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Journal articles on the topic 'Caputo's approach'

Author: Grafiati

Published: 28 June 2021

Last updated: 1 February 2022

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1

Al-Refai, Mohammed, Mohamed Ali Hajji, and Muhammad I. Syam. "An Efficient Series Solution for Fractional Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/891837.

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We introduce a simple and efficient series solution for a class of nonlinear fractional differential equations of Caputo's type. The new approach is a modified form of the well-known Taylor series expansion where we overcome the difficulty of computing iterated fractional derivatives, which do not compute in general. The terms of the series are determined sequentially with explicit formula, where only integer derivatives have to be computed. The efficiency of the new algorithm is illustrated through several examples. Comparison with other series methods such as the Adomian decomposition method and the homotopy perturbation method is made to indicate the efficiency of the new approach. The algorithm can be implemented for a wide class of fractional differential equations with different types of fractional derivatives.

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2

Alofi, Abdulaziz, Jinde Cao, Ahmed Elaiw, and Abdullah Al-Mazrooei. "Delay-Dependent Stability Criterion of Caputo Fractional Neural Networks with Distributed Delay." Discrete Dynamics in Nature and Society 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/529358.

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This paper is concerned with the finite-time stability of Caputo fractional neural networks with distributed delay. The factors of such systems including Caputo’s fractional derivative and distributed delay are taken into account synchronously. For the Caputo fractional neural network model, a finite-time stability criterion is established by using the theory of fractional calculus and generalized Gronwall-Bellman inequality approach. Both the proposed criterion and an illustrative example show that the stability performance of Caputo fractional distributed delay neural networks is dependent on the time delay and the order of Caputo’s fractional derivative over a finite time.

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3

Brouwer, Rein. "“Fragment of What Will Happen”." Religion and the Arts 23, no. 4 (October 10, 2019): 384–410. http://dx.doi.org/10.1163/15685292-02304003.

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Abstract The Swedish poet Tomas Tranströmer (1931–2015), winner of the 2011 Nobel Prize in Literature, is known for the perceptive metaphors in his poems “couched in deceptively spare language, crystalline descriptions of natural beauty and explorations of the mysteries of identity and creativity.” Although Tranströmer himself never made a secret of the religious tendency in his work, there is some discussion about the importance of the religious dimension in his poems, which are widely acclaimed in Sweden, a predominately secular country. This article discusses several discourses exploring the religious dimensions of Tranströmer’s poetry, and presents a new approach for understanding the religious and spiritual aspects of his art based on the work of philosopher of religion John D. Caputo. Caputo’s “hauntology” is claimed to be conducive in reading Tranströmer’s poetry as a religious text. A “hauntological” reading of the poetry of Tranströmer interprets the event that is haunting the poems, and suggests a new way of conceiving a religious insight in a work of modern art.

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4

Taïeb, Amele, and Zoubir Dahmani. "Generalized Isoperimetric FVPs Via Caputo Approach." Acta Mathematica 56 (2019): 23–40. http://dx.doi.org/10.4467/20843828am.19.003.12111.

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5

Jean-Claude, Trigeassou, Maamri Nezha, and Oustaloup Alain. "The Caputo Derivative And The Infinite State Approach." IFAC Proceedings Volumes 46, no. 1 (February 2013): 587–92. http://dx.doi.org/10.3182/20130204-3-fr-4032.00122.

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6

Koca, Ilknur, and Pelin Yaprakdal. "A new approach for nuclear family model with fractional order Caputo derivative." Applied Mathematics and Nonlinear Sciences 5, no. 1 (March 31, 2020): 393–404. http://dx.doi.org/10.2478/amns.2020.1.00037.

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AbstractA work on a mathematical modeling is very popular in applied sciences. Nowadays many mathematical models have been considered and new methods have been used for approaching of these models. In this paper we are considering mathematical modeling of nuclear family model with fractional order Caputo derivative. Also the existence and uniqueness results and numerical scheme are given with Adams-Bashforth scheme via fractional order Caputo derivative.

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7

Evirgen, Fırat, and Mehmet Yavuz. "An Alternative Approach for Nonlinear Optimization Problem with Caputo - Fabrizio Derivative." ITM Web of Conferences 22 (2018): 01009. http://dx.doi.org/10.1051/itmconf/20182201009.

