Fractional-order Derivative Operator Research Articles

A generalized dusty Brinkman type fluid of MHD free convection two phase flow between parallel plates

Fractional order derivative operators are regarded to be very complex mathematical tools for implementing more realistic solutions in a wide range of physics and engineering problems. Due to significance applications of fractional derivatives, the aim of this article is analyzed the combinational effect of the magnetic field and heat transfer on Magneto hydrodynamic, free convective, two-phase flow of generalized Brinkman type dusty fluid among two parallel plates. The buoyant force causes the flow, which is aided by natural convection to transfer heat. Furthermore, the left plate travels at a constant speed and the right plate remains fixed, and all spherically dust particles are homogeneously dispersed among the fluid. The problem is modeled mathematically via set of partial differential equations. The Caputo-Fabrizio fractional derivative is employed to generalize the derived PDE's. A combination of Laplace and finite sine Fourier transformations is utilized to get the closed form solution of the problem. The impacts of many factors on temperature, Brinkman fluid, and dust particle velocity have also been examined, including a Grashof number, magnetic parameter, Reynold number, Peclet number and Dusty fluid parameter. Mathcad-15 is used to plot the graphical findings for the dusty fluid, Brinkman fluid, and temperature profiles. For various embedded parameters, the Brinkman fluid and dusty fluid behave similarly. It is concluded that the fractional Brinkman type dusty fluid is more realistic feature as compared to the classical one. The rate of heat transfer can be enhance with the passage of time.

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor’s series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He’s and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results.

Nonlocal Integro-Multi-Point (k, ψ)-Hilfer Type Fractional Boundary Value Problems

In this paper we investigate the criteria for the existence of solutions for single-valued as well as multi-valued boundary value problems involving (k,ψ)-Hilfer fractional derivative operator of order in (1,2], equipped with nonlocal integral multi-point boundary conditions. For the single-valued case, we rely on fixed point theorems due to Banach and Krasnosel’skiĭ, and Leray–Schauder alternative to establish the desired results. The existence results for the multi-valued problem are obtained by applying the Leray–Schauder nonlinear alternative for multi-valued maps for convex-valued case, while the nonconvex-valued case is studied with the aid of Covit–Nadler’s fixed point theorem for multi-valued contractions. Numerical examples are presented for the illustration of the obtained results.

Time response optimal rational approximation: Improvement of time responses of MSBL based approximate fractional order derivative operators by using gradient descent optimization

Fractional calculus has become a popular topic in recent years. Its applications have been seen frequently in the analysis and solution of many engineering problems. Since, the exact realization of fractional order (FO) operators is not computationally feasible, several integer order approximations methods such as Oustaloup, Matsuda, Curve Fitting, modified stability boundary locus (MSBL) are utilized for frequency domain approximation of FO operators. These methods present advantages in frequency response approximation. However, real-world engineering applications require more accurate time responses to implement FO operators. Thus, any improvement in the approximate transfer function of FO systems will positively contribute to realization performance of FO system models in real-world applications. In this work, time responses of MSBL based approximate FO derivative models are improved by using gradient descent optimization (GDO) technique. Frequency domain approximation methods do not guarantee optimal approximation performance for the step response of FO elements. To deal with this weakness, author proposes a new GDO based rational approximation scheme that can considerably enhance the step response performance of MSBL method without deteriorating frequency response approximation performance. Results for the improved approximate FO model obtained by GDO (GDO-MSBL) are compared with the results of approximate FO models for popular FO approximation methods. The analog circuit realization is performed for the derivative operator, and the performance improvements for the proposed method are shown on the FO-PID controller and FO filter.

Behavioural modelling of delayed imbalance dynamics in nature: a parametric modelling for simulation of delayed instability dynamics

Imbalance dynamics can develop very slowly, and real systems and structures may seem to be stable or balanced for long periods of time before signs of instability behaviour become apparent. This study presents two dynamic system modelling approaches for simulation of delayed instability: Firstly, frequency domain properties of the system instability are investigated, and a parametric model to represent delayed instability behaviour is formulated according to the system pole placement technique. Secondly, a new type of instability modelling approach, which is based on time-domain characteristics of fractional order derivative operators, is introduced by utilizing the finite convergence regions of the Binomial series. This special instability modelling technique essentially uses the region of convergence in the series expansion of impulse responses. Several illustrative modelling and simulation examples are illustrated for engineering problems such as slowly developing cracks in metals, the voltage collapse in power systems and the delayed instability in control systems.

