Nonlocal Fractional Operator Research Articles

On a fractional derivative operator with a singular kernel: definition, properties and numerical simulation

This paper is concerned with proposing a novel nonlocal fractional derivative operator with a singular kernel. We considered a fractional integral operator as a single integral of convolution type combined with a Mittag-Leffler kernel of Prabhakar type. The proposed singular fractional derivative operator is formulated as a proper inverse of the considered integral operator. We provided some useful features and relationships of the proposed derivative and introduced comparisons with the Caputo derivative which can be utilized for potential applications. Next, we presented numerical solutions for some nonlinear fractional order models incorporating the proposed derivative using a numerical algorithm developed in this paper. As a case study, we discussed the dynamic behavior of a fractional logistic model with the proposed derivative.

A MODERN TRAVELING WAVE SOLUTION FOR CAPUTO-FRACTIONAL KLEIN–GORDON EQUATIONS

This research paper introduces a novel approach to deriving traveling wave solutions (TWSs) for the Caputo-fractional Klein–Gordon equations. This research presents a distinct methodological advancement by introducing TWSs of a particular time-fractional partial differential equation, utilizing a non-local fractional operator, specifically the Caputo derivative. To achieve our goal, a novel transformation is considered, that converts a time-fractional partial differential equation into fractional ordinary differential equations, enabling analytical solutions through various analytical methods. This paper employs the homotopy analysis method to achieve the target objectives. To demonstrate the efficiency and applicability of the proposed transform and method, two examples are discussed and analyzed in figures.

A static memory sparse spectral method for time-fractional PDEs

We discuss a method which provides accurate numerical solutions to fractional-in-time partial differential equations posed on [0,T]×Ω with Ω⊂Rd without the excessive memory requirements associated with the nonlocal fractional derivative operator. Our approach combines recent advances in the development and utilization of multivariate sparse spectral methods as well as fast methods for the computation of Gauss quadrature nodes with recursive non-classical methods for the Caputo fractional derivative of general fractional order α>0. An attractive feature of the method is that it has minimal theoretical overhead when using it on any domain Ω⊂Rd on which an orthogonal polynomial basis is available. We discuss the memory requirements of the method, present several numerical experiments demonstrating the method's performance in solving time-fractional PDEs on intervals, triangles and disks and derive error bounds which suggest sensible convergence strategies. As an important model problem for this approach we consider a type of wave equation with time-fractional dampening related to acoustic waves in viscoelastic media with applications in the physics of medical ultrasound and outline future research steps required to use such methods for the reverse problem of image reconstruction from sensor data.

The initial value problem for the equations of motion of fractional compressible viscous fluids

In this paper we consider the initial value problem to the fractional compressible isentropic generalized Navier-Stokes equations for viscous fluids with one Levy diffusion process in which the viscosity term appeared in the fluid equations is described by the nonlocal fractional Laplace operator. We give one detailed spectrum analysis on a linearized operator and the decay law in time of the solution semigroup for the linearized fractional compressible isentropic generalized Navier-Stokes equations around a constant state by the Fourier analysis technique, which is shown that the order of the fractional derivatives plays a key role in the analysis so that the spectrum structure involved here is more complex than that of the classical compressible Navier-Stokes system. Based on this and the elaborate energy method, the global-in-time existence and one optimal decay rate in time of the smooth solution are obtained under the assumption that the initial data are given in a small neighborhood of a constant state.

Monotonicity properties of the eigenvalues of nonlocal fractional operators and their applications

Monotonicity properties of the eigenvalues of nonlocal fractional operators and their applications

Singular boundary behaviour and large solutions for fractional elliptic equations.

We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equationsposed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace-Poisson equationwith zero boundarydata. In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that, for equationsdriven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Greenfunction.

