66 results on '"Caputo derivative"'
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2. Darbo's Fixed-Point Theorem: Establishing Existence and Uniqueness Results for Hybrid Caputo–Hadamard Fractional Sequential Differential Equations.
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Yaseen, Muhammad, Mumtaz, Sadia, George, Reny, Hussain, Azhar, and Nabwey, Hossam A.
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HYBRID systems , *EXISTENCE theorems , *CAPUTO fractional derivatives , *FIXED point theory , *FRACTIONAL calculus , *FRACTIONAL differential equations , *DIFFERENTIAL equations - Abstract
This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo's fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The hybrid approach considered in this work combines the advantages of both the Caputo and Hadamard fractional derivatives, leading to a more comprehensive and versatile model for describing sequential processes. To address the problem of the existence and uniqueness of solutions for such hybrid fractional sequential differential equations, we turn to Darbo's fixed-point theorem, a powerful mathematical tool that establishes the existence of fixed points for certain types of mappings. By appropriately transforming the differential equation into an equivalent fixed-point formulation, we can exploit the properties of Darbo's theorem to analyze the solutions' existence and uniqueness. The outcomes of this research expand the understanding of HCHFSDEs and contribute to the growing body of knowledge in fractional calculus and fixed-point theory. These findings are expected to have significant implications in various scientific and engineering applications, where sequential processes are prevalent, such as in physics, biology, finance, and control theory. [ABSTRACT FROM AUTHOR]
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- 2024
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3. Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation.
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Alikhanov, Anatoly A., Asl, Mohammad Shahbazi, and Huang, Chengming
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FRACTIONAL calculus , *CAPUTO fractional derivatives , *WAVE equation , *FRACTIONAL integrals , *EQUATIONS , *NUMERICAL analysis - Abstract
This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0, 1). By providing an a priori estimate of the exact solution, we have established the continuous dependence on the initial data and uniqueness of the solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2-type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings. [ABSTRACT FROM AUTHOR]
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- 2024
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4. On collocation-Galerkin method and fractional B-spline functions for a class of stochastic fractional integro-differential equations.
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Masti, I. and Sayevand, K.
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FRACTIONAL calculus , *FUNCTIONAL equations , *INTEGRAL equations , *CAPUTO fractional derivatives , *FRACTIONAL integrals , *FRACTIONAL differential equations , *INTEGRO-differential equations - Abstract
In recent years, as detailed in several monographs, derivations of the fractional differential equations and fractional integral equations are based on random functional or stochastic equations, with the output that physical interpretation of the resulting fractional derivatives and fractional integrals has been elusive. In many different sciences and problems such as biological systems, environmental quality and natural resources engineering, and so on, stochastic equations and in some cases random functional have appeared. Despite the widespread use of stochastic fractional integro-differential equations (SFIDE), the analytical solution of this equation is not easy and in some cases it is impossible. Therefore, the existence of an efficient and appropriate numerical method can solve this problem. In this study and based on fractional derivative in the Caputo sense, we investigate a class of SFIDE due to Brownian motion by using fractional B-spline basis functions (FB-spline) and with the help of the collocation-Galerkin method as well as the trapezoidal integral law. In other words, a continuous operator problem (here was named as SFIDE) is transformed into a discrete problem by limited sets of basis functions with common assumptions and approximation methods. In the follow-up a system of linear equations is generated, which makes the analysis of the method be efficient. As an important advantage of this combined method is its flexible and easy implementation. Another advantage of the method is its ability to be implemented for different types of linear, non-linear and system of SFIDE, which are discussed in the body of manuscript. An accurate upper bound is obtained and some theorems are established on the stability and convergence analysis. The computational cost is estimated from the sum of the number arithmetic operations and a lower bound for approximation of this equation is formulated. Finally, by examining several examples, the computational performance of the proposed method effectively verifies the applicability and validity of the suggested scheme. [ABSTRACT FROM AUTHOR]
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- 2024
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5. A New Hybrid Optimal Auxiliary Function Method for Approximate Solutions of Non-Linear Fractional Partial Differential Equations.
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Ashraf, Rashid, Nawaz, Rashid, Alabdali, Osama, Fewster-Young, Nicholas, Ali, Ali Hasan, Ghanim, Firas, and Alb Lupaş, Alina
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FRACTIONAL differential equations , *PARTIAL differential equations , *NONLINEAR differential equations , *CAPUTO fractional derivatives , *NONLINEAR equations , *FRACTIONAL calculus , *HYBRID systems , *POISSON'S equation - Abstract
This study uses the optimal auxiliary function method to approximate solutions for fractional-order non-linear partial differential equations, utilizing Riemann–Liouville's fractional integral and the Caputo derivative. This approach eliminates the need for assumptions about parameter magnitudes, offering a significant advantage. We validate our approach using the time-fractional Cahn–Hilliard, fractional Burgers–Poisson, and Benjamin–Bona–Mahony–Burger equations. Comparative testing shows that our method outperforms new iterative, homotopy perturbation, homotopy analysis, and residual power series methods. These examples highlight our method's effectiveness in obtaining precise solutions for non-linear fractional differential equations, showcasing its superiority in accuracy and consistency. We underscore its potential for revealing elusive exact solutions by demonstrating success across various examples. Our methodology advances fractional differential equation research and equips practitioners with a tool for solving non-linear equations. A key feature is its ability to avoid parameter assumptions, enhancing its applicability to a broader range of problems and expanding the scope of problems addressable using fractional calculus techniques. [ABSTRACT FROM AUTHOR]
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- 2023
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6. A linear Galerkin numerical method for a quasilinear subdiffusion equation.
