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Journal articles on the topic 'Caputo derivatives'

Author: Grafiati

Published: 28 June 2021

Last updated: 6 February 2022

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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 science and engineering.

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2

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

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

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3

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 (October 31, 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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4

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

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

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5

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 (December 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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6

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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7

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 derivative. In the investigations, various fixed-point theories are exploited together with the concept of Piccard K-stability. We are then able to obtain the existence and uniqueness results for the models with both versions of derivatives. We conclude the analysis by performing some numerical approximations with both derivatives and graphical simulations being presented for some values of the derivative order γ. Similar behaviors are pointed out and they concur with the expected multisoliton solutions well known for the Sawada-Kotera equation. This great observation means either of both derivatives is suitable to describe the motion of shallow water waves.

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8

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

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

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9

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 (February 25, 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 the Caputo fractional derivative. In addition, we derive a new numerical method based on the Adams-Bashforth scheme. We show that the type of differential operators and the value of orders significantly influence the stability of the Lotka-Volterra system and numerical results demonstrate that different fractional operator derivatives of the nonlinear population model lead to different dynamical behaviors.

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10

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

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

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11

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 (June 3, 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 fluid velocity at the sidewall. The effects of magnetohydrodynamic (MHD) and porous medium are also considered. Triple fractional analysis is performed to solve the resulting three models, based on three different fractional operators. The Laplace transform is applied to each problem separately, and Zakian’s numerical algorithm is used for the Laplace inversion. The solutions are presented in various graphs with physical arguments. Results are computed and shown in various plots. The empirical results indicate that, for ramped temperature, the temperature field is highest for the ABC derivative, followed by the CF and Caputo fractional derivatives. In contrast, for isothermal temperature, the temperature field of C-derivative is higher than the CF and ABC derivatives, respectively. It was noticed that the velocity field for the ABC derivative is higher than the CF and Caputo fractional derivatives for ramped velocity. However, the velocity field for the Caputo fractional derivative is lower than the ABC and CF for isothermal velocity.

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12

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 (May 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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13

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 (January 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 models. Exact solutions with integer-order derivative (fractional parameter α = 1) are also obtained for both temperature and velocity distributions as a special case. A graphical illustration is made to discuss the effect of Prandtl number Pr and time t on the temperature field. Similarly, the effects of Maxwell fluid parameter λ and other flow parameters on the velocity field are presented graphically, as well as in tabular form.

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14

Odibat, Zaid, and Dumitru Baleanu. "On a New Modification of the Erdélyi–Kober Fractional Derivative." Fractal and Fractional 5, no. 3 (September 13, 2021): 121. http://dx.doi.org/10.3390/fractalfract5030121.

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

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15

WU, CONG. "A GENERAL COMPARISON PRINCIPLE FOR CAPUTO FRACTIONAL-ORDER ORDINARY DIFFERENTIAL EQUATIONS." Fractals 28, no. 04 (June 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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16

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 (February 11, 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 and the fractional derivative order have significant effects on the beam deflection, whereas the difference between the results of the two fractional derivative models is expressed by the determination of the correlation coefficient.

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17

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 (December 23, 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 based on the Caputo–Fabrizio fractional derivative, traditional Tzou, Tzou based on the classical Caputo fractional derivative, and Tzou based on the Caputo–Fabrizio fractional derivative. The results show that the force due to static prestress, the fractional order parameter, the isothermal value of natural frequency, and the beam’s length significantly affect the thermal quality factor. The two types of fractional derivatives applied have different and significant effects on the thermal quality factor.

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18

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 study of those examples verifies the theoretical results of global existence and uniqueness of solution. Moreover, numerical results underline the difference between solutions of both examples.

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19

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 solutions approach to classical solutions as the order of the fractional derivatives approaches 1.

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20

FARID1, G. "On Caputo Fractional Derivatives via Convexity." Kragujevac Journal of Mathematics 44, no. 3 (September 2020): 393–99. http://dx.doi.org/10.46793/kgjmat2003.393f.

