Non-local Kernel Research Articles

A robust study to conceptualize the interactions of CD4+ T-cells and human immunodeficiency virus via fractional-calculus

It is eminent that the infection of HIV is a censorious problem to public health for the last forty years. This viral infection is the cause of Acquired Immunodeficiency Virus (AIDS) which deteriorated the human population around the world and having a high economic burden on different infected regions. In HIV infection, the virus attack on the CD4+ T-cells in the body, this viral infection makes the body of the host open for the attack of other infections. Therefore, it is valuable to study and examine the interactions of CD4+ T-cells and HIV viruses for a better understanding of the infection. In this study, we structure the intricate interactions of HIV and CD4+ T-cells through the fractional framework to interrogate this viral infection. We use Atangana-Baleanu (AB) fractional operator to highlight the overall picture of HIV infection with the nonlocal and nonsingular kernel. We present the fundamental theory of the fractional operator for the analysis of our system. We have proved results for the unique solution of the AB system of HIV infection. A numerical method is introduced to highlight the chaotic and dynamical behavior of the HIV infection with the variation of the input parameters. We perform different simulations to conceptualize the role of input factor on the system and highlight the dynamics of HIV infection through numerically. The chaos and oscillatory behavior is strongly associated and is due to the nonlinearity of the system. In addition to this, our analysis recommends the most dominant parameters for the control and prevention of the infection.

Displacement-driven approach to nonlocal elasticity

This study presents a physically-consistent displacement-driven reformulation of the concept of action-at-a-distance, which is at the foundation of nonlocal elasticity. In contrast to the class of existing approaches that adopts an integral stress–strain constitutive relation, the displacement-driven approach is predicated on an integral strain–displacement relation. One of the most remarkable consequence of this reformulation is that the (total) strain energy is guaranteed to be a convex and positive-definite function without imposing any constraint on the symmetry of the nonlocal kernel. This feature is critical to enable the application of nonlocal formulations to general continua exhibiting asymmetric interactions; ultimately a manifestation of material heterogeneity. Remarkably, the proposed approach also enables a strong (point-wise) satisfaction of the locality recovery condition and of the laws of thermodynamics, which are not foregone conclusions in most classical nonlocal elasticity theories. Additionally, the formulation is frame-invariant and the nonlocal operator remains physically consistent at material interfaces and domain boundaries. The study is complemented by a detailed analysis of the dynamic response of the nonlocal continuum and of its intrinsic dispersion, leading to the consideration that the choice of a nonlocal kernel should depend on the specific material. Examples of either exponential or power-law kernels are presented in order to demonstrate the applicability of the method to different classes of nonlocal media. The ability to admit generalized kernels reinforces the generalized nature of the displacement-driven approach over existing integral methodologies, which typically lead to simplified differential models based on exponential kernels. The theoretical formulation is also leveraged to perform numerical simulations of the linear static response of nonlocal beams and plates further illustrating the intrinsic consistency of the approach, which is free from unwanted and unrealistic boundary effects.

A mathematical study on a fractional COVID-19 transmission model within the framework of nonsingular and nonlocal kernel

In this work, a mathematical model consisting of a compartmentalized coupled nonlinear system of fractional order differential equations describing the transmission dynamics of COVID-19 is studied. The fractional derivative is taken in the Atangana-Baleanu-Caputo sense. The basic dynamic properties of the fractional model such as invariant region, existence of equilibrium points as well as basic reproduction number are briefly discussed. Qualitative results on the existence and uniqueness of solutions via a fixed point argument as well as stability of the model solutions in the sense of Ulam-Hyers are furnished. Furthermore, the model is fitted to the COVID-19 data circulated by Nigeria Centre for Disease Control and the two-step Adams-Bashforth method incorporating the noninteger order parameter is used to obtain an iterative scheme from which numerical results for the model can be generated. Numerical simulations for the proposed model using Adams-Bashforth iterative scheme are presented to describe the behaviors at distinct values of the fractional index parameter for of each of the system state variables. It was shown numerically that the value of fractional index parameter has a significant effect on the transmission behavior of the disease however, the infected population (the exposed, the asymptomatic infectious, the symptomatic infectious) shrinks with time when the basic reproduction number is less than one irrespective of the value of fractional index parameter.