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Płociniczak, Łukasz
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GALERKIN methods , *FRACTIONAL calculus , *CAPUTO fractional derivatives , *EQUATIONS , *NONLINEAR functions - Abstract
We couple the L1 discretization for the Caputo derivative in time with the spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear functions of the solution. We prove the stability and convergence of the method with spectral accuracy in space. The temporal order depends on the regularity of the solution in time. Furthermore, we support our results with numerical simulations that utilize parallelism for spatial discretization. Moreover, as a side result, we find exact asymptotic values of the error constants along with their remainders for discretizations of the Caputo derivative and fractional integrals. These constants are the smallest possible, which improves previously established results from the literature. [ABSTRACT FROM AUTHOR]
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- 2023
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7. An analytic and numerical study for two classes of differential equations of fractional order involving Caputo and Khalil derivatives.
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Rakah, Mahdi, Anber, Ahmed, Dahmani, Zoubir, and Jebril, Iqbal
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MATHEMATICAL research , *DIFFERENTIAL equations , *FRACTIONAL calculus , *CAPUTO fractional derivatives , *KORTEWEG-de Vries equation - Abstract
In this paper, we study new classes of differential equations of fractional order. The first considered problem involves the derivative of Caputo, while the second one involves conformable Khalil derivative. For the first class, we prove an existence and uniqueness result, then, we discuss an example to show the applicability of the result. For the second one, we apply the exp-function method to find new traveling wave solutions for a generalised conformable fractional partial differential equation, then some examples on Ostrovsky and KdV equations are given to illustrate the efficiency and accuracy of the method. Some graphs are plotted and discussed to show more the importance of the obtained results. [ABSTRACT FROM AUTHOR]
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- 2023
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8. Time-Delay Fractional Variable Order Adaptive Synchronization and Anti-Synchronization between Chen and Lorenz Chaotic Systems Using Fractional Order PID Control.
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Padron, Joel Perez, Perez, Jose P., Diaz, Jose Javier Perez, and Astengo-Noguez, Carlos
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CAPUTO fractional derivatives , *FRACTIONAL calculus , *SYNCHRONIZATION , *CHAOS synchronization , *PROBLEM solving , *SYSTEMS theory - Abstract
In this research work, time-delay adaptive synchronization and adaptive anti-synchronization of chaotic fractional order systems are analyzed via the Caputo fractional derivative, and the prob-lem of synchronization and anti-synchronization of chaotic systems of variable fractional order is solved by using the fractional order PID control law, the adaptive laws of variable-order frac-tional calculus, and a control law deduced from Lyapunov's theory extended to systems of time-delay variable-order fractional calculus. In this research work, two important problems are solved in the control area: The first problem is described in which deals with syn-chro-nization of chaotic systems of adaptive fractional order with time delay, this problem is solved by using the fractional order PID control law and adaptative laws. The second problem is de-scribed in which deals with anti-synchronization of chaotic systems of adaptive frac-tional order with time delay, and this problem is solved by using the fractional order PID con-trol law and adaptative laws. [ABSTRACT FROM AUTHOR]
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- 2023
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9. Solution method for the time‐fractional hyperbolic heat equation.
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Dassios, Ioannis and Font, Francesc
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BOUNDARY value problems , *CAPUTO fractional derivatives , *SEPARATION of variables , *FOURIER series , *FRACTIONAL calculus , *HEAT equation , *ANALYTICAL solutions - Abstract
In this article, we propose a method to solve the time‐fractional hyperbolic heat equation. We first formulate a boundary value problem for the standard hyperbolic heat equation in a finite domain and provide an analytical solution by means of separation of variables and Fourier series. Then, we consider the same boundary value problem for the fractional hyperbolic heat equation. The fractional problem is solved using three different definitions of the fractional derivative: the Caputo fractional derivative and two recently defined alternative versions of this derivative, the Caputo–Fabrizio and the Atangana–Baleanu. A closed form of the solution is provided for each case. Finally, we compare the solutions of the fractional and the standard problem and show numerically that the solution of the standard hyperbolic heat equation can be retrieved from the solution of the fractional equation in the limit γ→2, where γ represents the exponent of the fractional derivative. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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10. New problem of sequential differential equations with nonlocal conditions.
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Bekkouche, Z. and Dahmani, Z.
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DIFFERENTIAL equations , *FRACTIONAL calculus , *CAPUTO fractional derivatives , *NONLINEAR differential equations , *FRACTIONAL integrals - Abstract
In this paper, we study a nonlinear differential equation of Lane Emden type. The considered problem, with its nonlocal conditions, involves Caputo fractional derivative and Riemann-Liouville integral in its nonlinearities. We investigate the existence and uniqueness of solution for our problem. Then, we study the Ulam-Hyers stability of its solutions. [ABSTRACT FROM AUTHOR]
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- 2021
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11. On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann-Liouville and Caputo Fractional Derivatives.
