorporates Atangana–Baleanu Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion are presented here, where we used a simple nonstandard kernel fun

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35

Almeida, Ricardo, Ravi P. Agarwal, Snezhana Hristova, and Donal O’Regan. "Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks." Axioms 10, no. 4 (2021): 322. http://dx.doi.org/10.3390/axioms10040322.

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A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful in

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36

Hassouna, Meryeme, El Hassan El Kinani, and Abdelaziz Ouhadan. "Global Existence and Uniqueness of Solution of Atangana–Baleanu Caputo Fractional Differential Equation with Nonlinear Term and Approximate Solutions." International Journal of Differential Equations 2021 (July 5, 2021): 1–11. http://dx.doi.org/10.1155/2021/5675789.

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In this paper, a class of fractional order differential equation expressed with Atangana–Baleanu Caputo derivative with nonlinear term is discussed. The existence and uniqueness of the solution of the general fractional differential equation are expressed. To present numerical results, we construct approximate scheme to be used for producing numerical solutions of the considered fractional differential equation. As an illustrative numerical example, we consider two Riccati fractional differential equations with different derivatives: Atangana–Baleanu Caputo and Caputo derivatives. Finally, the

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Ajmal, Muhammad. "Caputo Fractional Derivative Inequalities for modified h , m −Convex Functions." Journal of Corrosion and Materials 48, no. 1 (2024): 101–15. http://dx.doi.org/10.61336/jcm2023-11.

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Fractional calculus has emerged as a powerful tool in various branches of science and engineering, including mathematical modeling of complex phenomena. In particular, the Caputo k-fractional derivative has been extensively used to model various real-world problems. In this paper, we focus on developing Hadamard type inequalities for modified (h,m)−convex functions via the Caputo k−fractional derivatives. The main objective of this paper is to provide a new approach to estimating the fractional derivative of modified (h,m)−convex functions through the use of two integral identities involving t

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Taqbibt, Abdellah, Latifa El Bezdaoui, M'hamed Elomari, and Lalla Saadia Chadli. "Generalized solutions of the Cauchy problem involving $\Phi$-Caputo fractional derivatives." Boletim da Sociedade Paranaense de Matemática 42 (May 21, 2024): 1–12. http://dx.doi.org/10.5269/bspm.66650.

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The main objective of this research paper is to embed $\Phi$-Caputo fractional derivative in the Colombeau algebra of generalized functions and we investigate the existence and uniqueness of the Cauchy problem involving $\Phi$-Caputo fractional derivatives in the extended Colombeau algebras. Finally, we give anexample of how the ideas presented in the document can be applied.

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Mehandiratta, Vaibhav, Mani Mehra, and Günter Leugering. "Distributed optimal control problems driven by space-time fractional parabolic equations." Control and Cybernetics 51, no. 2 (2022): 191–226. http://dx.doi.org/10.2478/candc-2022-0014.

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Abstract We study distributed optimal control problems, governed by space-time fractional parabolic equations (STFPEs) involving time-fractional Caputo derivatives and spatial fractional derivatives of Sturm-Liouville type. We first prove existence and uniqueness of solutions of STFPEs on an open bounded interval and study their regularity. Then we show existence and uniqueness of solutions to a quadratic distributed optimal control problem. We derive an adjoint problem using the right-Caputo derivative in time and provide optimality conditions for the control problem. Moreover, we propose a f

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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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Nosheen, Ammara, Maria Tariq, Khuram Ali Khan, Nehad Ali Shah, and Jae Dong Chung. "On Caputo Fractional Derivatives and Caputo–Fabrizio Integral Operators via (s, m)-Convex Functions." Fractal and Fractional 7, no. 2 (2023): 187. http://dx.doi.org/10.3390/fractalfract7020187.

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This paper contains a variety of new integral inequalities for (s,m)-convex functions using Caputo fractional derivatives and Caputo–Fabrizio integral operators. Various generalizations of Hermite–Hadamard-type inequalities containing Caputo–Fabrizio integral operators are derived for those functions whose derivatives are (s,m)-convex. Inequalities involving the digamma function and special means are deduced as applications.

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Youssef, Hamdy M., Alaa A. El-Bary, and Eman A. N. Al-Lehaibi. "Characterization of the Quality Factor Due to the Static Prestress in Classical Caputo and Caputo–Fabrizio Fractional Thermoelastic Silicon Microbeam." Polymers 13, no. 1 (2020): 27. http://dx.doi.org/10.3390/polym13010027.

