Type Fractional Derivative Research Articles

Numerical study for a novel variable-order multiple time delay awareness programs mathematical model

In the present work, a novel variable-order awareness programs mathematical model with multiple time delay is presented. Two types of variable-order fractional derivatives of Atangana-Baleanu-Caputo definitions are given. The properties of the proposed model such as the basic reproduction number and the stability of the equilibrium points are studied analytically and numerically. Two numerical methods are introduced to study the proposed models. These methods are the predictor-corrector method and the fifth order Adams-Moulton method. In order to validate the theoretical results, numerical simulations and comparative studies are given.

Initial-value / Nonlocal Cauchy Problems for Fractional Differential Equations Involving ψ-Hilfer Multivariable Operators

Abstract In this paper, we investigate two types of problems (the initial-value problem and nonlocal Cauchy problem) for fractional differential equations involving ψ-Hilfer derivative in multivariable case (ψ-m-Hilfer derivative). First we propose and discuss ψ-fractional integral, ψ-fractional derivative and ψ-Hilfer type fractional derivative of a multivariable function f : ℝ m → ℝ (m is a positive integer). Then, using the properties of the ψ-m-Hilfer fractional derivative with m = 1 (the ψ-Hilfer derivative), we derive an equivalent relationship between solutions to the initial-value (Cauchy) problem and solutions to some integral equations, and also present an existence and uniqueness theorem. Based on the equivalency relationship, we establish new and general existence results for the nonlocal Cauchy problem of fractional differential equations involving ψ-Hilfer multivariable operators in the space of weighted continuous functions. Moreover, we obtain a new Gronwall-type inequality with singular kernel, and derive the dependence of the solution on the order and the initial condition for the fractional Cauchy problem with the help of this Gronwall-type inequality. Finally, some examples are given to illustrate our results. Compared with the recent paper [2] and other previous works, the novelties in this paper are in treating the multivariable case of operators (f : ℝ m → ℝ, m is a positive integer).

Computational and theoretical modeling of the transmission dynamics of novel COVID-19 under Mittag-Leffler Power Law

In the current article, we studied the novel corona virus (2019-nCoV or COVID-19) which is a threat to the whole world nowadays. We consider a fractional order epidemic model which describes the dynamics of COVID-19 under nonsingular kernel type of fractional derivative. An attempt is made to discuss the existence of the model using the fixed point theorem of Banach and Krasnoselskii’s type. We will also discuss the Ulam-Hyers type of stability of the mentioned problem. For semi analytical solution of the problem the Laplace Adomian decomposition method (LADM) is suggested to obtain the required solution. The results are simulated via Matlab by graphs. Also we have compare the simulated results with some reported real data for Commutative class at classical order.

Study of evolution problem under Mittag–Leffler type fractional order derivative

The current work is devoted to investigate evolution problem of nonlocal Cauchy type under “Mittag–Leffler” type fractional derivative which works as nonsingular and nonlocal kernel. For the considered problem, we establish some appropriate results devoted to the qualitative theory of existence and stability analysis. Upon using Banach and Krasnoselskii’s fixed point theorems, we establish the required analysis. Further stability theory of Ulam’s type is constructed by sing nonlinear analysis. Some interesting examples are provided to support our established results.

Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control.

In this paper, a nonlinear fractional order epidemic model for HIV transmission is proposed and analyzed by including extra compartment namely exposed class to the basic SIR epidemic model. Also, the infected class of female sex workers is divided into unaware infectives and the aware infectives. The focus is on the spread of HIV by female sex workers through prostitution, because in the present world sexual transmission is the major cause of the HIV transmission. The exposed class contains those susceptible males in the population who have sexual contact with the female sex workers and are exposed to the infection directly or indirectly. The Caputo type fractional derivative is involved and generalized Adams-Bashforth-Moulton method is employed to numerically solve the proposed model. Model equilibria are determined and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle. Analysis of the model demonstrates that the population is free from the disease if and disease spreads in the population if . Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Furthermore, for the fractional optimal control problem associated with the control strategies such as condom use for exposed class, treatment for aware infectives, awareness about disease among unaware infectives and behavioral change for susceptibles, we formulated a fractional optimality condition for the proposed model. The existence of fractional optimal control is analyzed and the Euler-Lagrange necessary conditions for the optimality of fractional optimal control are obtained. The effectiveness of control strategies is shown through numerical simulations and it can be seen through simulation, that the control measures effectively increase the quality of life and age limit of the HIV patients. It significantly reduces the number of HIV/AIDS patients during the whole epidemic.