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21

Gómez-Aguilar, J. F. "Fractional Meissner–Ochsenfeld effect in superconductors." Modern Physics Letters B 33, no. 26 (September 20, 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, Caputo–Fabrizio–Caputo, Atangana–Baleanu–Caputo, fractional conformable derivative in Liouville–Caputo sense and Atangana–Koca–Caputo fractional-order derivatives. Numerical simulations were obtained for different values of the fractional-order.

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22

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

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In this paper, we unify and extend recent developments in Lyapunov stability theory to present techniques to determine the asymptotic stability of six types of fractional dynamical systems. These differ by being modeled with one of the following fractional derivatives: the Caputo derivative, the Caputo distributed order derivative, the variable order derivative, the conformable derivative, the local fractional derivative, or the distributed order conformable derivative (the latter defined in this work). Additionally, we apply these results to study the consensus of a fractional multiagent system, considering all of the aforementioned fractional operators. Our analysis covers multiagent systems with linear and nonlinear dynamics, affected by bounded external disturbances and described by fixed directed graphs. Lastly, examples, which are solved analytically and numerically, are presented to validate our contributions.

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23

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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24

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

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

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25

Zhao, Jinchao, Saad Ihsan Butt, Jamshed Nasir, Zhaobo Wang, and Iskander Tlili. "Hermite–Jensen–Mercer Type Inequalities for Caputo Fractional Derivatives." Journal of Function Spaces 2020 (March 24, 2020): 1–11. http://dx.doi.org/10.1155/2020/7061549.

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In this article, certain Hermite–Jensen–Mercer type inequalities are proved via Caputo fractional derivatives. We established some new inequalities involving Caputo fractional derivatives, such as Hermite–Jensen–Mercer type inequalities, for differentiable mapping hn whose derivatives in the absolute values are convex.

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26

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

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

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27

Zhou, Mei-Xiu, A. S. V. Ravi Kanth, K. Aruna, K. Raghavendar, Hadi Rezazadeh, Mustafa Inc, and Ayman A. Aly. "Numerical Solutions of Time Fractional Zakharov-Kuznetsov Equation via Natural Transform Decomposition Method with Nonsingular Kernel Derivatives." Journal of Function Spaces 2021 (July 22, 2021): 1–17. http://dx.doi.org/10.1155/2021/9884027.

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In this paper, we have studied the time-fractional Zakharov-Kuznetsov equation (TFZKE) via natural transform decomposition method (NTDM) with nonsingular kernel derivatives. The fractional derivative considered in Caputo-Fabrizio (CF) and Atangana-Baleanu derivative in Caputo sense (ABC). We employed natural transform (NT) on TFZKE followed by inverse natural transform, to obtain the solution of the equation. To validate the method, we have considered a few examples and compared with the actual results. Numerical results are in accordance with the existing results.

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28

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 experimental study using real world dataset. The experiment has revealed that the Caputo fractional derivative is the most appropriate tool for fitting the model, since it has produced the smallest error rate of 1.74% corresponding to the fractional order of derivative α = 1.005. Caputo–Fabrizio was the second best since it yielded an error rate value of 1.97% for a fractional order of derivative α = 1.042, and finally the classical method produced an error rate of 4.36%.

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29

Culbreth, Garland, Mauro Bologna, Bruce J. West, and Paolo Grigolini. "Caputo Fractional Derivative and Quantum-Like Coherence." Entropy 23, no. 2 (February 9, 2021): 211. http://dx.doi.org/10.3390/e23020211.

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We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

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30

Ahmad, Mushtaq, Muhammad Imran, Dumitru Baleanu, and Ali Alshomrani. "Thermal analysis of magnetohydrodynamic viscous fluid with innovative fractional derivative." Thermal Science 24, Suppl. 1 (2020): 351–59. http://dx.doi.org/10.2298/tsci20351a.