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In this study, a fractional mathematical model with steepest descent direction is proposed to find optimal solutions for a class of nonlinear programming problem. In this sense, Caputo-Fabrizio derivative is adapted to the mathematical model. To demonstrate the solution trajectory of the mathematical model, we use the multistage variational iteration method (MVIM). Numerical simulations and comparisons on some test problems show that the mathematical model generated using Caputo-Fabrizio fractional derivative is both feasible and efficient to find optimal solutions for a certain class of equality constrained optimization problems.

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8

Hasan, Nabaa N., and Zainab John. "Analytic Approach for Solving System of Fractional Differential Equations." Al-Mustansiriyah Journal of Science 32, no. 1 (February 21, 2021): 14. http://dx.doi.org/10.23851/mjs.v32i1.929.

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In this paper, Sumudu transformation (ST) of Caputo fractional derivative formulae are derived for linear fractional differential systems. This formula is applied with Mittage-Leffler function for certain homogenous and nonhomogenous fractional differential systems with nonzero initial conditions. Stability is discussed by means of the system's distinctive equation.

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9

Hoa, Ngo Van, Ho Vu, and Tran Minh Duc. "Fuzzy fractional differential equations under Caputo–Katugampola fractional derivative approach." Fuzzy Sets and Systems 375 (November 2019): 70–99. http://dx.doi.org/10.1016/j.fss.2018.08.001.

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10

Albadarneh, Ramzi B., Iqbal Batiha, A. K. Alomari, and Nedal Tahat. "Numerical approach for approximating the Caputo fractional-order derivative operator." AIMS Mathematics 6, no. 11 (2021): 12743–56. http://dx.doi.org/10.3934/math.2021735.

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<abstract><p>This work aims to propose a new simple robust power series formula with its truncation error to approximate the Caputo fractional-order operator $ D_{a}^{\alpha}y(t) $ of order $ m-1 < \alpha < m $, where $ m\in\mathbb{N} $. The proposed formula, which are derived with the help of the weighted mean value theorem, is expressed ultimately in terms of a fractional-order series and its reminder term. This formula is used successfully to provide approximate solutions of linear and nonlinear fractional-order differential equations in the form of series solution. It can be used to determine the analytic solutions of such equations in some cases. Some illustrative numerical examples, including some linear and nonlinear problems, are provided to validate the established formula.</p></abstract>

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11

HERRMANN, RICHARD. "INFRARED SPECTROSCOPY OF DIATOMIC MOLECULES — A FRACTIONAL CALCULUS APPROACH." International Journal of Modern Physics B 27, no. 06 (February 5, 2013): 1350019. http://dx.doi.org/10.1142/s0217979213500197.

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The eigenvalue spectrum of the fractional quantum harmonic oscillator is calculated numerically, solving the fractional Schrödinger equation based on the Riemann and Caputo definition of a fractional derivative. The fractional approach allows a smooth transition between vibrational and rotational type spectra, which is shown to be an appropriate tool to analyze IR spectra of diatomic molecules.

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12

Defez, Emilio, Michael M. Tung, Benito M. Chen-Charpentier, and José M. Alonso. "On the Inverse of the Caputo Matrix Exponential." Mathematics 7, no. 12 (November 21, 2019): 1137. http://dx.doi.org/10.3390/math7121137.

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Matrix exponentials are widely used to efficiently tackle systems of linear differential equations. To be able to solve systems of fractional differential equations, the Caputo matrix exponential of the index α > 0 was introduced. It generalizes and adapts the conventional matrix exponential to systems of fractional differential equations with constant coefficients. This paper analyzes the most significant properties of the Caputo matrix exponential, in particular those related to its inverse. Several numerical test examples are discussed throughout this exposition in order to outline our approach. Moreover, we demonstrate that the inverse of a Caputo matrix exponential in general is not another Caputo matrix exponential.

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13

Altan Koç, Dilara, and Mustafa Gülsu. "Numerical approach for solving time fractional diffusion equation." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 7, no. 3 (November 21, 2017): 281–87. http://dx.doi.org/10.11121/ijocta.01.2017.00492.

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In this article one of the fractional partial differential equations was solved by finite difference scheme based on five point and three point central space method with discretization in time. We use between the Caputo and the Riemann-Liouville derivative definition and the Grünwald-Letnikov operator for the fractional calculus. The stability analysis of this scheme is examined by using von-Neumann method. A comparison between exact solutions and numerical solutions is made. Some figures and tables are included.