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Delgado, Briceyda B. and Macías-Díaz, Jorge E.
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FRACTIONAL calculus , *DIRAC equation , *PARTIAL differential equations , *WAVE equation , *CAPUTO fractional derivatives - Abstract
In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann-Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann-Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann-Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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12. Spline Collocation for Multi-Term Fractional Integro-Differential Equations with Weakly Singular Kernels.
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Pedas, Arvet and Vikerpuur, Mikk
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CAPUTO fractional derivatives , *FRACTIONAL calculus , *KERNEL (Mathematics) , *KERNEL functions , *REPRODUCING kernel (Mathematics) - Abstract
We consider general linear multi-term Caputo fractional integro-differential equations with weakly singular kernels subject to local or non-local boundary conditions. Using an integral equation reformulation of the proposed problem, we first study the existence, uniqueness and regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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13. An Approximate Solution of the Space Fractional-Order Heat Equation by the Non-Polynomial Spline Functions.
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Hasan, Nabaa N. and Salim, Omar H.
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CAPUTO fractional derivatives , *FRACTIONAL calculus , *HEAT equation , *SPLINE theory , *APPROXIMATION theory - Abstract
The linear non-polynomial spline is used here to solve the fractional partial differential equation (FPDE). The fractional derivatives are described in the Caputo sense. The tensor products are given for extending the one-dimensional linear nonpolynomial spline S1 to a two-dimensional spline S1 ⊗ S2 to solve the heat equation. In this paper, the convergence theorem of the method used to the exact solution is proved and the numerical examples show the validity of the method. All computations are implemented by Mathcad15. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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14. Article Continuity Result on the Order of a Nonlinear Fractional Pseudo-Parabolic Equation with Caputo Derivative.
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Ho Duy Binh, Luc Nguyen Hoang, Baleanu, Dumitru, and Ho Thi Kim Van
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FRACTALS , *CAPUTO fractional derivatives , *FRACTIONAL calculus , *PARABOLIC differential equations , *PARTIAL differential equations - Abstract
In this paper, we consider a problem of continuity fractional-order for pseudo-parabolic equations with the fractional derivative of Caputo. Here, we investigate the stability of the problem with respect to derivative parameters and initial data. We also show that uω' → uω in an appropriate sense as ω' → ω where ω is the fractional order. Moreover, to test the continuity fractional-order, we present several numerical examples to illustrate this property. [ABSTRACT FROM AUTHOR]
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- 2021
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15. Numerical Solution of Fuzzy Fractional Differential Equation By Haar Wavelet.
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Khakrangin, Sakineh, Allahviranloo, Tofigh, Mikaeilvand, Nasser, and Abbasbandy, Saeid
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FRACTIONAL calculus , *CAPUTO fractional derivatives , *HILBERT space , *FRACTIONAL differential equations , *WAVELET transforms - Abstract
In this paper, we deal with a wavelet operational method based on Haar wavelet to solve the fuzzy fractional differential equation in the Caputo derivative sense. To this end, we derive the Haar wavelet operational matrix of the fractional order integration. The given approach provides an efficient method to find the solution and its upper bond error. To complete the discussion, the convergence theorem is subsequently expressed in detail. So far, no paper has used the Harr wavelet method using generalized difference and fuzzy derivatives, and this is the first time we have done so. Finally, the presented examples reflect the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2021
16. Applying fractional calculus to analyze final consumption and gross investment influence on GDP.
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Badík, A. and Fečkan, M.
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CAPUTO fractional derivatives , *REGRESSION analysis , *LEAST squares , *FRACTIONAL calculus , *GROSS domestic product - Abstract
This paper points out the possibility of suitable use of Caputo fractional derivative in regression model. Fitting historical data using a regression model seems to be useful in many fields, among other things, for the short-term prediction of further developments in the state variable. Therefore, it is important to fit the historical data as accurately as possible using the given variables. Using Caputo fractional derivative, this accuracy can be increased in the model described in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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17. Numerical solution of linear fractional weakly singular integro-differential equations with integral boundary conditions.
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Pedas, Arvet, Tamme, Enn, and Vikerpuur, Mikk
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SINGULAR integrals , *FRACTIONAL calculus , *FRACTIONAL differential equations , *INTEGRO-differential equations , *BOUNDARY value problems , *CAPUTO fractional derivatives , *INTEGRAL equations - Abstract
We consider a class of boundary value problems for linear fractional weakly singular integro-differential equations with Caputo fractional derivatives and integral boundary conditions. Using an integral equation reformulation of the boundary value problem, we first study the regularity of the exact solution and its Caputo derivative. Based on the obtained regularity properties and by using suitable smoothing transformations along with spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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18. Experimental Study of Fractional-Order RC Circuit Model Using the Caputo and Caputo-Fabrizio Derivatives.