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The thermal quality factor is the most significant parameter of the micro/nanobeam resonator. Less energy is released by vibration and low damping, which results in greater efficiency. Thus, for a simply supported microbeam resonator made of silicon (Si), a thermal analysis of the thermal quality factor was introduced. A force due to static prestress was considered. The governing equations were constructed in a unified system. This system generates six different models of heat conduction; the traditional Lord–Shulman, Lord–Shulman based on classical Caputo fractional derivative, Lord–Shulman b

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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 (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,

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Boyko, K. V. "LINEAR AND QUASILINEAR EQUATIONS WITH SEVERAL GERASIMOV - CAPUTO DERIVATIVES." Челябинский физико-математический журнал 9, no. 1 (2024): 5–22. http://dx.doi.org/10.47475/2500-0101-2024-9-1-5-22.

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A representation of a solution of the Cauchy problem for a linear inhomogeneous equation solved with respect to the oldest derivative with several fractional Gerasimov - Caputo derivatives and with a sectorial pencil of linear closed operators at them in the case of the Holder function in the right-hand side of the equation is obtained; the uniqueness of the solution is proved. This result is used to reduce the Cauchy problem for the corresponding quasilinear equation to an integro-differential equation. The existence of a unique local solution is proved by the method of contraction operators

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Gómez-Aguilar, J. F. "Fractional Meissner–Ochsenfeld effect in superconductors." Modern Physics Letters B 33, no. 26 (2019): 1950316. http://dx.doi.org/10.1142/s0217984919503160.

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Fractional calculus (FC) is a valuable tool in the modeling of many phenomena, and it has become a topic of great interest in science and engineering. This mathematical tool has proved its efficiency in modeling the intermediate anomalous behaviors observed in different physical phenomena. The Meissner–Ochsenfeld effect describes the levitation of superconductors in a nonuniform magnetic field if they are cooled below critical temperature. This paper presents analytical solutions of the fractional London equation that describes the Meissner–Ochsenfeld effect considering the Liouville–Caputo, C

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

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

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Cui, Zhoujin. "Solutions of some typical nonlinear differential equations with Caputo-Fabrizio fractional derivative." AIMS Mathematics 7, no. 8 (2022): 14139–53. http://dx.doi.org/10.3934/math.2022779.

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<abstract><p>In this paper, the solutions of some typical nonlinear fractional differential equations are discussed, and the implicit analytical solutions are obtained. The fractional derivative concerned here is the Caputo-Fabrizio form, which has a nonsingular kernel. The calculation results of different fractional orders are compared through images. In addition, by comparing the results obtained in this paper with those under Caputo fractional derivative, it is found that the solutions change relatively gently under Caputo-Fabrizio fractional derivative. It can be concluded that

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Abu-Alshaikh, Ibrahim M., and Amro A. Almbaidin. "Analytical responses of functionally graded beam under moving mass using Caputo and Caputo–Fabrizio fractional derivative models." Journal of Vibration and Control 26, no. 19-20 (2020): 1859–67. http://dx.doi.org/10.1177/1077546320908103.

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In this article, a functionally graded simply supported Euler–Bernoulli beam subjected to moving mass is considered in which the beam-damping is described using fractional Kelvin–Voigt model. A comparison between Caputo and Caputo–Fabrizio fractional derivatives for obtaining the analytical dynamic response of the beam is carried out. The equation of motion is solved by the decomposition method with the cooperation of the Laplace transform. Two verification studies were performed to check the validity of the solutions. The results show that the grading order, the velocity of the moving mass an

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O’Regan, Donal, Ravi P. Agarwal, Snezhana Hristova, and Mohamed I. Abbas. "Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative." Mathematics 12, no. 2 (2024): 233. http://dx.doi.org/10.3390/math12020233.

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An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportio

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Bayrak, Mine Aylin, and Ali Demir. "On a fractional operator of adjoint hybrid fractional derivative operator." Annals of the University of Craiova Mathematics and Computer Science Series 51, no. 1 (2024): 21–39. http://dx.doi.org/10.52846/ami.v51i1.1678.

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The achievement of this paper is to propose a new kind of fractional derivative which is called New Constant Proportional Caputo (NCPC) operator and to construct the solution of time-fractional initial value problem (TFIVPs) with NCPC derivative by taking the combination of Laplace transform (LT) and Homotopy Analysis method (HAM) into account. Later, the obtained solution is compared with the solutions of TFIVPs with Caputo and Constant Proportional Caputo (CPC) derivatives. The gained results reveal that the combination of LT and HAM together form an efficient method to build the approximate

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