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14

Hristov, Jordan. "Steady-state heat conduction in a medium with spatial non-singular fading memory: Derivation of Caputo-Fabrizio space-fractional derivative from Cattaneo concept with Jeffrey`s Kernel and analytical solutions." Thermal Science 21, no. 2 (2017): 827–39. http://dx.doi.org/10.2298/tsci160229115h.

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Starting from the Cattaneo constitutive relation with a Jeffrey's kernel the derivation of a transient heat diffusion equation with relaxation term expressed through the Caputo-Fabrizio time fractional derivative has been developed. This approach allows seeing the physical back ground of the newly defined Caputo-Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories.

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15

Rashid, Saima, Rehana Ashraf, Ahmet Ocak Akdemir, Manar A. Alqudah, Thabet Abdeljawad, and Mohamed S. Mohamed. "Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels." Fractal and Fractional 5, no. 3 (September 8, 2021): 113. http://dx.doi.org/10.3390/fractalfract5030113.

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This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the time-fractional Fornberg–Whitham equation (FWE) under gH-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter ζ∈[0,1] and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

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16

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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17

Piotrowska, Ewa, and Łukasz Sajewski. "Analysis of an Electrical Circuit Using Two-Parameter Conformable Operator in the Caputo Sense." Symmetry 13, no. 5 (April 29, 2021): 771. http://dx.doi.org/10.3390/sym13050771.

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The problem of voltage dynamics description in a circuit containing resistors, and at least two fractional order elements such as supercapacitors, supplied with constant voltage is addressed. A new operator called Conformable Derivative in the Caputo sense is used. A state solution is proposed. The considered operator is a generalization of three derivative definitions: classical definition (integer order), Caputo fractional definition and the so-called Conformable Derivative (CFD) definition. The proposed solution based on a two-parameter Conformable Derivative in the Caputo sense is proven to be better than the classical approach or the one-parameter fractional definition. Theoretical considerations are verified experimentally. The cumulated matching error function is given and it reveals that the proposed CFD–Caputo method generates an almost two times lower error compared to the classical method.

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18

Ali, Zeeshan, Shayan Naseri Nia, Faranak Rabiei, Kamal Shah, and Ming Kwang Tan. "A Semianalytical Approach to the Solution of Time-Fractional Navier-Stokes Equation." Advances in Mathematical Physics 2021 (July 16, 2021): 1–13. http://dx.doi.org/10.1155/2021/5547804.

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In this manuscript, a semianalytical solution of the time-fractional Navier-Stokes equation under Caputo fractional derivatives using Optimal Homotopy Asymptotic Method (OHAM) is proposed. The above-mentioned technique produces an accurate approximation of the desired solutions and hence is known as the semianalytical approach. The main advantage of OHAM is that it does not require any small perturbations, linearization, or discretization and many reductions of the computations. Here, the proposed approach’s reliability and efficiency are demonstrated by two applications of one-dimensional motion of a viscous fluid in a tube governed by the flow field by converting them to time-fractional Navier-Stokes equations in cylindrical coordinates using fractional derivatives in the sense of Caputo. For the first problem, OHAM provides the exact solution, and for the second problem, it performs a highly accurate numerical approximation of the solution compare with the exact solution. The presented simulation results of OHAM comparison with analytical and numerical approaches reveal that the method is an efficient technique to simulate the solution of time-fractional types of Navier-Stokes equation.

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19

El-Ajou, Ahmad, Moa’ath N. Oqielat, Zeyad Al-Zhour, and Shaher Momani. "A class of linear non-homogenous higher order matrix fractional differential equations: Analytical solutions and new technique." Fractional Calculus and Applied Analysis 23, no. 2 (April 28, 2020): 356–77. http://dx.doi.org/10.1515/fca-2020-0017.

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AbstractIn this paper, our formulation generalizes the fractional power series to the matrix form and a new version of the matrix fractional Taylor’s series is also considered in terms of Caputo’s fractional derivative. Moreover, several significant results have been realignment to these generalizations. Finally, to demonstrate the capability and efficiency of our theoretical results, we present the solutions of three linear non-homogenous higher order (m − 1 < α ≤ m, m ∈ N) matrix fractional differential equations by using our new approach.