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Lin, Da, Liao, Xiaozhong, Dong, Lei, Yang, Ruocen, Yu, Samson S., Iu, Herbert Ho-Ching, Fernando, Tyrone, and Li, Zhen
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RC circuits , *ANALOG circuits , *CAPUTO fractional derivatives , *IDEAL sources (Electric circuits) , *RESISTOR-inductor-capacitor circuits - Abstract
This study employs the Caputo-Fabrizio fractional derivative to determine the model of fractional-order RC circuits with arbitrary voltage input which can be widely used in a variety of electrical systems. Analog circuit implementation of fractional-order RC circuits defined by Caputo-Fabrizio fractional derivative is presented and verified by comparing with the model proposed in this work. For the purpose of judging whether the fractional-order model defined by the Caputo-Fabrizio derivative is practical, the comparison experiments are carried out. By using Laplace transform, the analytical solutions of fractional-order RC circuits based on the Caputo-Fabrizio derivatives with constant and periodic voltage sources are deduced. Fractional-order model of RC circuits with arbitrary input are also calculated using the convolution formula. The correctness of the derivation of the model using the Caputo-Fabrizio derivative is verified. Through discussing the impedance model of capacitor in frequency domain, the analog realization of fractional capacitor based on the Caputo-Fabrizio derivative is derived. The fractional-orders of the RC circuits models defined by the Caputo and Caputo-Fabrizio fractional derivatives are fitted respectively through repeated charging and discharging experiment data. The fractional-order models based on the Caputo and Caputo-Fabrizio derivatives, and the integer-order model are all compared with the experiment data. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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19. Theorems on Some Families of Fractional Differential Equations and Their Applications.
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Bozkurt, Gülçin, Albayrak, Durmuş, and Dernek, Neşe
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CAPUTO fractional derivatives , *FRACTIONAL differential equations , *LAPLACE transformation , *FRACTIONAL calculus - Abstract
We use the Laplace transform method to solve certain families of fractional order differential equations. Fractional derivatives that appear in these equations are defined in the sense of Caputo fractional derivative or the Riemann-Liouville fractional derivative. We first state and prove our main results regarding the solutions of some families of fractional order differential equations, and then give examples to illustrate these results. In particular, we give the exact solutions for the vibration equation with fractional damping and the Bagley-Torvik equation. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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20. Fractional model of the dielectric dispersion.
- Author
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Ortega, A., Rosales, J.J., Cruz-Duarte, J.M., and Guía, M.
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EXAMPLE , *CAPUTO fractional derivatives - Abstract
Abstract This work deals with the generalization of the Lorentz model. It is made using the Caputo fractional derivative of order 0 < γ ≤ 1 in the frequency domain. The fractional complex dielectric function, index of refraction, reflectance, electric susceptibility were obtained. It is shown that all of them depend not only on the frequency of applied field, but also on the fractional order of the differential equation 0 < γ ≤ 1. This opens the possibility to describe more accurately the experimental results in complex systems. The ordinary results are obtained when γ = 1. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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21. A new definition of fractional derivative.
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Zheng, Zhibao, Zhao, Wei, and Dai, Hongzhe
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FRACTIONAL differential equations , *INTEGRAL transforms , *DEFINITIONS , *CAPUTO fractional derivatives - Abstract
Abstract In this paper, a new fractional derivative of the Caputo type is proposed and some basic properties are studied. The form of the definition shows that the new derivative is the natural extension of the Caputo one, and that it yields the Caputo derivative with designated memory length. By adaptively changing the memory length, the new definition is capable of capturing local memory effect in a distinct way, which is critical in modelling complex systems where the short memory properties has to be considered. Another attractive property of the new derivative is that it is naturally associated with the Riemann–Liouville definition and as a result, the well established Grünwald–Letnikov approach for numerically solving the fractional differential equation can be readily embedded to approximate the solution of differential equation that involves the new derivatives. Numerical simulations demonstrate the changeable memory effect of the new definition. Highlights • The paper presents a new fractional derivative of the Caputo type. • Some important properties and integral transform of the new derivative is derived. • The new derivative offers particular value for capturing the local memory effect. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
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22. Numerical solution of space fractional diffusion equation by the method of lines and splines.
- Author
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Salehi, Younes, Darvishi, Mohammad T., and Schiesser, William E.
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NUMERICAL analysis , *FRACTIONAL calculus , *CAPUTO fractional derivatives , *SPLINE theory , *APPROXIMATION theory , *BOUNDARY value problems - Abstract
This paper is devoted to the application of the method of lines to solve one-dimensional diffusion equation where the classical (integer) second derivative is replaced by a fractional derivative of the Caputo type of order α less than 2 as the space derivative. A system of initial value problems approximates the solution of the fractional diffusion equation with spline approximation of the Caputo derivative. The result is a numerical approach of order O ( Δ x 2 + Δ t m ) , where Δx and Δt denote spatial and temporal step-sizes, and 1 ≤ m ≤ 5 is an integer which is set by an ODE integrator that we used. The convergence and numerical stability of the method are considered, and numerical tests to investigate the efficiency and feasibility of the scheme are provided. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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23. A new approach for space-time fractional partial differential equations by residual power series method.
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Aylin Bayrak, Mine and Demir, Ali
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FRACTIONAL differential equations , *FRACTIONAL powers , *CAPUTO fractional derivatives , *FRACTIONAL calculus , *SPACE-time configurations - Abstract
In this paper, the approximate analytic solution of any order space-time fractional differential equations is constructed by means of semi-analytical method, named as residual power series method (RPSM). The first step is to reduce space-time fractional differential equation to either a space fractional differential equations or a time fractional differential equations before applying RSPM. The main step is to obtain fractional power series solutions by RSPM. At the final step, it is shown that RPSM is very efficacious, plain and powerful for obtaining the solution of any-order space-time fractional differential equations in the form of fractional power series by illustrative examples. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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24. Fractional thermoelasticity problem for a plane with a line crack under heat flux loading.