An extensive note on various fractional-order type operators and some of their effects to certain holomorphic functions

Abstract The aim of this paper is to present background information in relation with some fractional-order type operators in the complex plane, which is designed by the fractional-order derivative operator(s). Next we state various implications of that operator and then we show some interesting-special results of those applications.

Approximate and Exact Solutions to Fractional Order Cauchy Reaction-Diffusion Equations by New Combine Techniques

In this paper, we present a simple and efficient novel semianalytic method to acquire approximate and exact solutions for the fractional order Cauchy reaction-diffusion equations (CRDEs). The fractional order derivative operator is measured in the Caputo sense. This novel method is based on the combinations of Elzaki transform method (ETM) and residual power series method (RPSM). The proposed method is called Elzaki residual power series method (ERPSM). The proposed method is based on the new form of fractional Taylor’s series, which constructs solution in the form of a convergent series. As in the RPSM, during establishing the coefficients for a series, it is required to compute the fractional derivatives every time. While ERPSM only requires the concept of the limit at zero in establishing the coefficients for the series, consequently scarce calculations give us the coefficients. The recommended method resolves nonlinear problems deprived of utilizing Adomian polynomials or He’s polynomials which is the advantage of this method over Adomain decomposition method (ADM) and homotopy-perturbation method (HTM). To study the effectiveness and reliability of ERPSM for partial differential equations (PDEs), absolute errors of three problems are inspected. In addition, numerical and graphical consequences are also recognized at diverse values of fractional order derivatives. Outcomes demonstrate that our novel method is simple, precise, applicable, and effectual.

An effective analog circuit design of approximate fractional-order derivative models of M-SBL fitting method

There is a growing interest in fractional calculus and Fractional Order (FO) system modeling in many fields of science and engineering. Utilization of FO models in real-world applications requires practical realization of FO elements. This study performs an analog circuit realization of approximate FO derivative models based on Modified Stability Boundary Locus (M-SBL) fitting method. This study demonstrates a low-cost and accurate analog circuit implementation of M-SBL fitting based approximate model of FO derivative elements for industrial electronics. For this purpose, a 4th order approximate derivative transfer function model of the M-SBL method is decomposed into the sum of first order low-pass filters form by using Partial Fraction Expansion (PFE) method, and the analog circuit design of the approximate FO derivative model is performed. Firstly, by using the final value theorem, authors theoretically show that the time response of the sum of first order low-pass filter form can converge to the time response of fractional order derivative operators. Then, the approximation performance of proposed FO derivative circuit design is validated for various input waveforms such as sinusoidal, square and sawtooth waveforms via Multisim simulations. Results indicate an accurate realization of the FO derivative in time response (an RMSE of 0.0241). The derivative circuit realization of the M-SBL fitting model in the form of the sum of first order low pass filters can yield a better time response approximation performance compared to the Continued Fraction Expansion (CFE) based ladder network realization of the approximate derivative circuit.

On One Limit Theorem Related to the Cauchy Problem Solution for the Schrödinger Equation with a Fractional Derivative Operator of Order $$ \upalpha\ \upepsilon \bigcup_{m=3}^{\infty}\left(m-1,m\right) $$

We prove a limit theorem on the convergence of mathematical expectations of functionals of sums of independent random variables to the Cauchy problem solution for the nonstationary Schr¨odinger equation with a symmetric fractional derivative operator of order $$ \upalpha \upepsilon \bigcup_{m=3}^{\infty}\left(m-1,m\right) $$ in the righthand side.