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20

Liu, Kui, Michal Fečkan, and JinRong Wang. "A Fixed-Point Approach to the Hyers–Ulam Stability of Caputo–Fabrizio Fractional Differential Equations." Mathematics 8, no. 4 (April 22, 2020): 647. http://dx.doi.org/10.3390/math8040647.

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In this paper, we study Hyers–Ulam and Hyers–Ulam–Rassias stability of nonlinear Caputo–Fabrizio fractional differential equations on a noncompact interval. We extend the corresponding uniqueness and stability results on a compact interval. Two examples are given to illustrate our main results.

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21

Awadalla, Muath, Yves Yannick Yameni Noupoue, and Kinda Abu Asbeh. "Psi-Caputo Logistic Population Growth Model." Journal of Mathematics 2021 (July 26, 2021): 1–9. http://dx.doi.org/10.1155/2021/8634280.

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This article studies modeling of a population growth by logistic equation when the population carrying capacity K tends to infinity. Results are obtained using fractional calculus theories. A fractional derivative known as psi-Caputo plays a substantial role in the study. We proved existence and uniqueness of the solution to the problem using the psi-Caputo fractional derivative. The Chinese population, whose carrying capacity, K, tends to infinity, is used as evidence to prove that the proposed approach is appropriate and performs better than the usual logistic growth equation for a population with a large carrying capacity. A psi-Caputo logistic model with the kernel function x + 1 performed the best as it minimized the error rate to 3.20% with a fractional order of derivative α = 1.6455.

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22

Atangana, Abdon, and Kolade M. Owolabi. "New numerical approach for fractional differential equations." Mathematical Modelling of Natural Phenomena 13, no. 1 (2018): 3. http://dx.doi.org/10.1051/mmnp/2018010.

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In the present case, we propose the correct version of the fractional Adams-Bashforth methods which take into account the nonlinearity of the kernels including the power law for the Riemann-Liouville type, the exponential decay law for the Caputo-Fabrizio case and the Mittag-Leffler law for the Atangana-Baleanu scenario.The Adams-Bashforth method for fractional differentiation suggested and are commonly use in the literature nowadays is not mathematically correct and the method was derived without taking into account the nonlinearity of the power law kernel. Unlike the proposed version found in the literature, our approximation, in all the cases, we are able to recover the standard case whenever the fractional powerα= 1. Numerical results are finally given to justify the effectiveness of the proposed schemes.

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23

Zhang, Hai, Jinde Cao, and Wei Jiang. "Reachability and Controllability of Fractional Singular Dynamical Systems with Control Delay." Journal of Applied Mathematics 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/567089.

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This paper is concerned with the reachability and controllability of fractional singular dynamical systems with control delay. The factors of such systems including the Caputo’s fractional derivative, control delay, and singular coefficient matrix are taken into account synchronously. The state structure of fractional singular dynamical systems with control delay is characterized by analysing the state response and reachable set. A set of sufficient and necessary conditions of controllability for such systems are established based on the algebraic approach. Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed criteria.

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24

Alharbi, Weam, and Snezhana Hristova. "New Series Solution of the Caputo Fractional Ambartsumian Delay Differential Equationation by Mittag-Leffler Functions." Mathematics 9, no. 2 (January 13, 2021): 157. http://dx.doi.org/10.3390/math9020157.

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The fractional generalization of the Ambartsumian delay equation with Caputo’s fractional derivative is considered. The Ambartsumian delay equation is very difficult to be solved neither in the case of ordinary derivatives nor in the case of fractional derivatives. In this paper we combine the Laplace transform with the Adomian decomposition method to solve the studied equation. The exact solution is obtained as a series which terms are expressed by the Mittag-Leffler functions. The advantage of the present approach over the known in the literature ones is discussed.

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25

Gómez Aguilar, José Francisco, Juan Rosales García, Jesus Bernal Alvarado, and Manuel Guía. "Mathematical modelling of the mass-spring-damper system - A fractional calculus approach." Acta Universitaria 22, no. 5 (August 15, 2012): 5–11. http://dx.doi.org/10.15174/au.2012.328.

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In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter characterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

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26

Devi, A., and M. Jakhar. "A New Computational Approach for Solving Fractional Order Telegraph Equations." Journal of Scientific Research 13, no. 3 (September 1, 2021): 715–32. http://dx.doi.org/10.3329/jsr.v13i3.50659.

