Variable-order Fractional Derivative Research Articles

Variable-order fractional derivative under Hadamard’s finite-part integral: Leibniz-type rule and its applications

Variable-order fractional derivative under Hadamard’s finite-part integral: Leibniz-type rule and its applications

Fractional viscoelastic models with novel variable and constant order fractional derivative operators

This paper deals with the application of a novel variable-order and constant-order fractional derivative without singular kernel of Atangana-Koca type to describe the fractional viscoelastic models, namely, fractional Maxwell model, fractional Kelvin-Voigt model, fractional Zener model and fractional Poynting-Thomson model. For each fractional viscoelastic model, the stress relaxation modulus and creep compliance are derived analytically under the variable-order and constant-order fractional derivative without singular kernel. Our results show that the relaxation modulus and creep compliance exhibit viscoelastic behaviors producing temporal fractality at different scales. For each viscoelastic model, the stress relaxation modulus and creep compliance are derived analytically under novel variable-order and constant-order fractional derivative with no singular kernel.

Optimal control of hybrid variable-order fractional coronavirus (2019-nCov) mathematical model; numerical treatments

A novel coronavirus is a serious global issue and has a negative impact on the economy of Egypt. According to the publicly reported data, the first case of the novel corona virus in Egypt was reported on 14 February 2020. Total of 96753 cases were recorded in Egypt from the beginning of the pandemic until the eighteenth of August, where 96, 581 individuals were Egyptians and 172 were foreigners. Recently, many mathematical models have been considered to better understand coronavirus infection. Most of these models are based on classical integer-order derivatives which can not capture the fading memory and crossover behavior found in many biological phenomena. Therefore, we study the coronavirus disease in this paper by exploring the dynamics of COVID-19 infection using new variable-order fractional derivatives. This paper presents an optimal control problem of the hybrid variable-order fractional model of Coronavirus. The variable-order fractional operator is modified by an auxiliary parameter in order to satisfy the dimensional matching between the both sides of the resultant variable-order fractional equations. Existence, uniqueness, boundedness, positivity, local and global stability of the solutions are proved. Two control variables are considered to reduce the transmission of infection into healthy people. To approximate the new hybrid variable-order operator, Grünwald-Letnikov approximation is used. Finite difference method with a hybrid variable-order operator and generalized fourth order Runge-Kutta method are used to solve the optimality system. Numerical examples and comparative studies for testing the applicability of the utilized methods and to show the simplicity of these approximation approaches are presented. Moreover, by using the proposed methods we can concluded that, the model given in this paper describes well the confirmed real data given by WHO about Egypt.

A mathematical model for SARS-CoV-2 in variable-order fractional derivative.

A new coronavirus mathematical with hospitalization is considered with the consideration of the real cases from March 06, 2021 till the end of April 30, 2021. The essential mathematical results for the model are presented. We show the model stability when {mathcal {R}}_0<1 in the absence of infection. We show that the system is stable locally asymptotically when {mathcal {R}}_0<1 at infection free state. We also show that the system is globally asymptotically stable in the disease absence when {mathcal {R}}_0<1. Data have been used to fit accurately to the model and found the estimated basic reproduction number to be {mathcal {R}}_0= 1.2036. Some graphical results for the effective parameters are drawn for the disease elimination. In addition, a variable-order model is introduced, and so as to handle the outbreak effectively and efficiently, a genetic algorithm is used to produce high-quality control. Numerical simulations clearly show that decision-makers may develop helpful and practical strategies to manage future waves by implementing optimum policies.

Numerical approximation for a nonlinear variable-order fractional differential equation via a collocation method

We study a numerical approximation for a nonlinear variable-order fractional differential equation via a collocation method. Due to the lack of monotonicity of the discretization coefficients of the variable-order fractional derivative in standard approximation schemes, existing numerical analysis techniques do not apply directly. By an approximate inversion technique, the proposed model is transformed as a second kind Volterra integral equation, based on which a collocation method under uniform or graded mesh is developed and analyzed. In particular, the mesh grading parameter we used is consistent with respect to the regularity of the solutions, and the convergence rates are characterized in terms of the initial value of the variable order, which serve as improvements compared with the existing results in the literature.