SOME FURTHER EXTENSIONS CONSIDERING DISCRETE PROPORTIONAL FRACTIONAL OPERATORS

In this paper, some attempts have been devoted to investigating the dynamic features of discrete fractional calculus (DFC). To date, discrete fractional systems with complex dynamics have attracted the most consideration. By considering discrete [Formula: see text]-proportional fractional operator with nonlocal kernel, this study contributes to the major consequences of the certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete [Formula: see text]-proportional fractional sums, as presented. The proposed system has an intriguing feature not investigated in the literature so far, it is characterized by the nabla [Formula: see text] fractional sums. Novel special cases are reported with the intention of assessing the dynamics of the system, as well as to highlighting the several existing outcomes. In terms of applications, we can employ the derived consequences to investigate the existence and uniqueness of fractional difference equations underlying worth problems. Finally, the projected method is efficient in analyzing the complexity of the system.

Learning the structure of wind: A data-driven nonlocal turbulence model for the atmospheric boundary layer

We develop a novel data-driven approach to modeling the atmospheric boundary layer. This approach leads to a nonlocal, anisotropic synthetic turbulence model which we refer to as the deep rapid distortion (DRD) model. Our approach relies on an operator regression problem that characterizes the best fitting candidate in a general family of nonlocal covariance kernels parameterized in part by a neural network. This family of covariance kernels is expressed in Fourier space and is obtained from approximate solutions to the Navier–Stokes equations at very high Reynolds numbers. Each member of the family incorporates important physical properties such as mass conservation and a realistic energy cascade. The DRD model can be calibrated with noisy data from field experiments. After calibration, the model can be used to generate synthetic turbulent velocity fields. To this end, we provide a new numerical method based on domain decomposition which delivers scalable, memory-efficient turbulence generation with the DRD model as well as others. We demonstrate the robustness of our approach with both filtered and noisy data coming from the 1968 Air Force Cambridge Research Laboratory Kansas experiments. Using these data, we witness exceptional accuracy with the DRD model, especially when compared to the International Electrotechnical Commission standard.

Analysis of COVID-19 Fractional Model Pertaining to the Atangana-Baleanu-Caputo Fractional Derivatives

The transmission dynamics of a COVID-19 pandemic model with vertical transmission is extended to nonsingular kernel type of fractional differentiation. To study the model, Atangana-Baleanu fractional operator in Caputo sense with nonsingular and nonlocal kernels is used. By using the Picard-Lindel method, the existence and uniqueness of the solution are investigated. The Hyers-Ulam type stability of the extended model is discussed. Finally, numerical simulations are performed based on real data of COVID-19 in Indonesia to show the plots of the impacts of the fractional order derivative with the expectation that the proposed model approximation will be better than that of the established classical model.

Dynamic PET image reconstruction incorporating a median nonlocal means kernel method

In dynamic positron emission tomography (PET) imaging, the reconstructed image of a single frame often exhibits high noise due to limited counting statistics of projection data. This study proposed a median nonlocal means (MNLM)-based kernel method for dynamic PET image reconstruction. The kernel matrix is derived from median nonlocal means of pre-reconstructed composite images. Then the PET image intensities in all voxels were modeled as a kernel matrix multiplied by coefficients and incorporated into the forward model of PET projection data. Then, the coefficients of each feature were estimated by the maximum likelihood method. Using simulated low-count dynamic data of Zubal head phantom, the quantitative performance of the proposed MNLM kernel method was investigated and compared with the maximum-likelihood method, conventional kernel method with and without median filter, and nonlocal means (NLM) kernel method. Simulation results showed that the MNLM kernel method achieved visual and quantitative accuracy improvements (in terms of the ensemble mean squared error, bias versus variance, and contrast versus noise performances). Especially for frame 2 with the lowest count level of a single frame, the MNLM kernel method achieves lower ensemble mean squared error (10.43%) than the NLM kernel method (13.68%), conventional kernel method with and without median filter (11.88% and 23.50%), and MLEM algorithm (24.77%). The study on real low-dose 18F-FDG rat data also showed that the MNLM kernel method outperformed other methods in visual and quantitative accuracy improvements (in terms of regional noise versus intensity mean performance).