On the initial value problems for the Caputo–Fabrizio impulsive fractional differential equations

This paper is devoted to initial value problems for impulsive fractional differential equations of Caputo–Fabrizio type fractional derivative. By means of Banach’s fixed point theorem and Schaefer’s fixed point theorem, the existence and uniqueness results are obtained. Finally, an example is given to illustrate one of the main results.

Comparison of the Probabilistic Ant Colony Optimization Algorithm and Some Iteration Method in Application for Solving the Inverse Problem on Model With the Caputo Type Fractional Derivative.

This paper presents the algorithms for solving the inverse problems on models with the fractional derivative. The presented algorithm is based on the Real Ant Colony Optimization algorithm. In this paper, the examples of the algorithm application for the inverse heat conduction problem on the model with the fractional derivative of the Caputo type is also presented. Based on those examples, the authors are comparing the proposed algorithm with the iteration method presented in the paper: Zhang, Z. An undetermined coefficient problem for a fractional diffusion equation. Inverse Probl. 2016, 32.

Existence and Integral Representation of Scalar Riemann-Liouville Fractional Differential Equations with Delays and Impulses

Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and impulses are studied and initial conditions and impulsive conditions are set up in an appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence of the delay in the equation. We study the case of a fixed lower limit of the fractional derivative and the case of a changeable lower limit at each impulsive time. Integral representations of the solutions in all considered cases are obtained. Existence results on finite time intervals are proved using the Banach principle.

Fractional Derivatives and Dynamical Systems in Material Instability

Loss of stability is studied extensively in nonlinear investigations, and classified as generic bifurcations. It requires regularity, being connected with non-locality. Such behavior comes from gradient terms in constitutive equations. Most fractional derivatives are non-local, thus by using them in defining strain, a non-local strain appears. In such a way, various versions of non-localities are obtained by using various types of fractional derivatives. The study aims for constitutive modeling via instability phenomena, that is, by observing the way of loss of stability of material, we can be informed about the proper form of its mathematical model.

On more general forms of proportional fractional operators

Abstract In this article, more general types of fractional proportional integrals and derivatives are proposed. Some properties of these operators are discussed.

Analysis of some generalized [formula omitted] – Fractional logistic models

In this article, some logistic models in the settings of Caputo fractional operators with multi-parametered Mittag-Leffler kernels (ABC) are studied. This study mainly focuses on modified quadratic and cubic logistic models in the presence of a Caputo type fractional derivative. Existence and uniqueness theorems are proved and stability analysis is discussed by perturbing the equilibrium points. Numerical illustrative examples are discussed for the studied models.

New Travelling Wave Solutions for KdV6 Equation Using Sub Equation Method

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

Asymptotic Behavior in a Laminated Beams Due Interfacial Slip with a Boundary Dissipation of Fractional Derivative Type

We consider a laminated beams due interfacial slip with control boundary conditions of fractional derivative type. We show the existence and uniqueness of solutions. Furthermore, concerning the asymptotic behavior we show the lack of exponential stability and the polynomial decay rate of the corresponding semigroup by using the classic theorem of Borichev and Tomilov.

A new truncated M-fractional derivative for air pollutant dispersion

In this paper, we study the potential of fractional derivatives to model air pollution. We introduce an M-fractional truncated derivative type for $$\alpha $$ -differentiable functions that generalizes other types of fractional derivatives. We denote this new differential operator by $$_{i}D_{\mathrm{M}}^{\alpha ,\beta }$$ , where the parameters $$\alpha $$ and $$\beta $$ , associated with the order of the derivative are such that $$0<\alpha <1$$ , $$\beta >1$$ and M is the notation to indicate that the function to be derived involves the truncated function of Mittag-Leffler with a parameter. The definition of this type of truncated M-fractional derivative satisfies the properties of the integer calculation. Based on this observation, we solved these models and we compared the solutions with the data obtained from the Copenhagen experiment. Fractional derivative models work much better than the traditional Gaussian model and the computed values are in good agreement with experimental ones.

Solitary solutions for time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative.

