Browsing by Author "Gomez-Aguilar, J. F."

Now showing 1 - 17 of 17

Citation - WoS: 100
Citation - Scopus: 119
Analytical and Numerical Study of the Dna Dynamics Arising in Oscillator-Chain of Peyrard-Bishop Model
(Pergamon-elsevier Science Ltd, 2020) Cattani, Carlo; Gomez-Aguilar, J. F.; Baleanu, Dumitru; Osman, M. S.; Ali, Khalid K.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this work, we introduce a numerical and analytical study of the Peyrard-Bishop DNA dynamic model equation. This model is studied analytically by hyperbolic and exponential ansatz methods and numerically by finite difference method. A comparison between the results obtained by the analytical methods and the numerical method is investigated. Furthermore, some figures are introduced to show how accurate the solutions will be obtained from the analytical and numerical methods. (C) 2020 Elsevier Ltd. All rights reserved.
Citation - WoS: 104
Citation - Scopus: 106
Beta-Derivative and Sub-Equation Method Applied To the Optical Solitons in Medium With Parabolic Law Nonlinearity and Higher Order Dispersion
(Elsevier Gmbh, Urban & Fischer verlag, 2018) Gomez-Aguilar, J. F.; Baleanu, Dumitru; Yepez-Martinez, H.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
By using the sub-equation method, we construct the analytical solutions of the space-time generalized nonlinear Schrodinger equation involving the beta-derivative. This equation describing the propagation of ultra-short optical solitons through parabolic law medium. Nonlinear perturbations of higher-order and self-steepening terms are taken into account. As a result, some new exact solutions are constructed under constraint conditions. (C) 2017 Elsevier GmbH. All rights reserved.
Citation - WoS: 9
Citation - Scopus: 11
Correcting Dimensional Mismatch in Fractional Models With Power, Exponential and Proportional Kernel: Application To Electrical Systems
(Elsevier, 2022) Correa-Escudero, I. L.; Gomez-Aguilar, J. F.; Lopez-Lopez, M. G.; Alvarado-Martinez, V. M.; Baleanu, D.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Fractional calculus is a powerful tool for describing diffusion phenomena, anomalous behaviors, and in general, systems with highly complex dynamics. However, the application of fractional operators for modeling purposes, produces a dimensional problem. In this paper, the fractional models of the RC, RL, RLC electrical circuits, a supercapacitor, a bank of supercapacitors, a LiFePO4 battery and a direct current motor are presented. A correction parameter is included in their formulation in order to preserve dimensionality in the physical equations. The optimal value of this parameter was determined via particle swarm optimization algorithm using numerical simulations and experimental data. Thus, a direct and effective approach for the construction of dimensionally corrected fractional models with power, exponential-decay and constant proportional Caputo hybrid derivative is presented. To show the effectiveness of the procedure, the time-response of the models is compared with experimental data and the modeling error is computed. The numerical solutions of the models were obtained using a numerical method based on the Adams methods.
Citation - WoS: 9
Citation - Scopus: 8
Derivation of Operational Matrix of Rabotnov Fractional-Exponential Kernel and Its Application To Fractional Lienard Equation
(Elsevier, 2020) Gomez-Aguilar, J. F.; Lavin-Delgado, J. E.; Baleanu, D.; Kumar, Sachin; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Our motive in this contribution is to find out the operational matrix of fractional derivative having non singular kernel namely Rabotnov fractional-exponential (RFE) kernel which is recently introduced and seeking numerical solution of non-linear Lienard equation which have Rabotnov fractional-exponential kernel fractional derivative. First we derive an approximation formula of the fractional order derivative of polynomial function z(k) in term of RFE kernel. Using this formula and some properties of shifted Legendre polynomials, we find out the operational matrix of fractional order differentiation. In the author of knowledge this operational matrix of RFE kernel fractional derivative is derived first time. We solve a new class of fractional partial differential equation (FPDEs) by implementation of this newly derived operational matrix. We show that our newly derived operational matrix is valid by taking an fractional derivative of a polynomial. Also, we study a new model of Lienard equation with RFE kernel fractional derivative and we can easily predict the feasibility of our numerical method to this new model. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.