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In this study, an attempt is made to investigate a fractional model of unsteady and an incompressible MHD viscous fluid with heat transfer. The fluid is lying over a vertical and moving plate in its own plane. The problem is modeled by using the constant proportional Caputo fractional derivatives with suitable boundary conditions. The non-dimensional governing equations of problem have been solved analytically with the help of Laplace transform techniques and explicit expressions for respective field variable are obtained. The transformed solutions for energy and momentum balances are appeared in terms of series form. The analytical results regarding velocity and temperature are plotted graphically by MATHCAD software to see the influence of physical parameters. Some graphic comparisons are also mad among present results with hybrid fractional derivatives and the published results that have been obtained by Caputo. It is found that the velocity and temperature with constant proportional Capu?to fractional derivative are portrait better decay than velocities and temperatures that obtained with Caputo and Caputo-Fabrizio derivative. Further, rate of heat transfer and skin friction can be enhanced with smaller values of fractional parameter.

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31

Ahmad, Mushtaq, Muhammad Imran, Dumitru Baleanu, and Ali Alshomrani. "Thermal analysis of magnetohydrodynamic viscous fluid with innovative fractional derivative." Thermal Science 24, Suppl. 1 (2020): 351–59. http://dx.doi.org/10.2298/tsci20s1351a.

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In this study, an attempt is made to investigate a fractional model of unsteady and an incompressible MHD viscous fluid with heat transfer. The fluid is lying over a vertical and moving plate in its own plane. The problem is modeled by using the constant proportional Caputo fractional derivatives with suitable boundary conditions. The non-dimensional governing equations of problem have been solved analytically with the help of Laplace transform techniques and explicit expressions for respective field variable are obtained. The transformed solutions for energy and momentum balances are appeared in terms of series form. The analytical results regarding velocity and temperature are plotted graphically by MATHCAD software to see the influence of physical parameters. Some graphic comparisons are also mad among present results with hybrid fractional derivatives and the published results that have been obtained by Caputo. It is found that the velocity and temperature with constant proportional Capu?to fractional derivative are portrait better decay than velocities and temperatures that obtained with Caputo and Caputo-Fabrizio derivative. Further, rate of heat transfer and skin friction can be enhanced with smaller values of fractional parameter.

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32

Medved, Milan, and Michal Pospisil. "ASYMPTOTIC INTEGRATION OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH INTEGRODIFFERENTIAL RIGHT-HAND SIDE." Mathematical Modelling and Analysis 20, no. 4 (July 20, 2015): 471–89. http://dx.doi.org/10.3846/13926292.2015.1068233.

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In this paper we deal with the problem of asymptotic integration of a class of fractional differential equations of the Caputo type. The left-hand side of such type of equation is the Caputo derivative of the fractional order r ∈ (n − 1, n) of the solution, and the right-hand side depends not only on ordinary derivatives up to order n − 1 but also on the Caputo derivatives of fractional orders 0 < r 1 < · · · < r m < r, and the Riemann–Liouville fractional integrals of positive orders. We give some conditions under which for any global solution x(t) of the equation, there is a constant c ∈ R such that x(t) = ctR + o(tR) as t → ∞, where R = max{n − 1, r m }.

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33

Agarwal, Ravi P., Donal O’Regan, and Snezhana Hristova. "Strict stability with respect to initial time difference for Caputo fractional differential equations by Lyapunov functions." Georgian Mathematical Journal 24, no. 1 (March 1, 2017): 1–13. http://dx.doi.org/10.1515/gmj-2016-0080.

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AbstractThe strict stability properties are generalized to nonlinear Caputo fractional differential equations in the case when both initial points and initial times are changeable. Using Lyapunov functions, some criteria for strict stability, eventually strict stability and strict practical stability are obtained. A brief overview of different types of derivatives in the literature related to the application of Lyapunov functions to Caputo fractional equations are given, and their advantages and disadvantages are discussed with several examples. The Caputo fractional Dini derivative with respect to to initial time difference is used to obtain some sufficient conditions.

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Tang, Ruihua, Sadique Rehman, Aamir Farooq, Muhammad Kamran, Muhammad Imran Qureshi, Asfand Fahad, and Jia-Bao Liu. "A Comparative Study of Natural Convection Flow of Fractional Maxwell Fluid with Uniform Heat Flux and Radiation." Complexity 2021 (August 30, 2021): 1–16. http://dx.doi.org/10.1155/2021/9401655.