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In this work, a modified decomposition method namely Sumudu-Adomian Decomposition Method (SADM) is implemented to find the exact and approximate solutions of fractional order telegraph equations. The derivatives of fractional-order are expressed in terms of caputo operator. Some numerical examples are illustrated to examine the efficiency of the proposed technique. Solutions of fractional order telegraph equations are obtained in the form of a series solution. It is observed that the solutions of fractional order telegraph equations converge towards the solution of an integer-order problem, which confirmed the reliability of the suggested method.

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27

Ahmad, Shabir, Aman Ullah, Ali Akgül, and Manuel De la Sen. "A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel." Journal of Function Spaces 2021 (September 20, 2021): 1–11. http://dx.doi.org/10.1155/2021/8770488.

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In this paper, we introduce the Yang transform homotopy perturbation method (YTHPM), which is a novel method. We provide formulae for the Yang transform of Caputo-Fabrizio fractional order derivatives. We derive an algorithm for the solution of Caputo-Fabrizio (CF) fractional order partial differential equation in series form and show its convergence to the exact solution. To demonstrate the novel approach, we include some examples with detailed solutions. We use tables and graphs to compare the exact and approximate solutions.

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28

Quezada-Téllez, L. A., and L. Franco-Pérez. "A fractional logistic approach for economic growth." International Journal of Modern Physics C 29, no. 12 (December 2018): 1850123. http://dx.doi.org/10.1142/s0129183118501231.

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A discrete logistic model is proposed for the Gross Domestic Product of an economy and its parameters are adjusted for some specific countries, taken from the Organization for Economic Co-operation and Development databases. Although this model fits “adequately” for some countries analyzed in this work, others show that it could be enhanced. A Caputo like fractional discrete model as a fractional version of the logistic one is presented, with the parameters stated previously. It is shown, by using the mean squared error, that this fractional version fits better for an economy whose behavior shows over time an isolated and unexpected boom period in its economy growth followed by a crisis period. For this type of dynamics, the fractional-order logistic equation improves the modeling given by an integer order logistic equation.

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29

Pandolfi, L. "A Lavrent'ev-type approach to the on-line computation of Caputo fractional derivatives." Inverse Problems 24, no. 1 (January 17, 2008): 015014. http://dx.doi.org/10.1088/0266-5611/24/1/015014.

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30

Khan, Yasir, Muhammad Altaf Khan, Fatmawati, and Naeem Faraz. "A fractional Bank competition model in Caputo-Fabrizio derivative through Newton polynomial approach." Alexandria Engineering Journal 60, no. 1 (February 2021): 711–18. http://dx.doi.org/10.1016/j.aej.2020.10.003.

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31

Surkov, Platon G. "Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach." Fractional Calculus and Applied Analysis 24, no. 3 (June 1, 2021): 895–922. http://dx.doi.org/10.1515/fca-2021-0038.

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Abstract A specific formulation of the “classical” problem of mathematical analysis is considered. This is the problem of calculating the derivative of a function. The purpose of this work is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on the methods of control theory. The input data of the algorithm is represented by inaccurate measured function values at discrete, frequently enough, times. The proposed algorithm is based on two aspects: a local modification of the Tikhonov regularization method from the theory of ill-posed problems and the Krasovskii extremal shift method from the guaranteed control theory, both of which ensure the stability to informational noises and computational errors. Numerical experiments were carried out to illustrate the operation of the algorithm.

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32

Izadi, Mohammad, and Hari M. Srivastava. "A Discretization Approach for the Nonlinear Fractional Logistic Equation." Entropy 22, no. 11 (November 21, 2020): 1328. http://dx.doi.org/10.3390/e22111328.

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The present study aimed to develop and investigate the local discontinuous Galerkin method for the numerical solution of the fractional logistic differential equation, occurring in many biological and social science phenomena. The fractional derivative is described in the sense of Liouville-Caputo. Using the upwind numerical fluxes, the numerical stability of the method is proved in the L∞ norm. With the aid of the shifted Legendre polynomials, the weak form is reduced into a system of the algebraic equations to be solved in each subinterval. Furthermore, to handle the nonlinear term, the technique of product approximation is utilized. The utility of the present discretization technique and some well-known standard schemes is checked through numerical calculations on a range of linear and nonlinear problems with analytical solutions.