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Povstenko, Yuriy and Kyrylych, Tamara
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THERMOELASTICITY , *FRACTIONAL calculus , *HEAT flux , *CAPUTO fractional derivatives , *HEAT conduction - Abstract
Symmetric stress distribution in an infinite isotropic plane with a line crack, in which surfaces are exposed to the heat flux loading is considered in the framework of fractional thermoelasticity. The heat conduction is described by the time-fractional heat conduction equations with the Caputo derivative of fractional order α. The solution is obtained using the integral transform technique and is expressed in terms of the Mittag-Leffler function. The stress intensity factor is calculated for different values of the order of fractional derivative. The numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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25. A new approximate solution for the Telegraph equation of space-fractional order derivative by using Sumudu method.
- Author
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AL-Safi, Mohammed G. S., AL-Hussein, Wurood R. Abd, and Al-Shammari, Ayad Ghazi Naser
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FRACTIONAL calculus , *FRACTIONAL differential equations , *BOUNDARY value problems , *HOMOTOPY theory , *CAPUTO fractional derivatives - Abstract
In this work, we are concerned with how to find an explicit approximate solution (AS) for the telegraph equation of space-fractional order (TESFO) using Sumudu transform method (STM). In this method, the space-fractional order derivatives are defined in the Caputo idea. The Sumudu method (SM) is established to be reliable and accurate. Three examples are discussed to check the applicability and the simplicity of this method. Finally, the Numerical results are tabulated and displayed graphically whenever possible to make comparisons between the AS and exact solution (ES). [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
26. A fractional dynamical model for honeybee colony population.
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Akman Yıldız, Tuğba
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COLONY collapse disorder of honeybees , *FRACTIONAL calculus , *CAPUTO fractional derivatives , *MATHEMATICAL models , *DIFFERENTIATION (Mathematics) - Abstract
Bees are the main contributors of pollination and it is predicted that disruption of pollination causes some serious problems in economics, agriculture and ecology. It has been reported that honeybee colony collapse has increased dramatically for some time in different parts of the world. In order to investigate the causes of colony collapse, we establish a fractional multi-order honeybee colony population model. Two different models with Caputo and Caputo–Fabrizio differentiation operators have been compared. We justify the existence and uniqueness of the solution. Stability analysis has been presented. We find the numerical values of the fractional order ( p , q ) so that the numerical results are close to the experimental findings more than the integer-order case. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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27. A GENERALIZED DEFINITION OF CAPUTO DERIVATIVES AND ITS APPLICATION TO FRACTIONAL ODES.
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LEI LI and JIAN-GUO LIU
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CAPUTO fractional derivatives , *DIFFERENTIAL equations , *MATHEMATICAL convolutions , *FRACTIONAL calculus , *INTEGRAL equations - Abstract
We propose a generalized definition of Caputo derivatives from t = 0 of order γ∈in (0, 1) using a convolution group, and we build a convenient framework for studying initial value problems of general nonlinear time fractional differential equations. Our strategy is to define a modified Riemann-Liouville fractional calculus which agrees with the traditional Riemann-Liouville definition for t > 0 but includes some singularities at t = 0 so that the group property holds. Then, making use of this fractional calculus, we introduce the generalized definition of Caputo derivatives. The new definition is consistent with various definitions in the literature while revealing the underlying group structure. The underlying group property makes many properties of Caputo derivatives natural. In particular, it allows us to deconvolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general comparison principle with the most general conditions. This then allows for a priori energy estimates of fractional PDEs. Since the new definition is valid for locally integrable functions that can blow up in finite time, it provides a framework for solutions to fractional ODEs and fractional PDEs. Many fundamental results for fractional ODEs are revisited within this framework under very weak conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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28. Applying Fractional Calculus to Analyze Economic Growth Modelling.
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LUO, D., WANG, J. R., and FEČKAN, M.
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FRACTIONAL calculus , *MATHEMATICAL models of economic development , *LEAST squares , *GENETIC algorithms , *CAPUTO fractional derivatives - 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]. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
29. Polynomial solutions for a class of fractional differential equations and systems.
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Dahmani, Zoubir, Abdelaoui, Mohamed Amin, and Houas, Mohamed
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POLYNOMIALS , *FRACTIONAL calculus , *CAPUTO fractional derivatives , *DIFFERENTIAL equations , *FRACTIONAL differential equations - Abstract
This paper studies the existence and uniqueness of polynomial solutions for a class of fractional differential equations and systems involving Caputo derivatives. Several illustrative examples with explicit solutions are also presented. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
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30. A High-Order Accurate Numerical Scheme for the Caputo Derivative with Applications to Fractional Diffusion Problems.