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This paper focuses on the comparative study of natural convection flow of fractional Maxwell fluid having uniform heat flux and radiation. The well-known Maxwell fluid equation with an integer-order derivative has been extended to a non-integer-order derivative, i.e., fractional derivative. The explicit expression for the temperature and velocity is acquired by utilizing the Laplace transform (LT) technique. The two fractional derivative concepts are used (Caputo and Caputo–Fabrizio derivatives) in the formulation of the problem. Utilizing the Mathcad programming, the effect of certain embedded factors and fractional parameters on temperature and velocity profile is graphically presented.

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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 (May 1, 2014): 41–46. http://dx.doi.org/10.1515/jmbm-2014-0006.

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

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Li, Changpin, and Min Cai. "High-Order Approximation to Caputo Derivatives and Caputo-type Advection–Diffusion Equations: Revisited." Numerical Functional Analysis and Optimization 38, no. 7 (February 10, 2017): 861–90. http://dx.doi.org/10.1080/01630563.2017.1291521.

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37

Li, Hefeng, Jianxiong Cao, and Changpin Li. "High-order approximation to Caputo derivatives and Caputo-type advection–diffusion equations (III)." Journal of Computational and Applied Mathematics 299 (June 2016): 159–75. http://dx.doi.org/10.1016/j.cam.2015.11.037.

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Dlamini, Anastacia, Emile F. Doungmo Goufo, and Melusi Khumalo. "On the Caputo-Fabrizio fractal fractional representation for the Lorenz chaotic system." AIMS Mathematics 6, no. 11 (2021): 12395–421. http://dx.doi.org/10.3934/math.2021717.

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<abstract><p>The widespread application of chaotic dynamical systems in different fields of science and engineering has attracted the attention of many researchers. Hence, understanding and capturing the complexities and the dynamical behavior of these chaotic systems is essential. The newly proposed fractal-fractional derivative and integral operators have been used in literature to predict the chaotic behavior of some of the attractors. It is argued that putting together the concept of fractional and fractal derivatives can help us understand the existing complexities better since fractional derivatives capture a limited number of problems and on the other side fractal derivatives also capture different kinds of complexities. In this study, we use the newly proposed Caputo-Fabrizio fractal-fractional derivatives and integral operators to capture and predict the behavior of the Lorenz chaotic system for different values of the fractional dimension $ q $ and the fractal dimension $ k $. We will look at the well-posedness of the solution. For the effect of the Caputo-Fabrizio fractal-fractional derivatives operator on the behavior, we present the numerical scheme to study the graphical numerical solution for different values of $ q $ and $ k $.</p></abstract>

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39

Firozja, Mohamad Adabitabar, and Bahram Agheli. "Approximate method for solving strongly fractional nonlinear problems using fuzzy transform." Nonlinear Engineering 9, no. 1 (September 25, 2019): 72–80. http://dx.doi.org/10.1515/nleng-2018-0123.

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AbstractIn this research work, we have shown that it is possible to use fuzzy transform method (FTM) for approximate solution of strongly fractional nonlinear problems. In numerical methods, in order to approximate a function on a particular interval, only a restricted number of points are employed. However, what makes the F-transform preferable to other methods is that it makes use of all points in this interval. The comparison of the time used in minutes is given for two derivatives Caputo derivative and Caputo-Fabrizio derivative.

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Hattaf, Khalid. "A New Generalized Definition of Fractional Derivative with Non-Singular Kernel." Computation 8, no. 2 (May 21, 2020): 49. http://dx.doi.org/10.3390/computation8020049.

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This paper proposes a new definition of fractional derivative with non-singular kernel in the sense of Caputo which generalizes various forms existing in the literature. Furthermore, the version in the sense of Riemann–Liouville is defined. Moreover, fundamental properties of the new generalized fractional derivatives in the sense of Caputo and Riemann–Liouville are rigorously studied. Finally, an application in epidemiology as well as in virology is presented.

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Almeida, Ricardo. "Caputo–Hadamard Fractional Derivatives of Variable Order." Numerical Functional Analysis and Optimization 38, no. 1 (November 14, 2016): 1–19. http://dx.doi.org/10.1080/01630563.2016.1217880.