Dynamical Analysis of a Fractional Order HIV/AIDS Model

This article discusses a dynamical analysis of the fractional-order model of HIV/AIDS. Biologically, the rate of subpopulation growth also depends on all previous conditions/memory effects. The dependency of the growth of subpopulations on the past conditions is considered by applying fractional derivatives. The model is assumed to consist of susceptible, HIV infected, HIV infected with treatment, resistance, and AIDS. The fractional-order model of HIV/AIDS with Caputo fractional-order derivative operators is constructed and then, the dynamical analysis is performed to determine the equilibrium points, local stability and global stability of the equilibrium points. The dynamical analysis results show that the model has two equilibrium points, namely the disease-free equilibrium point and endemic equilibrium point. The disease-free equilibrium point always exists and is globally asymptotically stable when the basic reproduction number is less than one. The endemic equilibrium point exists if the basic reproduction number is more than one and is globally asymptotically stable unconditionally. To illustrate the dynamical analysis, we perform some numerical simulation using the Predictor-Corrector method. Numerical simulation results support the analytical results.

Qualitative analysis of 2019-nCoV mathematical model via an efficient computational technique

In the present scenario, the whole world struggling with Corona Virus Disease (COVID-19) which is a extremely transmittable disease stimulate by the novel corona virus (2019-nCoV) that believed to be originated in Wuhan, central China. Here, a mathematical analysis of 2019-nCoV mathematical model is considered with Caputo fractional order derivative operator. This model is made up of six subclasses i.e. susceptible people, exposed people, infected people, asymptotically infected people, recovered people and reservoir people. Authors obtained the series solutions of each subclasses of the proposed mathematical model via Sumudu Transform Homotopy Perturbation Method (STHPM). The results that have come are correct to show these authors have provided graphical interpretations.

Modeling and simulation of fractional order COVID-19 model with quarantined-isolated people.

The dynamics of diseases and effectiveness of control policies play important role in the prevention of epidemic diseases. To this end, this paper is concerned with the design of fractional order coronavirus disease (COVID‐19) model with Caputo Fabrizio fractional derivative operator of order for the COVID‐19. Verify the nonnegative special solution and convergence of the scheme with in the domain. Caputo‐Fabrizio technique apply with Sumudu transformation method is used to solve the fractional order COVID‐19 model. Fixed point theory and Picard Lindelof approach are used to provide the stability and uniqueness of the results. Numerical simulations conspicuously demonstrate that by applying the proposed fractional order model, governments could find useful and practical ways for control of the disease.

Numerical approach for approximating the Caputo fractional-order derivative operator

<abstract><p>This work aims to propose a new simple robust power series formula with its truncation error to approximate the Caputo fractional-order operator $ D_{a}^{\alpha}y(t) $ of order $ m-1 < \alpha < m $, where $ m\in\mathbb{N} $. The proposed formula, which are derived with the help of the weighted mean value theorem, is expressed ultimately in terms of a fractional-order series and its reminder term. This formula is used successfully to provide approximate solutions of linear and nonlinear fractional-order differential equations in the form of series solution. It can be used to determine the analytic solutions of such equations in some cases. Some illustrative numerical examples, including some linear and nonlinear problems, are provided to validate the established formula.</p></abstract>

A New Multimodel Representation of Fractional-Order Systems in Both Time and Frequency Domains

In order to deal with some difficult problems in fractional-order systems, like high computational load of fractional-order operator, fractional-order transfer function is commonly approximated by an integer order model. However, the dimension of this model increases with its accuracy, which can make the design of a controller more difficult. In this paper, a new approach for modelling of fractional-order systems is investigated. Exploiting the multimodel technique, the suggested method replaces the unique fractional-order model by a set of simpler integer order models. The determination of the different models is based on an approximation of the fractional-order derivative operator sα. Then the global model is obtained through a fusion of the simple models weighted by their respective relevance degrees calculated by optimizing a constrained least squares problem. The resulting final model can represent adequately the fractional-order systems both in time and in frequency domains. Simulations and comparative studies carried out on academic examples indicate the interest, the clarity and the improvement in accuracy in both time and frequency domains of the proposed modelling method, compared to modelling by a single model based on the approximation of Oustaloup.

New numerical solutions of fractional-order Korteweg-de Vries equation

We present new solutions of fractional-order Korteweg-de Vries (KdV) equation by employing a method that utilizes advantages of both techniques of fictititous time integration and group preserving. Proposed method converts the KdV to a new system of equations which is approximately solved by a semi-discretization numerical scheme. Caputo type fractional-order derivative operators are used. We apply the method to some specific cases of the KdV equation. Computational results indicate that the method gives new and highly efficient solutions of the KdV equation.