Variational problems of variable fractional order involving arbitrary kernels

&lt;abstract&gt;&lt;p&gt;The aim of this work is to study several problems of the calculus of variations, where the dynamics of the state function is given by a generalized fractional derivative. This derivative combines two well-known concepts: fractional derivative with respect to another function and fractional derivative of variable order. We present the Euler–Lagrange equation, which is a necessary condition that every optimal solution of the problem must satisfy. Other problems are also studied: with integral and holonomic constraints, with higher order derivatives, and the Herglotz variational problem.&lt;/p&gt;&lt;/abstract&gt;

Fibonacci collocation pseudo-spectral method of variable-order space-fractional diffusion equations with error analysis

&lt;abstract&gt;&lt;p&gt;In this article, we evaluated the approximate solutions of one-dimensional variable-order space-fractional diffusion equations (sFDEs) by using a collocation method. This method depends on operational matrices for fractional derivatives and the integration of generalized Fibonacci polynomials. In this method, a Caputo fractional derivative of variable order is applied. Some properties of these polynomials (using boundary conditions) are presented to simplify and transform sFDEs into a system of equations with the expansion coefficients of the solution. Also, we discuss the convergence and error analysis of the generalized Fibonacci expansion. Finally, we compare the obtained results with those obtained via the other methods.&lt;/p&gt;&lt;/abstract&gt;

Haar wavelet method for solution of variable order linear fractional integro-differential equations

&lt;abstract&gt;&lt;p&gt;In this paper, we developed a computational Haar collocation scheme for the solution of fractional linear integro-differential equations of variable order. Fractional derivatives of variable order is described in the Caputo sense. The given problem is transformed into a system of algebraic equations using the proposed Haar technique. The results are obtained by solving this system with the Gauss elimination algorithm. Some examples are given to demonstrate the convergence of Haar collocation technique. For different collocation points, maximum absolute and mean square root errors are computed. The results demonstrate that the Haar approach is efficient for solving these equations.&lt;/p&gt;&lt;/abstract&gt;

Uniqueness of continuous solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order

In this paper, the authors introduced a novel definition based on Hilfer fractional derivative, which name $q$-Hilfer fractional derivative of variable order. And the uniqueness of solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order of the form \eqref{eq:varorderfrac} with $0 &lt; \alpha(t) &lt; 1$, $0 \leq \beta \leq 1$, and $0 &lt; q &lt; 1$ is studied. Moreover, an example is provided to demonstrate the result.

Orthonormal shifted discrete Legendre polynomials for the variable-order fractional extended Fisher–Kolmogorov equation

This paper presents a numerical technique for solving the variable-order fractional extended Fisher–Kolmogorov equation. The method suggested to solve this problem is based on the orthonormal shifted discrete Legendre polynomials and the collocation method. First, we expand the unknown solution of the problem using the these polynomialss. Also, we approximate the second- and fourth-order classical derivatives, as well as the variable-order fractional derivatives by these basis functions. Then, we substitute these approximations in the equation. Next, we utilize the classical and fractional derivative matrices together with the collocation method to convert the main equation into a system containing nonlinear algebraic equations. We show the correctness of the proposed scheme by providing several numerical examples.

Enhancement of heat and mass transfer of a physical model using Generalized Caputo fractional derivative of variable order and modified Laplace transform method

In this paper, we use a model of non-Newtonian second grade fluid which having three partial differentialequations of momentum, heat and mass transfer with initial condition and boundary condition. Wedevelop the modified Laplace transform of this model with fractional order generalized Caputo fractional operator.We find out the solutions for temperature, concentration and velocity fields by using modified Laplacetransform and investigated the impact of the order α and ρ on temperature, concentration and velocity fieldsrespectively. From the graphical results, we have seen that both the α and ρ have reverse effect on the fluidflow properties. In consequence, it is observed that flow properties of present model can be enhanced nearthe plate for smaller and larger values of ρ. Furthermore, we have compared the present results with theexisting literature for the validation and found that they are in good agreement.