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42

Murio, Diego A. "Time fractional IHCP with Caputo fractional derivatives." Computers & Mathematics with Applications 56, no. 9 (November 2008): 2371–81. http://dx.doi.org/10.1016/j.camwa.2008.05.015.

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

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

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Gouari, Yazid, Zoubir Dahmani, and Ameth Ndiaye. "A generalized sequential problem of Lane-Emden type via fractional calculus." Moroccan Journal of Pure and Applied Analysis 6, no. 2 (December 1, 2020): 168–83. http://dx.doi.org/10.2478/mjpaa-2020-0013.

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AbstractIn this paper, we combine the Riemann-Liouville integral operator and Caputo derivative to investigate a nonlinear time-singular differential equation of Lane Emden type. The considered problem involves n fractional Caputo derivatives under the conditions that neither commutativity nor semi group property is satisfied for these derivatives. We prove an existence and uniqueness analytic result by application of Banach contraction principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions related to this result are established by means of the fixed point theorem of Schaefer. We end the paper by presenting to the reader some illustrative examples.

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

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

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Alzabut, Jehad, Weerawat Sudsutad, Zeynep Kayar, and Hamid Baghani. "A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative." Mathematics 7, no. 8 (August 15, 2019): 747. http://dx.doi.org/10.3390/math7080747.

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New versions of a Gronwall–Bellman inequality in the frame of the generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their features. We prove our main result in light of some efficient comparison analyses. The Gronwall–Bellman inequality in the case of weighted function is also obtained. By the help of the new proposed inequalities, examples of Riemann–Liouville and Caputo proportional fractional initial value problems are presented to emphasize the solution dependence on the initial data and on the right-hand side.

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Aguilar, José Francisco Gómez, and Margarita Miranda Hernández. "Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/283019.

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An alternative construction for the space-time fractional diffusion-advection equation for the sedimentation phenomena is presented. The order of the derivative is considered as0<β,γ≤1for the space and time domain, respectively. The fractional derivative of Caputo type is considered. In the spatial case we obtain the fractional solution for the underdamped, undamped, and overdamped case. In the temporal case we show that the concentration has amplitude which exhibits an algebraic decay at asymptotically large times and also shows numerical simulations where both derivatives are taken in simultaneous form. In order that the equation preserves the physical units of the system two auxiliary parametersσxandσtare introduced characterizing the existence of fractional space and time components, respectively. A physical relation between these parameters is reported and the solutions in space-time are given in terms of the Mittag-Leffler function depending on the parametersβandγ. The generalization of the fractional diffusion-advection equation in space-time exhibits anomalous behavior.

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Zeng, Shengda, Stanisław Migórski, Van Thien Nguyen, and Yunru Bai. "Maximum principles for a class of generalized time-fractional diffusion equations." Fractional Calculus and Applied Analysis 23, no. 3 (June 25, 2020): 822–36. http://dx.doi.org/10.1515/fca-2020-0041.

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AbstractTwo significant inequalities for generalized time fractional derivatives at extreme points are obtained. Then, we apply the inequalities to establish the maximum principles for multi-term time-space fractional variable-order operators. Finally, we employ the principles to investigate two kinds of diffusion equations involving generalized time-fractional Caputo derivatives and space-fractional Riesz-Caputo derivatives.

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

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

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Gomez, Francisco, and Khaled Saad. "Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense." Revista Mexicana de Física 64, no. 5 (August 31, 2018): 539. http://dx.doi.org/10.31349/revmexfis.64.539.

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In this paper, we have generalized the fractional cubic isothermal auto-catalytic chemical system (FCIACS) with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional time derivatives, respectively. We apply the Homotopy Analysis Transform Method (HATM) to compute the approximate solutions of FCIACS using these fractional derivatives. We study the convergence analysis of HATM by computing the residual error function. Also, we find the optimal values of h so we assure the convergence of the approximate solutions. Finally we show the behavior of the approximate solutions graphically. The results obtained are very effectiveness and accuracy.

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