Citation - WoS: 3
Citation - Scopus: 2
Double-Quasi Numerical Method for the Variable-Order Time Fractional and Riesz Space Fractional Reaction-Diffusion Equation Involving Derivatives in Caputo-Fabrizio Sense
(World Scientific Publ Co Pte Ltd, 2020) Pandey, Prashant; Gomez-Aguilar, J. F.; Baleanu, D.; Kumar, Sachin; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Our motive in this scientific contribution is to deal with nonlinear reaction-diffusion equation having both space and time variable order. The fractional derivatives which are used are non-singular having exponential kernel. These derivatives are also known as Caputo-Fabrizio derivatives. In our model, time fractional derivative is Caputo type while spatial derivative is variable-order Riesz fractional type. To approximate the variable-order time fractional derivative, we used a difference scheme based upon the Taylor series formula. While approximating the variable order spatial derivatives, we apply the quasi-wavelet-based numerical method. Here, double-quasi-wavelet numerical method is used to investigate this type of model. The discretization of boundary conditions with the help of quasi-wavelet is discussed. We have depicted the efficiency and accuracy of this method by solving the some particular cases of our model. The error tables and graphs clearly show that our method has desired accuracy.
Citation - WoS: 41
Citation - Scopus: 48
An Efficient Technique for Solving the Space-Time Fractional Reaction-Diffusion Equation in Porous Media
(Elsevier, 2020) Kumar, Sachin; Gomez-Aguilar, J. F.; Baleanu, D.; Pandey, Prashant; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper, we obtained the approximate numerical solution of space-time fractional-order reaction-diffusion equation using an efficient technique homotopy perturbation technique using Laplace transform method with fractional-order derivatives in Caputo sense. The solution obtained is very useful and significant to analyze the many physical phenomenons. The present technique demonstrates the coupling of the homotopy perturbation technique and Laplace transform using He's polynomials for finding the numerical solution of various non-linear fractional complex models. The salient features of the present work are the graphical presentations of the approximate solution of the considered porous media equation for different particular cases and reflecting the presence of reaction terms presented in the equation on the physical behavior of the solute profile for various particular cases.
Citation - WoS: 15
Citation - Scopus: 17
Fractional Dynamics of an Erbium-Doped Fiber Laser Model
(Springer, 2019) Saad, K. M.; Baleanu, D.; Gomez-Aguilar, J. F.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper we investigate the model of the time-fractional dynamics of an erbium-doped fiber laser model (TFDEFL) with Liouville-Caputo (LC), Caputo-Fabrizio-Caputo (CFC) and Atangana-Baleanu-Caputo (ABC) time-fractional derivatives. We employ the homotopy analysis transform method (HATM) to calculate approximate solutions for the TFDEFL model. This method gives the solution in the form of a rapidly convergent series that can ensure the convergence in solving the resultant series. We study the convergence analysis of HATM by computing the interval of convergence through the h-curves, the residual error function and the average residual error, respectively. We also show the effectiveness and accuracy of this method by comparing the approximate solutions based upon the LC, CFC and ABC time-fractional derivatives.
Citation - WoS: 69
Citation - Scopus: 91
Fractional Lienard Type Model of a Pipeline Within the Fractional Derivative Without Singular Kernel
(Springeropen, 2016) Torres, L.; Yepez-Martinez, H.; Baleanu, D.; Reyes, J. M.; Sosa, I. O.; Gomez-Aguilar, J. F.; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
This paper presents the procedure to obtain analytical solutions of Li,nard type model of a fluid transmission line represented by the Caputo-Fabrizio fractional operator. For such a model, we derive a new approximated analytical solution by using the Laplace homotopy analysis method. Both the efficiency and the accuracy of the method are verified by comparing the obtained solutions with the exact analytical solution. Good agreement between them is confirmed.