Nonlinear fractional stochastic heat equation driven by Gaussian noise rough in space

In this paper we consider a class of nonlinear fractional stochastic heat equation∂∂tu(t,x)=Dδαu(t,x)+σ(u(t,x))W˙(t,x),(t,x)∈[0,T]×R, in (1+1)-dimension with T>0, where Dδα is a nonlocal fractional differential operator with α∈(1,2) and |δ|≤2−α. The diffusion coefficient σ(⋅) is a measurable function. W˙ is a Gaussian noise which is white in time and behaves as a fractional Brownian motion with Hurst index H satisfying 3−α4<H<12 in the space variable. Under some mild assumptions, we prove the existence and uniqueness of the mild solution in some function spaces. Along the way, we study the moment bounds for the solution and show that the solution is weakly full intermittent.

Well-Posedness and Global Attractors for Viscous Fractional Cahn–Hilliard Equations with Memory

We examine a viscous Cahn–Hilliard phase-separation model with memory and where the chemical potential possesses a nonlocal fractional Laplacian operator. The existence of global weak solutions is proven using a Galerkin approximation scheme. A continuous dependence estimate provides uniqueness of the weak solutions and also serves to define a precompact pseudometric. This, in addition to the existence of a bounded absorbing set, shows that the associated semigroup of solution operators admits a compact connected global attractor in the weak energy phase space. The minimal assumptions on the nonlinear potential allow for arbitrary polynomial growth.

A thermodynamic two-temperature model with distinct fractional derivative operators for an infinite body with a cylindrical cavity and varying properties

The current work presents a novel model in the field of thermoelasticity to study an infinitely cylindrical cavity solid medium with variable thermal properties. The governing equations are based on a two-temperature heat conduction model with local and non-local fractional differential operators (Caputo-Fabrizio (CF) and Atangana-Baleanu (AB)). The surface surrounding the cylindrical cavity is due to decreasing heat flow and transport at a constant speed. The effects of different fractional differential operators have been shown and analyzed by making some comparisons and presenting them in tables. The numerical results and figures show the effects of the fractional arrangement parameter and the heat flux velocity significantly on the stress and deformation patterns, but they have little effect on the dynamic temperature and conductivity changes. The thermoelastic model with two temperatures and fractional operators is suitable for describing the phenomenon of heat transfer within elastic and viscous media.

Fractional dynamics and analysis for a lana fever infectious ailment with Caputo operator

This study provides a deep insight into the complex process of the lana fever infectious disease model by employing the advantages of a non-local fractional operator called Caputo derivative. The fractional-type lana fever model possessing memory effect and hereditary properties allows us to comprehend the dynamics of the disease transmission in detail. For this purpose, we introduce some crucial theoretical and numerical results for the suggested disease model with the aid of the Caputo fractional derivative. The existence and uniqueness of the solution of lana fever system are shown through the fixed point theorem. Moreover, we find the most sensitive parameters under the forward sensitivity index in relation to reproduction number R0. Also, some basic properties of the lana fever model such as virus-free equilibrium point, reproduction number, virus endemic equilibrium point, local and global stability are presented by using the Caputo derivative. On the other hand, further analysis of the fractional version of the lana fever model shows that the disease-free equilibrium is locally asymptotically stable when R0<1, and also the endemic equilibrium of the model under investigation is globally asymptotically stable when R0>1. Additionally, numerical results are presented to observe the advantages of the non-local fractional differential operator of Caputo.

Comparative analysis of suitability of fractional derivatives in modelling the practical capacitor

PurposeThe purpose of this paper is to compare the suitability of fractional derivatives in the modelling of practical capacitors. Such suitability refers to ability to provide the analytical capacitance function that matches the experimental ones of each fractional derivative.Design/methodology/approachThe analytical capacitance functions based on various fractional derivatives of both local and nonlocal types including the author’s have been derived. The derived capacitance functions have been simulated and compared with the experimental ones of aluminium electrolytic and electrical double layer capacitors (EDLCs).FindingsThis paper has found that any local fractional derivative with fractional power law-based relationship with the conventional one is suitable for modelling the aluminium electrolytic capacitor (AEC) by incorporating with the conventional capacitance definition. On the other hand, the author’s nonlocal fractional derivatives have been found to be more suitable than the others for modelling the EDLC by incorporating with the revisited definition of capacitance.Originality/valueThe proposed comparative analysis has been originally presented in this work. The criterion for local fractional derivative, to be suitable for modelling the AEC, has been found. The nonlocal fractional operators which are most suitable for modelling the EDLC have been derived where the unsuitable one has been pointed out.