A new fractional-order compartmental disease model

In this paper, we propose a new SEIRS model and are concerned with stability and numerical solutions of the model. The model is generated under certain assumptions such as individuals are vaccinated or have a special treatment but do not carry lifelong immunity. After generating a new SEIRS model, we perturb the model into fractional time derivative form where Caputo type fractional-order derivative operators are employed. After showing existence and uniqueness of the non-negative solutions, we determine disease free steady-state point and basic reproduction number. We also determine endemic steady state points and study on stability of the fractional system in these equilibrium points. We solve fractional-order system approximately with an efficient Euler type numerical method. We conclude that the proposed system may serve as a kernel for understanding, analysis and computational solutions of a wide range of disease models in epidemiology.

Enhanced FPGA realization of the fractional-order derivative and application to a variable-order chaotic system

The efficiency of the hardware implementations of fractional-order systems heavily relies on the efficiency of realizing the fractional-order derivative operator. In this work, a generic hardware implementation of the fractional-order derivative based on the Grunwald–Letnikov’s approximation is proposed and verified on a field-programmable gate array. The main advantage of this particular realization is its flexibility in applications which enable easy real-time configuration of the values of the fractional orders, step sizes, and/or other system parameters without changing the hardware architecture. Different approximation techniques are used to improve the hardware performance including piece-wise linear/quadratic methods. As an application, a variable-order chaotic oscillator is implemented and verified using fractional orders that vary in time.

Mathematical Analysis of Fractional Order Co-Infection TB and HIV Model

A mathematical model of HIV/AIDS and TB including its co-infections is formulated. We find the Equilibrium points and with the help of numerical simulation, we have analyze that the sub-models of TB, HIV/AIDS and its co-infections. The Caputo and Caputo Febrizo fractional derivative operator of order α ∈ (0, 1] is employed to obtain the system of fractional differential equations. Laplace Adomian Decomposition Method was successfully used for solving the different differential equations.Laplace transform is a perfect technique in various field of biological science,engineering,pure and applied mathematics. The latest technique Laplace Adomian Decomposition Method is employed on the developed fractional order model for the numerical solutions. Finally numerical simulations are also established to investigate the influence of the system parameter on the spread of disease and which show effect of fractional parameter α on our obtained solutions.

On the oscillation of differential equations in frame of generalized proportional fractional derivatives

In this paper, sufficient conditions are established for the oscillation of all solutions of generalized proportional fractional differential equations of the form $ \begin{equation} \left\{ \begin{array}{l} {_{a}D}^{\alpha, \rho}x(t) + \xi_1(t,x(t)) = \mu(t) + \xi_2(t,x(t)),\quad t \gt a \ge 0,\\[0.3cm] \lim\limits_{t\to a^{+}} {_{a}I}^{j-\alpha, \rho}x(t) = b_j,\quad j = 1,2,\ldots,n, \end{array} \right. \end{equation} $ where $n = \lceil \alpha \rceil$, ${_{a}D}^{\alpha, \rho}$ is the generalized proportional fractional derivative operator of order $\alpha\in \mathbb{C}$, $Re(\alpha)\ge 0$, $0 \lt \rho\le 1$ in the Riemann-Liouville setting and ${_{a}I}^{\alpha, \rho}$ is the generalized proportional fractional integral operator. The results are also obtained for the generalized proportional fractional differential equations in the Caputo setting. Numerical examples are provided to illustrate the applicability of the main results.

Some new mathematical models of the fractional-order system of human immune against IAV infection.

Fractional derivative operators of non-integer order can be utilized as a powerful tool to model nonlinear fractional differential equations. In this paper, we propose numerical solutions for simulating fractional-order derivative operators with the power-law and exponential-law kernels. We construct the numerical schemes with the help the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. These schemes are applied to simulate the dynamical fractional-order model of the immune response (FMIR) to the uncomplicated influenza A virus (IAV) infection, which focuses on the control of the infection by the innate and adaptive immunity. Numerical results are then presented to show the applicability and efficiency on the FMIR.