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33

Alizadeh, Ali, and Sohrab Effati. "An iterative approach for solving fractional optimal control problems." Journal of Vibration and Control 24, no. 1 (March 9, 2016): 18–36. http://dx.doi.org/10.1177/1077546316633391.

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In this work, the variational iteration method (VIM) is used to solve a class of fractional optimal control problems (FOCPs). New Lagrange multipliers are determined and some new iterative formulas are presented. The fractional derivative (FD) in these problems is in the Caputo sense. The necessary optimality conditions are achieved for FOCPs in terms of associated Euler–Lagrange equations and then the VIM is used to solve the resulting fractional differential equations. This technique rapidly provides the convergent successive approximations of the exact solution and the solutions approach the classical solutions of the problem as the order of the FDs approaches 1. To achieve the solution of the FOCPs using VIM, four illustrative examples are included to demonstrate the validity and applicability of the proposed method.

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34

Navickas, Zenonas, Tadas Telksnys, Inga Timofejeva, Romas Marcinkevičius, and Minvydas Ragulskis. "AN OPERATOR-BASED APPROACH FOR THE CONSTRUCTION OF CLOSED-FORM SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS." Mathematical Modelling and Analysis 23, no. 4 (October 9, 2018): 665–85. http://dx.doi.org/10.3846/mma.2018.040.

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An operator-based approach for the construction of closed-form solutions to fractional differential equations is presented in this paper. The technique is based on the analysis of Caputo and Riemann-Liouville algebras of fractional power series. Explicit solutions to a class of linear fractional differential equations are obtained in terms of Mittag-Leffler and fractionally-integrated exponential functions in order to demonstrate the viability of the proposed technique.

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35

Agarwal, Ravi, Snezhana Hristova, and Donal O’Regan. "Applications of Lyapunov Functions to Caputo Fractional Differential Equations." Mathematics 6, no. 11 (October 30, 2018): 229. http://dx.doi.org/10.3390/math6110229.

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One approach to study various stability properties of solutions of nonlinear Caputo fractional differential equations is based on using Lyapunov like functions. A basic question which arises is the definition of the derivative of the Lyapunov like function along the given fractional equation. In this paper, several definitions known in the literature for the derivative of Lyapunov functions among Caputo fractional differential equations are given. Applications and properties are discussed. Several sufficient conditions for stability, uniform stability and asymptotic stability with respect to part of the variables are established. Several examples are given to illustrate the theory.

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36

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

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

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37

Ceretani, Andrea N. "A note on models for anomalous phase-change processes." Fractional Calculus and Applied Analysis 23, no. 1 (February 25, 2020): 167–82. http://dx.doi.org/10.1515/fca-2020-0006.

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AbstractWe review some fractional free boundary problems that were recently considered for modeling anomalous phase-transitions. All problems are of Stefan type and involve fractional derivatives in time according to Caputo’s definition. We survey the assumptions from which they are obtained and observe that the problems are nonequivalent though all of them reduce to a classical Stefan problem when the order of the fractional derivatives is replaced by one. We further show that a simple heuristic approach built upon a fractional version of the energy balance and the classical Fourier’s law leads to a natural generalization of the classical Stefan problem in which time derivatives are replaced by fractional ones.

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Sakar, Mehmet Giyas, and Onur Saldır. "A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems." Fractal and Fractional 4, no. 4 (November 24, 2020): 53. http://dx.doi.org/10.3390/fractalfract4040053.

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In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to α and β are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms.

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Khan, Muhammad Altaf. "The dynamics of a new chaotic system through the Caputo–Fabrizio and Atanagan–Baleanu fractional operators." Advances in Mechanical Engineering 11, no. 7 (July 2019): 168781401986654. http://dx.doi.org/10.1177/1687814019866540.

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The aim of this article is to analyze the dynamics of the new chaotic system in the sense of two fractional operators, that is, the Caputo–Fabrizio and the Atangana–Baleanu derivatives. Initially, we consider a new chaotic model and present some of the fundamental properties of the model. Then, we apply the Caputo–Fabrizio derivative and implement a numerical procedure to obtain their graphical results. Further, we consider the same model, apply the Atangana–Baleanu operator, and present their analysis. The Atangana–Baleanu model is used further to present a numerical approach for their solutions. We obtain and discuss the graphical results to each operator in details. Furthermore, we give a comparison of both the operators applied on the new chaotic model in the form of various graphical results by considering many values of the fractional-order parameter [Formula: see text]. We show that at the integer case, both the models (in Caputo–Fabrizio sense and the Atangana–Baleanu sense) give the same results.