DEVELOPMENT AND ANALYSIS OF A SEIR MODEL FOR COVID-19 EPIDEMIC WITH VACCINATION AND NONSINGULAR KERNEL

We propose a new model to investigate the recent coronavirus (COVID-19) epidemic. The fundamental reproductive number is determined using the next-generation matrix, and the model’s soundness is tested using stability theory. The local asymptotic stability conditions for the disease-free equilibrium are determined. The model is extended to the fractional domain, and its properties are investigated using the Caputo–Fabrizio (CFC) robust nonsingular and nonlocal fractional operator. We focused on numerous exciting elements of the expanded model, such as solution positivity. In numerical aspects, different values of numerous parameters are employed to view certain physical behavior of the model.

Chaos and bifurcations in a discretized fractional model of quasi-periodic plasma perturbations

Abstract The nonlinear dynamics of a discretized form of quasi-periodic plasma perturbations model (Q-PPP) with nonlocal fractional differential operator possessing singular kernel are investigated. For example, the conditions for the stability and occurrence of Neimark–Sacker (NS) and flip bifurcations in the proposed discretized equations are provided. Moreover, analysis of nonlinearities such as the existence of chaos in this map is proved numerically via bifurcation diagrams, Lyapunov exponents and analytically via Marotto’s Theorem. Also, some simulation results are utilized to confirm the theoretical results and to show that the obtained map exhibits double routes to chaos: one is via flip bifurcation and the other is via NS bifurcation. Furthermore, more complex dynamical phenomena such as existence of closed invariant curves, homoclinic orbits, homoclinic connections, period 3 and period 4 attractors are shown. This kind of research is useful for physicists who work with tokamak models.

Mathematical Analysis of Oxygen Uptake Rate in Continuous Process under Caputo Derivative

In this paper, the wastewater treatment model is investigated by means of one of the most robust fractional derivatives, namely, the Caputo fractional derivative. The growth rate is assumed to obey the Contois model, which is often used to model the growth of biomass in wastewaters. The characteristics of the model under consideration are derived and evaluated, such as equilibrium, stability analysis, and steady-state solutions. Further, important characteristics of the fractional wastewater model allow us to understand the dynamics of the model in detail. To this end, we discuss several important analyses of the fractional variant of the model under consideration. To observe the efficiency of the non-local fractional differential operator of Caputo over its counter-classical version, we perform numerical simulations.

Respiratory mechanics models in the frame of non-local fractional operators

This study is intended to provide an exhaustive comparison of various non-local fractional operators when analyzing the mechanics of breathing performed by a ventilator. We present the solutions of the fractional respiratory mechanics model mainly for allowing the health care team to monitor patients' conditions in detail. The aim is to introduce the underlying model flexibly by making use of the advantages of the non-integer order operators with one, two, and three parameters. On the other hand, since we concern the usage of different types of fractional operators, most often having virtue in the application, it can be clearly observed the similarities and differences between these operators. Moreover, we observe the mobility of the solutions curves of the above-stated fractional model for different values of the parameters $\alpha$, $\rho$, and $\gamma$ with the help of many graphs.

Analysis of novel fractional COVID-19 model with real-life data application

The current work is of interest to introduce a detailed analysis of the novel fractional COVID-19 model. Non-local fractional operators are one of the most efficient tools in order to understand the dynamics of the disease spread. For this purpose, we intend as an attempt at investigating the fractional COVID-19 model through Caputo operator with order χ∈(0,1). Employing the fixed point theorem, it is shown that the solutions of the proposed fractional model are determined to satisfy the existence and uniqueness conditions under the Caputo derivative. On the other hand, its iterative solutions are indicated by making use of the Laplace transform of the Caputo fractional operator. Also, we establish the stability criteria for the fractional COVID-19 model via the fixed point theorem. The invariant region in which all solutions of the fractional model under investigation are positive is determined as the non-negative hyperoctant R+7. Moreover, we perform the parameter estimation of the COVID-19 model by utilizing the non-linear least squares curve fitting method. The sensitivity analysis of the basic reproduction number R0c is carried out to determine the effects of the proposed fractional model’s parameters on the spread of the disease. Numerical simulations show that all results are in good agreement with real data and all theoretical calculations about the disease.

Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator

&lt;p style='text-indent:20px;'&gt;The aim of this paper is to study the calcium profile governed by the advection diffusion equation. The mathematical and computational modeling has provided insights to understand the calcium signalling which depends upon cytosolic calcium concentration. Here the model includes the important physiological parameters like diffusion coefficient, flow velocity etc. The mathematical model is fractionalised using Hilfer derivative and appropriate boundary conditions have been framed. The use of fractional order derivative is more advantageous than the integer order because of the non-local property of the fractional order differentiation operator i.e. the next state of the system depends not only upon its current state but also upon all of its preceeding states. Analytic solution of the fractional advection diffusion equation arising in study of diffusion of cytosolic calcium in RBC is found using integral transform techniques. Since, the Hilfer derivative is generalisation of Riemann- Liouville and Caputo derivatives so, these two are also deduced as special cases. The numerical simulation has been done to observe the effects of the fractional order of the derivatives involved in the differential equation representing the model over the concentration of calcium which is function of time and distance. The concentration profile of calcium is significantly changed by the fractional order.&lt;/p&gt;

Existence of solutions for non-coercive variational-hemivariational inequalities involving the nonlocal fractional p-Laplacian

ABSTRACT In this paper, we study the existence of solutions for non-coercive variational-hemivariational inequalities involving nonlocal fractional p-Laplacian. Our approach is based on the theory of pseudomonotone operators in the sense of Brézis, recession analysis and the properties of nonlocal fractional p-Laplace operators recently established.The innovation of this paper is that we do not assume the usual coercivity or smallness assumptions as stated in most existing literature. The absence of such assumptions will make the existence of solutions very complicated. We have to proceed from the specific problem itself and seek specific methods based on the inherent characteristics of different problems.

The Effect of a Non-Local Fractional Operator in an Asymmetrical Glucose-Insulin Regulatory System: Analysis, Synchronization and Electronic Implementation

For studying biological conditions with higher precision, the memory characteristics defined by the fractional-order versions of living dynamical systems have been pointed out as a meaningful approach. Therefore, we analyze the dynamics of a glucose-insulin regulatory system by applying a non-local fractional operator in order to represent the memory of the underlying system, and whose state-variables define the population densities of insulin, glucose, and β-cells, respectively. We focus mainly on four parameters that are associated with different disorders (type 1 and type 2 diabetes mellitus, hypoglycemia, and hyperinsulinemia) to determine their observation ranges as a relation to the fractional-order. Like many preceding works in biosystems, the resulting analysis showed chaotic behaviors related to the fractional-order and system parameters. Subsequently, we propose an active control scheme for forcing the chaotic regime (an illness) to follow a periodic oscillatory state, i.e., a disorder-free equilibrium. Finally, we also present the electronic realization of the fractional glucose-insulin regulatory model to prove the conceptual findings.

Complex dynamics and control of a novel physical model using nonlocal fractional differential operator with singular kernel

Fractional calculus (FC) is widely used in many interdisciplinary branches of science due to its effectiveness in describing and investigating complicated phenomena. In this work, nonlinear dynamics for a new physical model using nonlocal fractional differential operator with singular kernel is introduced. New Routh-Hurwitz stability conditions are derived for the fractional case as the order lies in [0,2). The new and basic Routh-Hurwitz conditions are applied to the commensurate case. The local stability of the incommensurate orders is also discussed. A sufficient condition is used to prove that the solution of the proposed system exists and is unique in a specific region. Conditions for the approximating periodic solution in this model via Hopf bifurcation theory are discussed. Chaotic dynamics are found in the commensurate system for a wide range of fractional orders. The Lyapunov exponents and Lyapunov spectrum of the model are provided. Suppressing chaos in this system is also achieved via two different methods.