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Luo, Wei-Hua, Li, Changpin, Huang, Ting-Zhu, Gu, Xian-Ming, and Wu, Guo-Cheng
- Subjects
- *
SCHEMES (Algebraic geometry) , *CAPUTO fractional derivatives , *MATHEMATICAL functions , *DIFFUSION processes , *FRACTIONAL calculus , *NUMERICAL analysis - Abstract
In this paper, using the piecewise linear and quadratic Lagrange interpolation functions, we propose a novel numerical approximate method for the Caputo fractional derivative. For the obtained explicit recursion formula, the truncation error is investigated, which shows the involved convergence order isO(τ3−β) withβ∈(0,1). As an application, we use this proposed numerical approximation to solve the time fractional diffusion equations by the barycentric rational interpolations in space. The resultant systems of algebraic equations, truncation error, convergence, and stability are analyzed. Theoretical analysis and numerical examples show this constructed method enjoys accuracy of, wheredis the degree of the rational polynomial. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
31. A hybridizable discontinuous Galerkin method for a class of fractional boundary value problems.
- Author
-
Karaaslan, Mehmet Fatih, Celiker, Fatih, and Kurulay, Muhammet
- Subjects
- *
DISCONTINUOUS functions , *NUMERICAL solutions to boundary value problems , *GALERKIN methods , *FRACTIONAL calculus , *CAPUTO fractional derivatives - Abstract
In this paper, we present a hybridizable discontinuous Galerkin (HDG) method for solving a class of fractional boundary value problems involving Caputo derivatives. The HDG methods have the computational advantage of eliminating all internal degrees of freedom and the only globally coupled unknowns are those at the element interfaces. Furthermore, the global stiffness matrix is tridiagonal, symmetric, and positive definite. Internal degrees of freedom are recovered at an element-by-element postprocessing step. We carry out a series of numerical experiments to ascertain the performance of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
32. Time-Fractional Diffusion with Mass Absorption in a Half-Line Domain due to Boundary Value of Concentration Varying Harmonically in Time.
- Author
-
Povstenko, Yuriy and Kyrylych, Tamara
- Subjects
- *
FRACTIONAL calculus , *CAPUTO fractional derivatives , *LAPLACE transformation , *MATHEMATICAL transformations , *BOUNDARY value problems - Abstract
The time-fractional diffusion equation with mass absorption is studied in a half-line domain under the Dirichlet boundary condition varying harmonically in time. The Caputo derivative is employed. The solution is obtained using the Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate. The results of numerical calculations are illustrated graphically. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
33. RECONSTRUCTION OF THE ROBIN BOUNDARY CONDITION AND ORDER OF DERIVATIVE IN TIME FRACTIONAL HEAT CONDUCTION EQUATION.
- Author
-
BROCIEK, RAFAL and SLOTA, DAMIAN
- Subjects
- *
INVERSE problems , *CAPUTO fractional derivatives , *FRACTIONAL calculus , *HEAT conduction , *FINITE difference method , *TIKHONOV regularization , *MATHEMATICAL models - Abstract
This paper describes an algorithm for reconstruction the boundary condition and order of derivative for the heat conduction equation of fractional order. This fractional order derivative was applied to time variable and was defined as the Caputo derivative. The heat transfer coefficient, occurring in the boundary condition of the third kind, was reconstructed. Additional information for the considered inverse problem is given by the temperature measurements at selected points of the domain. The direct problem was solved by using the implicit finite difference method. To minimize functional defining the error of approximate solution an Artificial Bee Colony (ABC) algorithm and Nelder-Mead method were used. In order to stabilize the procedure the Tikhonov regularization was applied. The paper presents examples to illustrate the accuracy and stability of the presented algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
34. Preprocessing schemes for fractional-derivative problems to improve their convergence rates.
- Author
-
Stynes, Martin and Gracia, José Luis
- Subjects
- *
STOCHASTIC convergence , *FRACTIONAL calculus , *DERIVATIVES (Mathematics) , *BOUNDARY value problems , *CAPUTO fractional derivatives - Abstract
A simple and inexpensive preprocessing of an initial–boundary value problem with a Caputo time derivative is shown theoretically and numerically to yield an enhanced convergence rate for the L1 scheme. The same preprocessing can also be used with other methods for time-dependent problems. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
35. Numerical Solution for a Variable-Order Fractional Nonlinear Cable Equation via Chebyshev Cardinal Functions.
- Author
-
Irandoust-Pakchin, Safar, Abdi-Mazraeh, Somayeh, and Khani, Ali
- Subjects
- *
FRACTIONAL differential equations , *CHEBYSHEV series , *CAPUTO fractional derivatives , *BROWNIAN motion , *RIEMANN integral , *FRACTIONAL calculus - Abstract
In this paper, a variable-order fractional derivative nonlinear cable equation is considered. It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of class of fractional partial differential equation with variable coefficient of fractional differential equation in various continues functions of spatial and time orders. Our main aim is to generalize the Chebyshev cardinal operational matrix to the fractional calculus. Finally, illustrative examples are included to demonstrate the validity and applicability of the presented technique. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
36. General solutions of higher order impulsive fractional differential equations involved with the Caputo type generalized fractional derivatives and applications.