Thermodynamically consistent nonlocal kernel with boundary effect compensation and its application to the coupled phase field-nonlocal integral elasticity equations for modeling of martensitic transformations

In this article, the thermodynamically consistent compensated two-phase (CTP) kernel is employed for nonlocal integral elasticity based phase field modeling of stress-induced martensitic transformations. Utilizing a proper thermodynamic framework, the stress-strain relation with the CTP kernel is shown to be thermodynamically consistent. The coupled Ginzburg–Landau and local/nonlocal elasticity equations are solved using the finite element method. The advantages of the CTP kernel over previous kernels are shown through stress-induced martensitic growths in a simply connected region, in presence of a hole, in presence of a crack, and in a sample with a preexisting nucleus. In contrast to other widely used nonlocal kernels, for the CTP kernel, no ill-posedness is observed, the normalization and locality recovery conditions are satisfied and the boundary effects are entirely compensated. The numerical convergence of a phase field-nonlocal integral elasticity problem is studied, which indicates that the CTP kernel does not suffer from the numerical convergence issues of previous kernels. The present study provides a better insight into the CTP kernel and its application to the modeling of various phenomena at the nanoscale.

Smoothed peridynamics for the extremely large deformation and cracking problems: Unification of peridynamics and smoothed particle hydrodynamics

AbstractPeridynamics (PD) and smoothed particle hydrodynamics (SPH) are definitely attractive theories in amounts of numerical mechanical methods these years, and numerous researchers have studied the similarities of these two methods. In this paper, the smoothed peridynamics (SPD) is proposed to unify these two theories in the meshless view. The SPD employs an update‐Lagrangian (UL) method, which is useful for the extremely large deformation and cracking problem. The SPD governing equation, nonlocal interaction, micro‐bond modules for elastic material are derived. In addition, the choice of the nonlocal kernel and logarithmic stretch is investigated. Finally, numerical experiments are studied to confirm the ability of SPD. The numerical results show that SPD has an excellent performance and application prospect in computational solid mechanics. Since the SPD model formulations are derived for general 3D conditions, it can be straight forwardly extended for large‐scale practical applications across disciplines.

The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique

This research considers an inverse source problem for fractional diffusion equation that containing fractional derivative with non-singular and non-local kernel, namely, Atangana-Baleanu-Caputo fractional derivative. In our study, an explicit solution set is acquired via the expansion method and the overdetermination condition at a final time. The problem is ill-posed in the meaning of Hadamard and thus the solution does not continuously depend on the input data. We have applied the Tikhonov regularization method to regularize the unstable solution. For the estimation of convergence between the exact and the regularized solutions, we focus on two parameter choice rules, a-priori and a-posteriori parameter. In the end, a simulation example is utilized and discussed to affirm the presented theoretical results.

Effect of Local and Non-Local Kernels on Heat Transfer of Mixed Convection Flow of the Maxwell Fluid

The heat transfer study of mixed convection flow of the Maxwell fluid is carried out here. The fluid flow is demonstrated by the system of coupled partial differential equations in the dimensionless form firstly. Then, its fractional form is developed by using the new definition of the noninteger-order derivative with the singular kernel (Caputo/C) and nonsingular kernels (Caputo–Fabrizio/CF and Atangana–Baleanu (nonlocal)/ABC). The hybrid-form solutions are obtained by applying the Laplace transform, and for the inverse Laplace transform, the problem is tackled by the numerical algorithms of Stehfest and Tzou. The C, CF, and ABC solution comparison under the effects of considered different parameters is depicted. The physical aspects of the considered problem are well explained by C, CF, and ABC in comparison to the integer-order derivative due to its memory effects. Furthermore, the best fit model to explain the memory effects of velocity is CF. The solutions for the Newtonian fluid and ordinary Maxwell fluid are considered as a special case and found in the literature.

The fuzzy fractional SIQR model of computer virus propagation in wireless sensor network using Caputo Atangana–Baleanu derivatives

This work is devoted to study the uncertain attacking behavior of computer viruses in wireless sensor network involving fuzzy fractional derivatives with non-local Mittag-Leffler function kernel. Based on epidemic theory and fractional calculus, we propose a fuzzy fractional Susceptible – Infectious – Quarantine – Recovered (SIQR) model to describe dynamics of virus propagation with quarantine in the network. The concept of Atangana–Baleanu fuzzy fractional derivative in Caputo sense is proposed with some important properties to investigate the fractional SIQR model with fuzzy data. The fuzzy Laplace transform of Atangana–Baleanu derivative is proposed to represent the analytic mild solutions of the fractional SIQR model. Then, the existence and uniqueness of mild solutions is proved by using generalized contraction principle. An efficient numerical scheme to solve numerical solutions of the fractional SIQR model is introduced. Finally, some graphical representations are given to show the uncertain attack behavior of computer virus and dynamical behavior of the model.