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A. F. dos Santos, Maike. "Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels." Fractal and Fractional 2, no. 3 (July 29, 2018): 20. http://dx.doi.org/10.3390/fractalfract2030020.

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The investigation of diffusive process in nature presents a complexity associated withmemory effects. Thereby, it is necessary new mathematical models to involve memory conceptin diffusion. In the following, I approach the continuous time random walks in the context ofgeneralised diffusion equations. To do this, I investigate the diffusion equation with exponential andMittag–Leffler memory-kernels in the context of Caputo–Fabrizio and Atangana–Baleanu fractionaloperators on Caputo sense. Thus, exact expressions for the probability distributions are obtained,in that non-Gaussian distributions emerge. I connect the distribution obtained with a rich class ofdiffusive behaviour. Moreover, I propose a generalised model to describe the random walk processwith resetting on memory kernel context.

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Firoozjaee, M. A., H. Jafari, A. Lia, and D. Baleanu. "Numerical approach of Fokker–Planck equation with Caputo–Fabrizio fractional derivative using Ritz approximation." Journal of Computational and Applied Mathematics 339 (September 2018): 367–73. http://dx.doi.org/10.1016/j.cam.2017.05.022.

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Kheybari, Samad, Mohammad Taghi Darvishi, and Mir Sajjad Hashemi. "A semi-analytical approach to Caputo type time-fractional modified anomalous sub-diffusion equations." Applied Numerical Mathematics 158 (December 2020): 103–22. http://dx.doi.org/10.1016/j.apnum.2020.07.023.

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Haque, Ehsan Ul, Aziz Ullah Awan, Nauman Raza, Muhammad Abdullah, and Maqbool Ahmad Chaudhry. "A computational approach for the unsteady flow of maxwell fluid with Caputo fractional derivatives." Alexandria Engineering Journal 57, no. 4 (December 2018): 2601–8. http://dx.doi.org/10.1016/j.aej.2017.07.012.

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Mohamed Ali, Hegagi, and Ismail Gad Ameen. "An Efficient Approach for Solving Fractional Dynamics of a Predator-Prey System." Modern Applied Science 13, no. 11 (October 24, 2019): 116. http://dx.doi.org/10.5539/mas.v13n11p116.

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In this work, we execute a generally new analytical technique, the modified generalized Mittag-Leffler function method (MGMLFM) for solving nonlinear partial differential equations containing fractional derivative emerging in predator-prey biological population dynamics system. This dynamics system are given by a set of fractional differential equations in the Caputo sense. A new solution is constructed in a power series. The stability of equilibrium points is studied. Moreover, numerical solutions for different cases are given and the methodology is displayed. We conducted a comparing between the results obtained by our method with the results obtained by other methods to illustrate the reliability and effectiveness of our main results.

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Kumar, Sunil, R. P. Chauhan, Jagdev Singh, and Devendra Kumar. "A computational study of transmission dynamics for dengue fever with a fractional approach." Mathematical Modelling of Natural Phenomena 16 (2021): 48. http://dx.doi.org/10.1051/mmnp/2021032.

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Fractional derivatives are considered an influential weapon in terms of analysis of infectious diseases because of their nonlocal nature. The inclusion of the memory effect is the prime advantage of fractional-order derivatives. The main objective of this article is to investigate the transmission dynamics of dengue fever, we consider generalized Caputo-type fractional derivative (GCFD) (CD0β,σ) for alternate representation of dengue fever disease model. We discuss the existence and uniqueness of the solution of model by using fixed point theory. Further, an adaptive predictor-corrector technique is utilized to evaluate the considered model numerically.

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Nabil, Tamer. "Solvability of Fractional Differential Inclusion with a Generalized Caputo Derivative." Journal of Function Spaces 2020 (December 26, 2020): 1–11. http://dx.doi.org/10.1155/2020/2917306.

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This paper is devoted to the investigation of a kind of generalized Caputo semilinear fractional differential inclusions with deviated-advanced nonlocal conditions. Solvability of the problem is established by means of the Leray-Schauder’s alternative approach with the help of the Lagrange mean-value classical theorem. Finally, some examples are given to delineate the efficient of theoretical results.