- Author
-
YUJI LIU
- Subjects
- *
FRACTIONAL calculus , *CAPUTO fractional derivatives , *FRACTIONAL integrals , *RIEMANNIAN geometry , *HADAMARD matrices , *BOUNDARY value problems - Abstract
The generalized fractional integral operator and the Caputo type generalized fractional derivative operator are defined which contain the Riemann-Liouville integral operator, the Hadamard fractional integral operator, the Caputo fractional derivative operator and the Caputo type Hadamard fractional derivative operator as special cases. General solutions (the explicit solutions) of the impulsive Caputo type generalized fractional differential equations are given. Applying our results, existence results of solutions of boundary value problems for an impulsive fractional differential equations involved with the Caputo type generalized fractional derivatives are established. Examples and some remarks on recent published papers are presented to illustrate the main theorems. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
37. Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives.
- Author
-
Ferreira, M. and Vieira, N.
- Subjects
- *
EIGENFUNCTIONS , *LAPLACE'S equation , *DIRAC operators , *CAPUTO fractional derivatives , *FRACTIONAL calculus - Abstract
In this paper, we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operatorwhereand the fractional derivatives,,are in the Caputo sense. Applying integral transform methods, we describe a complete family of eigenfunctions and fundamental solutions of the operatorin classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag–Leffler function. From the family of fundamental solutions obtained, we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
38. Hermite Pseudospectral Method for the Time Fractional Diffusion Equation with Variable Coefficients.
- Author
-
Zeting Liu and Shujuan Lü
- Subjects
- *
PSEUDOSPECTRUM , *HEAT equation , *CAPUTO fractional derivatives , *FINITE differences , *FRACTIONAL calculus - Abstract
We consider the initial value problem of the time fractional diffusion equation on the whole line and the fractional derivative is described in Caputo sense. A fully discrete Hermite pseudospectral approximation scheme is structured basing Hermite-Gauss points in space and finite difference in time. Unconditionally stability and convergence are proved. Numerical experiments are presented and the results conform to our theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
39. Two Approaches to Obtaining the Space-Time Fractional Advection-Diffusion Equation.
- Author
-
Povstenko, Yuriy and Kyrylych, Tamara
- Subjects
- *
ADVECTION-diffusion equations , *FRACTIONAL calculus , *CAPUTO fractional derivatives , *CAUCHY problem , *LAPLACE transformation , *FOURIER transforms - Abstract
Two approaches resulting in two different generalizations of the space-time-fractional advection-diffusion equation are discussed. The Caputo time-fractional derivative and Riesz fractional Laplacian are used. The fundamental solutions to the corresponding Cauchy and source problems in the case of one spatial variable are studied using the Laplace transform with respect to time and the Fourier transform with respect to the spatial coordinate. The numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
40. Fractional optimal control problem for differential system with delay argument.
- Author
-
Bahaa, G.
- Subjects
- *
FRACTIONAL calculus , *OPTIMAL control theory , *HILBERT space , *CONSTRAINTS (Physics) , *CAPUTO fractional derivatives - Abstract
In this paper, we apply the classical control theory to a fractional differential system in a bounded domain. The fractional optimal control problem (FOCP) for differential system with time delay is considered. The fractional time derivative is considered in a Riemann-Liouville sense. We first study the existence and the uniqueness of the solution of the fractional differential system with time delay in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a FOCP is considered as a function of both state and control variables, and the dynamic constraints are expressed by a partial fractional differential equation. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of a right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in detail. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
41. Approximate solutions of Volterra-Fredholm integro-differential equations of fractional order.
- Author
-
Alkan, Sertan and Hatipoglu, Veysel Fuat
- Subjects
- *
APPROXIMATE solutions (Logic) , *INTEGRO-differential equations , *VOLTERRA equations , *FREDHOLM equations , *CAPUTO fractional derivatives , *FRACTIONAL calculus - Abstract
In this study, sinc-collocation method is introduced for solving Volterra-Fredholm integro-differential equations of fractional order. Fractional derivative is described in the Caputo sense. Obtained results are given to literature as a new theorem. Some numerical examples are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
42. Fractional heat conduction in a space with a source varying harmonically in time and associated thermal stresses.
- Author
-
Povstenko, Y.
- Subjects
- *
CAPUTO fractional derivatives , *FRACTIONAL calculus , *ELECTRICAL harmonics , *HEAT conduction , *THERMAL stresses - Abstract
Time-nonlocal generalization of the classical Fourier law with the “long-tail” power kernel can be interpreted in terms of fractional calculus (theory of integrals and derivatives of noninteger order) and leads to the time-fractional heat conduction equation with the Caputo derivative. Fractional heat conduction equation with the harmonic source term under zero initial conditions is studied. Different formulations of the problem for the standard parabolic heat conduction equation and for the hyperbolic wave equation appearing in thermoelasticity without energy dissipation are discussed. The integral transform technique is used. The corresponding thermal stresses are found using the displacement potential. [ABSTRACT FROM PUBLISHER]
- Published
- 2016
- Full Text
- View/download PDF
43. Theory and applications of a more general form for fractional power series expansion.
- Author
-
Jaradat, I., Al-Dolat, M., Al-Zoubi, K., and Alquran, M.