Citation - WoS: 88
Modeling and Simulation of the Fractional Space-Time Diffusion Equation
(Elsevier Science Bv, 2016) Miranda-Hernandez, M.; Lopez-Lopez, M. G.; Alvarado-Martinez, V. M.; Baleanu, D.; Gomez-Aguilar, J. F.; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper, the space-time fractional diffusion equation related to the electromagnetic transient phenomena in transmission lines is studied, three cases are presented; the diffusion equation with fractional spatial derivative, with fractional temporal derivative and the case with fractional space-time derivatives. For the study cases, the order of the spatial and temporal fractional derivatives are 0 < beta, gamma <= 2. respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation. The general solutions of the proposed equations are expressed in terms of the multivariate Mittag-Leffler functions; these functions depend only on the parameters beta and gamma and preserve the appropriated physical units for any value of the fractional derivative exponent. Furthermore, an analysis of the fractional time constant was made in order to indicate the change of the medium properties and the presence of dissipation mechanisms. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, irreversible thermodynamics, quantum optics or turbulent diffusion, thermal stresses, models of porous electrodes, the description of gel solvents and anomalous complex processes. (C) 2015 Elsevier B.V. All rights reserved.
Citation - WoS: 23
Citation - Scopus: 25
Modeling the Fractional Non-Linear Schrodinger Equation Via Liouville-Caputo Fractional Derivative
(Elsevier Gmbh, 2018) Morales-Delgado, V. F.; Gomez-Aguilar, J. F.; Taneco-Hernandez, M. A.; Baleanu, Dumitru; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper the modified homotopy analysis transform method is applied to obtain approximate analytical solutions of the time-fractional non-linear Schrodinger equation. The fractional derivative is described in the Liouville-Caputo sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. New exact solutions are constructed under constraint conditions. Employing theoretical parameters, we present some numerical simulations. (C) 2018 Elsevier GmbH. All rights reserved.
Citation - WoS: 11
Citation - Scopus: 19
A New Approach To Exact Optical Soliton Solutions for the Nonlinear Schrodinger Equation
(Springer Heidelberg, 2018) Gomez-Aguilar, J. F.; Baleanu, Dumitru; Morales-Delgado, V. F.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
By using the modified homotopy analysis transform method, we construct the analytical solutions of the space-time generalized nonlinear Schrodinger equation involving a new fractional conformable derivative in the Liouville-Caputo sense and the fractional-order derivative with the Mittag-Leffler law. Employing theoretical parameters, we present some numerical simulations and compare the solutions obtained.
Citation - WoS: 114
Citation - Scopus: 107
Numerical Solutions of the Fractional Fisher's Type Equations With Atangana-Baleanu Fractional Derivative by Using Spectral Collocation Methods
(Amer inst Physics, 2019) Khader, M. M.; Gomez-Aguilar, J. F.; Baleanu, Dumitru; Saad, K. M.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
The main objective of this paper is to investigate an accurate numerical method for solving a biological fractional model via Atangana-Baleanu fractional derivative. We focused our attention on linear and nonlinear Fisher's equations. We use the spectral collocation method based on the Chebyshev approximations. This method reduced the nonlinear equations to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them by using the finite difference method. This is the first time that this method is used to solve nonlinear equations in Atangana-Baleanu sense. We present the effectiveness and accuracy of the proposed method by computing the absolute error and the residual error functions. The results show that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels.
Citation - WoS: 24
Citation - Scopus: 30
On a More General Fractional Integration by Parts Formulae and Applications
(Elsevier, 2019) Gomez-Aguilar, J. F.; Jarad, Fahd; Abdeljawad, Thabet; Atangana, Abdon; 234808; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
The integration by part comes from the product rule of classical differentiation and integration. The concept was adapted in fractional differential and integration and has several applications in control theory. However, the formulation in fractional calculus is the classical integral of a fractional derivative of a product of a fractional derivative of a given function f and a function g. We argue that, this formulation could be done using only fractional operators: thus, we develop fractional integration by parts for fractional integrals, Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. We allow the left and right fractional integrals of order alpha > 0 to act on the integrated terms instead of the usual integral and then make use of the fractional type Leibniz rules to formulate the integration by parts by means of new generalized type fractional operators with binomial coefficients defined for analytic functions. In the case alpha = 1, our formulae of fractional integration by parts results in previously obtained integration by parts in fractional calculus. The two disciplines or branches of mathematics are built differently, while classical differentiation is built with the concept of rate of change of a given function, a fractional differential operator is a convolution. (C) 2019 Elsevier B.V. All rights reserved.