Numerical Solution of Variable-Order Fractional Differential Equations Using Bernoulli Polynomials

We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the Riemann–Liouville integral operator was used to give approximations for the unknown function and its variable-order derivatives. An operational matrix of variable-order fractional integration was introduced for the Bernoulli functions. By assuming that the solution of the problem is sufficiently smooth, we approximated a given order of its derivative using Bernoulli polynomials. Then, we used the introduced operational matrix to find some approximations for the unknown function and its derivatives. Using these approximations and some collocation points, the problem was reduced to the solution of a system of nonlinear algebraic equations. An error estimate is given for the approximate solution obtained by the proposed method. Finally, five illustrative examples were considered to demonstrate the applicability and high accuracy of the proposed technique, comparing our results with the ones obtained by existing methods in the literature and making clear the novelty of the work. The numerical results showed that the new method is efficient, giving high-accuracy approximate solutions even with a small number of basis functions and when the solution to the problem is not infinitely differentiable, providing better results and a smaller number of basis functions when compared to state-of-the-art methods.

Stabilization of impulsive fractional-order dynamic systems involving the Caputo fractional derivative of variable-order via a linear feedback controller

This paper investigates the stability and stabilization problem of variable-order fractional nonlinear dynamic systems with impulsive effects (VO-IFNDS) via a linear feedback controller. New inequalities on the VO Caputo fractional derivatives are established in this paper, which plays an essential role in the study of the stability theory of VO-IFNDS. Based on utilizing S-procedure and analytical technique, several sufficient criteria on Mittag-Leffler stability and asymptotical stability of VO-IFNDS are presented by means of the extension of the Lyapunov direct method. Finally, numerical examples are given to show the efficiency of the proposed method.

Jacobi spectral method for variable-order fractional Benney–Lin equation arising in falling film problems

In this paper, a variable-order fractional version of the Benney–Lin equation is defined using the variable-order fractional derivative in the Caputo type. A collocation method based on the shifted Jacobi polynomials is applied to deal with this problem. Some matrix relationships related to these polynomials are extracted and used in constructing the established method. The obtained relations cause the method calculations to be significantly reduced, which reduce the method execution time. The established technique converts solving the problem under study into solving an algebraic system of equations. The high accuracy and low computations of the presented scheme are investigated by solving some numerical examples.

An Improved Nonlinear Creep Model for Rock Applied to Tunnel Displacement Prediction

Traditional creep models cannot describe the nonlinear creep characteristics of rock; thus, considerable efforts have been made to establish better rock creep models. However, many of the previously presented improved models are too complicated, leading to difficulties in mathematical derivation and engineering application. To address this problem, this study proposes a simple model that can comprehensively describe rock creep characteristics and that can be effectively applied in practical engineering. Based on the variable-order fractional derivatives and continuum damage mechanics, an improved Burgers creep model is established. The use of this proposed model can be divided into two stages based on whether the accumulated creep strain of the rock reaches the threshold of starting accelerated creep. Then, both the one-dimensional and three-dimensional constitutive equations of the improved model are provided. In addition, the compression creep test results of six groups of rock samples are used to verify the effectiveness and reliability of the improved Burgers model proposed in this study. Furthermore, an analytical solution for tunnel displacement is derived on the basis of the improved model. This analytical solution is applied to the time-dependent displacement prediction of an auxiliary tunnel in Jinping hydropower station, and a good agreement between the theoretical results and monitoring data is achieved. The outcome of this study can provide useful guidance for related engineering designs.

Unique existence of solution to initial value problem for fractional differential equation involving with fractional derivative of variable order

We deal with unique existence of solution to initial value problem for fractional differential equation involving with fractional derivative of variable order. Some examples substantiate these theoretical results.

A viscoelastic-viscoplastic mechanical model of time-dependent materials based on variable-order fractional derivative

A viscoelastic-viscoplastic mechanical model of time-dependent materials based on variable-order fractional derivative

Variable-Order Fractional Models for Wall-Bounded Turbulent Flows.

Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+ to 0.1–0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.

On the sub–diffusion fractional initial value problem with time variable order

Abstract We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

A novel wavelets operational matrix method for the time variable-order fractional mobile–immobile advection–dispersion model

A novel computational technique for the solution of the variable-order fractional mobile–immobile advection–dispersion equation has been presented in this paper. Firstly, operational integration matrices and variable-order fractional derivatives were deduced using Boubaker wavelets to implement this proposed technique. Utilizing Boubaker wavelets basis for functions approximations and the operational matrices of integration and variable-order fractional derivative along with collocation points, the variable-order fractional mobile–immobile advection–dispersion equation is reduced into the system of algebraic equations. In addition, to determine the convergence analysis and error estimate of the proposed numerical technique, some useful theorems are discussed. Finally, to analyze the computational efficiency and applicability of the proposed numerical method, several numerical examples are presented.