- Subjects
- *
FRACTIONAL calculus , *POWER series , *CAPUTO fractional derivatives , *ITERATIVE methods (Mathematics) , *DIFFERENTIAL equations - Abstract
The latent potentialities and applications of fractional calculus present a mathematical challenge to establish its theoretical framework. One of these challenges is to have a compact and self-contained fractional power series representation that has a wider application scope and allows studying analytical properties. In this letter, we introduce a new more general form of fractional power series expansion, based on the Caputo sense of fractional derivative, with corresponding convergence property. In order to show the functionality of the proposed expansion, we apply the corresponding iterative fractional power series scheme to solve several fractional (integro-)differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
44. Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method.
- Author
-
Sakar, Mehmet Giyas, Uludag, Fatih, and Erdogan, Fevzi
- Subjects
- *
FRACTIONAL calculus , *NUMERICAL solutions to partial differential equations , *DELAY differential equations , *HOMOTOPY theory , *PERTURBATION theory , *CAPUTO fractional derivatives - Abstract
In this paper, homotopy perturbation method (HPM) is applied to solve fractional partial differential equations (PDEs) with proportional delay in t and shrinking in x . The method do not require linearization or small perturbation. The fractional derivatives are taken in the Caputo sense. The present method performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of α are presented graphically. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
45. Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line.
- Author
-
Hashemi, M.S. and Baleanu, D.
- Subjects
- *
CAPUTO fractional derivatives , *FRACTIONAL calculus , *GEOMETRIC function theory , *DEPENDENT variables , *ANALYTIC functions - Abstract
We propose a simple and accurate numerical scheme for solving the time fractional telegraph (TFT) equation within Caputo type fractional derivative. A fictitious coordinate ϑ is imposed onto the problem in order to transform the dependent variable u ( x , t ) into a new variable with an extra dimension. In the new space with the added fictitious dimension, a combination of method of line and group preserving scheme (GPS) is proposed to find the approximate solutions. This method preserves the geometric structure of the problem. Power and accuracy of this method has been illustrated through some examples of TFT equation. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
46. Nonlocal electrical diffusion equation.
- Author
-
Gómez-Aguilar, J. F., Escobar-Jiménez, R. F., Olivares-Peregrino, V. H., Benavides-Cruz, M., and Calderón-Ramón, C.
- Subjects
- *
HEAT equation , *FRACTIONAL calculus , *CAPUTO fractional derivatives , *ELECTROCHEMISTRY , *SEMICONDUCTORS , *ELECTRIC fields - Abstract
In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is and for the time domain is . We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
47. The finite difference method for Caputo-type parabolic equation with fractional Laplacian: One-dimension case.
- Author
-
Hu, Ye, Li, Changpin, and Li, Hefeng
- Subjects
- *
FINITE difference method , *CAPUTO fractional derivatives , *STOCHASTIC convergence , *LAPLACIAN operator , *FRACTIONAL calculus - Abstract
In this paper, we present the finite difference method for Caputo-type parabolic equation with fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0, 1) and the spatial derivative is the fractional Laplacian. The Caputo derivative is evaluated by the L 1 approximation, and the fractional Laplacian with respect to the space variable is approximated by the Caffarelli–Silvestre extension. The difference schemes are provided together with the convergence and error estimates. Finally, numerical experiments are displayed to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
48. Analysis of a class of boundary value problems depending on left and right Caputo fractional derivatives.
- Author
-
Antunes, Pedro R.S. and Ferreira, Rui A.C.
- Subjects
- *
BOUNDARY value problems , *CAPUTO fractional derivatives , *MATHEMATICAL regularization , *PROBLEM solving , *FRACTIONAL calculus - Abstract
In this work we study boundary value problems associated to a nonlinear fractional ordinary differential equation involving left and right Caputo derivatives. We discuss the regularity of the solutions of such problems and, in particular, give precise necessary conditions so that the solutions are C 1 ([0, 1]). Taking into account our analytical results, we address the numerical solution of those problems by the a u g m e n t e d -RBF method. Several examples illustrate the good performance of the numerical method. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
49. Projectile motion via Riemann-Liouville calculus.
- Author
-
Ahmad, Bashir, Batarfi, Hanan, Nieto, Juan, Otero-Zarraquiños, Óscar, and Shammakh, Wafa
- Subjects
- *
RIEMANNIAN geometry , *LIOUVILLE'S theorem , *CALCULUS , *PROBLEM solving , *CAPUTO fractional derivatives - Abstract
We present an analysis of projectile motion in view of fractional calculus. We obtain the solution for the problem using the Riemann-Liouville derivative, and then we compute some features of projectile motion in the framework of Riemann-Liouville fractional calculus. We compare the solutions using Caputo derivatives and Riemann-Liouville derivatives. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
50. Solution of fractional Drinfeld-Sokolov-Wilson equation using Homotopy perturbation transform method.
- Author
-
Singh, P. K., Vishal, K., and Som, T.
- Subjects
- *
NONLINEAR equations , *CAPUTO fractional derivatives , *FRACTIONAL calculus , *HOMOTOPY theory , *PERTURBATION theory - Abstract
In this article, the approximate solutions of the non-linear Drinfeld-Sokolov-Wilson equation with fractional time derivative have been obtained. The fractional derivative is described in the Caputo sense. He's polynomial is used to tackle the nonlinearity which arise in our considered problems. A time fractional nonlinear partial differential equation has been computed numerically. The numerical procedures illustrate the effectiveness and reliability of the method. Effects of fractional order time derivatives on the solutions for different particular cases are presented through graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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