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Nualsaard, Naravadee, Anirut Luadsong, and Nitima Aschariyaphotha. "The Numerical Solution of Fractional Black-Scholes-Schrodinger Equation Using the RBFs Method." Advances in Mathematical Physics 2020 (May 15, 2020): 1–17. http://dx.doi.org/10.1155/2020/1942762.

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In this paper, radial basis functions (RBFs) method was used to solve a fractional Black-Scholes-Schrodinger equation in an option pricing of financial problems. The RBFs method is applied in discretizing a spatial derivative process. The approximation of time fractional derivative is interpreted in the Caputo’s sense by a simple quadrature formula. This RBFs approach was theoretically proved with different problems of two numerical examples: time step arbitrage bubble case and time linear arbitrage bubble case. Then, the numerical results were compared with the semiclassical solution in case of fractional order close to 1. As a result, both numerical examples showed that the option prices from RBFs method satisfy the semiclassical solution.

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Billur İskender Eroğlu, Beyza, Derya Avcı, and Necati Özdemir. "Constrained Optimal Control of A Fractionally Damped Elastic Beam." International Journal of Nonlinear Sciences and Numerical Simulation 21, no. 3-4 (May 26, 2020): 389–95. http://dx.doi.org/10.1515/ijnsns-2018-0393.

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AbstractThis work presents the constrained optimal control of a fractionally damped elastic beam in which the damping characteristic is described with the Caputo fractional derivative of order 1/2. To achieve the optimal control that involves energy optimal control index with fixed endpoints, the fractionally damped elastic beam problem is first converted to a state space form of order 1/2 by using a change of coordinates. Then, the state and the costate equations are set in terms of Hamiltonian formalism and the constrained control law is acquired from Pontryagin Principle. The numerical solution of the problem is obtained with Grünwald-Letnikov approach by utilizing the link between the Riemann-Liouville and the Caputo fractional derivatives. Application of the formulations is demonstrated with an example and the illustrations are figured by MATLAB. Also, the effectiveness of the Grünwald-Letnikov approach is exhibited by comparing it with an iterative method which is one-step Adams-Bashforth-Moulton method.

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Al-Smadi, Mohammed, Asad Freihat, Hammad Khalil, Shaher Momani, and Rahmat Ali Khan. "Numerical Multistep Approach for Solving Fractional Partial Differential Equations." International Journal of Computational Methods 14, no. 03 (April 13, 2017): 1750029. http://dx.doi.org/10.1142/s0219876217500293.

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In this paper, we proposed a novel analytical technique for one-dimensional fractional heat equations with time fractional derivatives subjected to the appropriate initial condition. This new analytical technique, namely multistep reduced differential transformation method (MRDTM), is a simple amendment of the reduced differential transformation method, in which it is treated as an algorithm in a sequence of small intervals, in order to hold out accurate approximate solutions over a longer time frame compared to the traditional RDTM. The fractional derivatives are described in the Caputo sense, while the behavior of solutions for different values of fractional order [Formula: see text] compared with exact solutions is shown graphically. The analysis is accompanied by four test examples to demonstrate that the proposed approach is reliable, fully compatible with the complexity of these equations, and can be strongly employed for many other nonlinear problems in fractional calculus.

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Veeresha, Pundikala, Haci Mehmet Baskonus, and Wei Gao. "Strong Interacting Internal Waves in Rotating Ocean: Novel Fractional Approach." Axioms 10, no. 2 (June 16, 2021): 123. http://dx.doi.org/10.3390/axioms10020123.

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The main objective of the present study is to analyze the nature and capture the corresponding consequences of the solution obtained for the Gardner–Ostrovsky equation with the help of the q-homotopy analysis transform technique (q-HATT). In the rotating ocean, the considered equations exemplify strong interacting internal waves. The fractional operator employed in the present study is used in order to illustrate its importance in generalizing the models associated with kernel singular. The fixed-point theorem and the Banach space are considered to present the existence and uniqueness within the frame of the Caputo–Fabrizio (CF) fractional operator. Furthermore, for different fractional orders, the nature has been captured in plots. The realized consequences confirm that the considered procedure is reliable and highly methodical for investigating the consequences related to the nonlinear models of both integer and fractional order.

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