scholarly journals Numerical Solution of Fractional Model of HIV-1 Infection in Framework of Different Fractional Derivatives

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Sara Salem Alzaid ◽  
Badr Saad T. Alkahtani ◽  
Shivani Sharma ◽  
Ravi Shanker Dubey
Keyword(s):  
Mathematical Model ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Fractional Derivatives ◽  
Numerical Approach ◽  
Derivative Operator ◽  
Graphical Comparison ◽  
Uniqueness Of The Solution ◽  
Fractional Derivative Operators ◽  
Hiv 1

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.

scholarly journals Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 850-856 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Yun-Yun Xu
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Steady State Response ◽  
Vibration System ◽  
Derivative Operator ◽  
Vibration Equation ◽  
State Response ◽  
Frequency Relation

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator ${}_{-\infty} D_t^\beta,$where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

scholarly journals An Extension of Caputo Fractional Derivative Operator by Use of Wiman’s Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2238
Author(s):  
Rahul Goyal ◽  
Praveen Agarwal ◽  
Alexandra Parmentier ◽  
Clemente Cesarano
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Derivative Operator ◽  
The Family ◽  
Generating Relations ◽  
Fractional Derivative Operators ◽  
Extended Operator ◽  
Two Parameter

The main aim of this work is to study an extension of the Caputo fractional derivative operator by use of the two-parameter Mittag–Leffler function given by Wiman. We have studied some generating relations, Mellin transforms and other relationships with extended hypergeometric functions in order to derive this extended operator. Due to symmetry in the family of special functions, it is easy to study their various properties with the extended fractional derivative operators.

scholarly journals Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Moulay Rchid Sidi Ammi ◽  
Mostafa Tahiri ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Optimality Conditions ◽  
Fractional Derivatives ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Derivative Operator ◽  
Sir Epidemic ◽  
Non Local

<p style='text-indent:20px;'>The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators.</p>

scholarly journals FRACTIONAL DERIVATIVE AS FRACTIONAL POWER OF DERIVATIVE

2007 ◽  
Vol 18 (03) ◽  
pp. 281-299 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fourier Series ◽  
Fractional Derivative ◽  
Taylor Series ◽  
Fractional Derivatives ◽  
Fractional Power ◽  
Weyl Quantization ◽  
Fractional Powers ◽  
Derivative Operator ◽  
Quantization Procedure ◽  
Definition Of

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of selfadjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

scholarly journals Numerical approach of riemann-liouville fractional derivative operator

2021 ◽  
Vol 11 (6) ◽  
pp. 5367
Author(s):  
Ramzi B. Albadarneh ◽  
Iqbal M. Batiha ◽  
Ahmad Adwai ◽  
Nedal Tahat ◽  
A. K. Alomari
Keyword(s):  
Fractional Derivative ◽  
Nonlinear Problems ◽  
Numerical Approach ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Weighted Mean ◽  
Derivative Operator ◽  
Liouville Fractional Derivative ◽  
Linear And Nonlinear ◽  
Residual Errors

<p>This article introduces some new straightforward and yet powerful formulas in the form of series solutions together with their residual errors for approximating the Riemann-Liouville fractional derivative operator. These formulas are derived by utilizing some of forthright computations, and by utilizing the so-called weighted mean value theorem (WMVT). Undoubtedly, such formulas will be extremely useful in establishing new approaches for several solutions of both linear and nonlinear fractionalorder differential equations. This assertion is confirmed by addressing several linear and nonlinear problems that illustrate the effectiveness and the practicability of the gained findings.</p>

Numerical solution of fractionally damped beam by homotopy perturbation method

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Diptiranjan Behera ◽  
Snehashish Chakraverty
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Numerical Solution ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Analytic Solutions ◽  
Step Function ◽  
Homotopy Perturbation ◽  
Derivatives Of ◽  
Appl Math

AbstractThis paper investigates the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order. The Homotopy Perturbation Method (HPM) is used to obtain the dynamic response. Unit step function response is considered for the analysis. The obtained results are depicted in various plots. From the results obtained it is interesting to note that by increasing the order of the fractional derivative the beam suffers less oscillation. Similar observations have also been made by keeping the order of the fractional derivative constant and varying the damping ratios. Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan [Appl. Math. Mech. 28, 219 (2007)] to show the effectiveness and validation of this method.

scholarly journals Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator

2021 ◽  
Vol 5 (3) ◽  
pp. 64
Author(s):  
Igor V. Malyk ◽  
Mykola Gorbatenko ◽  
Arun Chaudhary ◽  
Shivani Sharma ◽  
Ravi Shanker Dubey
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Diffusion Equation ◽  
Numerical Solution ◽  
Analytical Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Derivative Operator ◽  
Uniqueness Of The Solution ◽  
Time Fractional Diffusion Equation

In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator.

scholarly journals Analysis of a fractional tumor-immune interaction model with exponential kernel

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 2023-2042
Author(s):  
Mustafa Dokuyucu ◽  
Hemen Dutta
Keyword(s):  
Fixed Point ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Interaction Model ◽  
Existence Of Solution ◽  
Fixed Point Method ◽  
Ulam Stability ◽  
Derivative Operator ◽  
Exponential Kernel ◽  
The Stability

In this paper, a tumor-immune interaction model has been analyzed via Caputo-Fabrizio fractional derivative operator with exponential kernel. Existence of solution of the model has been established with a fixed-point method and then it demonstrated the uniqueness of solution also. The stability of the model has been analyzed with the help of Hyers-Ulam stability approach and then numerical solution by using the Adam-Basford method. The results are further examined in detail with simulations for different fractional derivative values.

scholarly journals Alternative fractional derivative operator on non-newtonian calculus and its approaches

2021 ◽  
Vol 34 (02) ◽  
pp. 906-915
Author(s):  
Mohammad Momenzadeh ◽  
Sajedeh Norozpou
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Integral Operators ◽  
Derivative Operator ◽  
Geometric Calculus ◽  
Different Types ◽  
Fractional Differential ◽  
Fractional Derivative Operators

Nowadays, study on fractional derivative and integral operators is one of the hot topics of mathematics and lots of investigations and studies make their attentions in this field. Most of these concerns raised from the vast application of these operators in study of phenomena’s models. These operators interpreted by Newtonian calculus, however different types of calculi are existed and we introduce the fractional derivative operators focused on Bi-geometric calculus and also their fractional differential equations are studied.

scholarly journals A novel subclass of analytic functions specified by a family of fractional derivatives in the complex domain

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2837-2849
Author(s):  
Zainab Esa ◽  
H.M. Srivastava ◽  
Adem Kılıçman ◽  
Rabha Ibrahim
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Univalent Functions ◽  
Complex Domain ◽  
Open Unit ◽  
Coefficient Estimates ◽  
Fractional Derivative Operators ◽  
Coefficient Inequalities ◽  
Distortion Theorems ◽  
Analytic And Univalent Functions

In this paper, by making use of a certain family of fractional derivative operators in the complex domain, we introduce and investigate a new subclass P?,?(k,?,?) of analytic and univalent functions in the open unit disk U. In particular, for functions in the class P?,?(k,?,?), we derive sufficient coefficient inequalities and coefficient estimates, distortion theorems involving the above-mentioned fractional derivative operators, and the radii of starlikeness and convexity. In addition, some applications of functions in the class P?,?(k,?,?) are also pointed out.

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