In this paper, the time-fractional nonlinear dispersive (TFND) partial differential equations (PDEs) in the sense of conformable fractional derivative (CFD) are proposed and analyzed. Three types of TFND partial differential equations are considered in the sense of CFD, which are the TFND Boussinesq, TFND Klein-Gordon, and TFND B(2, 1, 1) PDEs. Solitary pattern solutions for this class of TFND partial differential equations based on the residual fractional power series method is constructed and discussed. Numerical and graphical results are also provided and conferred quantitatively to clarify the required solutions. The results suggest that the algorithm presented here offers solutions to problems in a rapidly convergent series leading to ideal solutions. Furthermore, the results obtained are like those in previous studies that used other types of fractional derivatives. In addition, the calculations used were much easier and shorter compared with other types of fractional derivatives.

Analysis of coupled systems of implicit impulsive fractional differential equations involving Hadamard derivatives

We present some results on the existence, uniqueness and Hyers–Ulam stability to the solution of an implicit coupled system of impulsive fractional differential equations having Hadamard type fractional derivative. Using a fixed point theorem of Kransnoselskii’s type, the existence and uniqueness results are obtained. Along these lines, different kinds of Hyers–Ulam stability are discussed. An example is given to illustrate the main theorems.

Existence results for a fraction hybrid differential inclusion with Caputo\u2013Hadamard type fractional derivative

In this manuscript, we talk over the existence of solutions of a class of hybrid Caputo–Hadamard fractional differential inclusions with Dirichlet boundary conditions. Our results are based on the Arzelá–Ascoli theorem and some suitable theorems of fixed point theory. As well, to illustrate our results, we confront the exceptional case of the fractional differential inclusions with examples.

Solutions for a fractional diffusion equation in heterogeneous media

We investigate diffusive phenomena governed by fractional diffusion equations in the presence of a spatial dependent diffusion coefficient and external forces. We consider a fractional operator which enables us to analyze, in a unified way, different types of fractional time derivatives. This approach opens the possibility of considering new situations related to memory effects. After determining exact analytical solutions, we show that different diffusive behaviors can be obtained, depending on the fractional time operator employed. In this framework, we also discuss the role of the anomalous diffusion processes.

Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel

Most disordered dielectrics especially solid dielectrics show non-Debye laws, and many empirical approximation models are proposed to describe these anomalous relaxation processes. Fractional calculus approach is a popular approach to analyze the anomalous relaxation processes and has been intensively studied in various dielectric materials. However, Capelas de Oliveira et al. proved that the memory kernels of Caputo type fractional derivatives must satisfy the initial value condition (IVC). But both kernels of the Caputo–Fabrizio derivative and the Atangana–Baleanu derivative do not satisfy this condition. In this paper, we prove that the Caputo type derivative with a Prabhakar-like kernel satisfies the IVC and this derivative is in the framework of general Caputo fractional derivative (GC derivative). Corresponding anomalous relaxation model and its solution are discussed. Analysis result shows our model, as a direct extension of the Cole–Cole model, contains the Debye model and the fractional relaxation model as particular cases.

Fractional operators with generalized Mittag-Leffler kernels and their iterated differintegrals.

Depending on a previous work about fractional operators of Riemann type (ABR) and Caputo type (ABC) with kernels of Mittag-Leffler in three parameters [Eα,μ γ(λ,t-s)], we derive the corresponding fractional integrals with arbitrary order by using the infinite binomial theorem, and study their semi-group properties and their action on the ABC type fractional derivatives to prove the existence and uniqueness theorem for the ABC-fractional initial value problems. In fact, as advantages to the obtained extension, we find that for μ≠1, we obtain a nontrivial solution for the linear ABC-type initial value problem with constant coefficient and prove a certain semigroup property in the parameters μ and γ simultaneously. Iterated type fractional differ-integrals are constructed by iterating fractional integrals of order (α,μ,1) to add a fourth parameter, and a semigroup property is derived under the existence of the fourth parameter. The Laplace transforms for the Atangana-Baleanu (AB) fractional integrals and the AB iterated fractional differ-integrals are calculated. An alternative representation of the ABR-derivatives is given and is compared, in the case γ=1, with the iterated AB differ-integrals with negative order (α,μ,1),-1. An example and several remarks are given to illustrate part of the proven results and to point out some particular cases. The obtained results generalized and improved some recent results.