Citation - WoS: 85
Citation - Scopus: 94
On Exact Solutions for Time-Fractional Korteweg-De Vries and Korteweg-De Vries-burger's Equations Using Homotopy Analysis Transform Method
(Elsevier, 2020) AL-Shareef, Eman H. F.; Alomari, A. K.; Baleanu, Dumitru; Gomez-Aguilar, J. F.; Saad, K. M.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper we consider the homotopy analysis transform method (HATM) to solve the time fractional order Korteweg-de Vries (KdV) and Korteweg-de Vries-Burger's (KdVB) equations. The HATM is a combination of the Laplace decomposition method (LDM) and the homotopy analysis method (HAM). The fractional derivatives are defined in the Caputo sense. This method gives the solution in the form of a rapidly convergent series with h-curves are used to determine the intervals of convergent. Averaged residual errors are used to find the optimal values of h. It is found that the optimal h accelerates the convergence of the HATM, with the rate of convergence depending on the parameters in the KdV and KdVB equations. The HATM solutions are compared with exact solutions and excellent agreement is found.
Citation - WoS: 4
Citation - Scopus: 4
On the Approximate Solution of Fractional-Order Whitham-Broer Equations
(World Scientific Publ Co Pte Ltd, 2021) Gomez-Aguilar, J. F.; Alderremy, A. A.; Aly, Shaban; Baleanu, Dumitru; Khan, Hassan; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper, the Homotopy perturbation Laplace method is implemented to investigate the solution of fractional-order Whitham-Broer-Kaup equations. The derivative of fractional-order is described in Caputo's sense. To show the reliability of the suggested method, the solution of certain illustrative examples are presented. The results of the suggested method are shown and explained with the help of its graphical representation. The solutions of fractional-order problems as well as integer-order problems are determined by using the present technique. It has been observed that the obtained solutions are in significant agreement with the actual solutions to the targeted problems. Computationally, it has been analyzed that the solutions at different fractional-orders have a higher rate of convergence to the solution at integer-order of the derivative. Due to the analytical analysis of the problems, this study can further modify the solution of other fractional-order problems.
Citation - WoS: 43
Schrodinger Equation Involving Fractional Operators With Non-Singular Kernel
(Taylor & Francis Ltd, 2017) Baleanu, Dumitru; Gomez-Aguilar, J. F.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
An alternative model of fractional Schrodinger equation involving Caputo-Fabrizio fractional operator and the new fractional operator based on the Mittag-Leffler function is proposed. We obtain the eigenvalues and eigenfunctions for a free particle moving in the infinite potential well. Numerical simulations of alternative models are presented for evaluating the effectiveness of these representations. We showed that fractional Schrodinger equation via Caputo-Fabrizio operator is a particular case of fractional Schrodinger equation obtained with the new fractional operator based in the Mittag-Leffler function.
Citation - WoS: 34
Citation - Scopus: 34
Travelling Waves Solution for Fractional-Order Biological Population Model
(Edp Sciences S A, 2021) Shah, Rasool; Gomez-Aguilar, J. F.; Shoaib; Baleanu, Dumitru; Kumam, Poom; Khan, Hassan; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
In this paper, we implemented the generalized (G'/G) and extended (G'/G) methods to solve fractional-order biological population models. The fractional-order derivatives are represented by the Caputo operator. The solutions of some illustrative examples are presented to show the validity of the proposed method. First, the transformation is used to reduce the given problem into ordinary differential equations. The ordinary differential equation is than solve by using modified (G'/G) method. Different families of traveling waves solutions are constructed to explain the different physical behavior of the targeted problems. Three important solutions, hyperbolic, rational and periodic, are investigated by using the proposed techniques. The obtained solutions within different classes have provided effective information about the targeted physical procedures. In conclusion, the present techniques are considered the best tools to analyze different families of solutions for any fractional-order problem.