Nonlocal hcp kernel functions based on ab initio calculations: Pertinent dislocation problems revisited

Eringen's nonlocal theory and an accurate determination of the nonlocal kernel functions for hexagonal close-packed (hcp) crystals are of interest. The kernel functions are closely related to the anisotropy as well as any crystalline symmetries. To this end, five new distinct nonlocal kernel functions which have the characteristics of discrete atomistic Green's functions in the stress space are obtained through consideration of the nonlocal dispersion relations associated with certain directions combined with ab initio Density Functional Perturbation Theory (DFPT) calculations of the pertinent phonon frequencies. This is the first work which provides the nonlocal hcp kernel functions pertinent to each component of the elastic moduli tensor of Eringen's nonlocal continuum theory based on quantum mechanical considerations. Moreover, the nonzero components of the classical elastic moduli as well as the equilibrium lattice parameters associated with the hcp crystals of Mg, Cd, Ti, and Zn have been computed using ab initio Density Functional Theory (DFT) and compared with the available experimental data. In order to show the importance of the anisotropy of the kernel functions, the nonlocal stress field of a straight screw dislocation and a straight edge dislocation inside an infinitely extended hcp medium have been examined for the case where the dislocation lies in the basal plane. Within this theory the unrealistic singular behavior of the classical stresses at the dislocation core is eliminated. In contrast to the classical anisotropy theory that predicts singular shear stress on the slip plane at the core of the dislocation, the corresponding nonlocal shear stress component predicted by the present theory is zero at the dislocation core. It is noteworthy to mention that the normalized nonlocal shear stress distribution and its maximum value are greatly influenced by the anisotropy of the crystalline solids.

Analytic solutions for delocalized heat transport

Comprehensive understanding of nonlocal transport is mandatory for many applications of laser–plasma interaction physics, such as inertial confinement fusion and modeling of astrophysical phenomena in the laboratory. Theoretical description is important for guiding numerical simulations and experiments. In this article, an analytic approach is developed: by using a simple integral differential model, we calculate the preheating of plasma upstream of the heat front associated with nonlocal transport. A detailed comparison of various transport configurations as a function of the boundary conditions and the nonlinearity parameter is presented in the context of laser–plasma interaction. Three important results are demonstrated in this paper. First, analytical expressions for all possible self-similar solutions of local nonlinear transport are obtained. Second, a systematic comparison of various nonlocal kernels and forms of the delocalization length is performed. Third, an analytic expression for the temperature profile upstream of the heat front is obtained.

Light propagation in a three-dimensional Rydberg gas with a nonlocal optical response.

We theoretically investigate the linear susceptibility and propagation of light in a three-dimensional (3-D) Rydberg gas under conditions of electromagnetically induced transparency. Rydberg atoms with two relevant S states are coupled via exchange interactions. When the gas is initially prepared in an entangled spin-wave state, this coupling induces a strong, nonlocal susceptibility whereby the photon field at one point of the medium acts as a source at a distant position. The nonlocal propagation occurs not only in the propagation direction but also in the paraxial direction. We discuss the absorption features and numerically simulate the 3-D propagation of probe laser light. Combined with the long-range exchange interaction, we show that the 3-D Rydberg gas is an ideal medium for studying nonlocal wave phenomena, in which the strength, range, and sign of the nonlocal interaction kernel can be widely tuned.

Fractional-calculus analysis of the transmission dynamics of the dengue infection.

In this research paper, a novel approach in dengue modeling with the asymptomatic carrier and reinfection via the fractional derivative is suggested to deeply interrogate the comprehensive transmission phenomena of dengue infection. The proposed system of dengue infection is represented in the Liouville-Caputo fractional framework and investigated for basic properties, that is, uniqueness, positivity, and boundedness of the solution. We used the next-generation technique in order to determine the basic reproduction number R0 for the suggested model of dengue infection; moreover, we conduct a sensitivity test of R0 through a partial rank correlation coefficient technique to know the contribution of input factors on the output of R0. We have shown that the infection-free equilibrium of dengue dynamics is globally asymptomatically stable for R0<1 and unstable in other circumstances. The system of dengue infection is then structured in the Atangana-Baleanu framework to represent the dynamics of dengue with the non-singular and non-local kernel. The existence and uniqueness of the solution of the Atangana-Baleanu fractional system are interrogated through fixed-point theory. Finally, we present a novel numerical technique for the solution of our fractional-order system in the Atangana-Baleanu framework. We obtain numerical results for different values of fractional-order ϑ and input factors to highlight the consequences of fractional-order ϑ and input parameters on the system. On the basis of our analysis, we predict the most critical parameters in the system for the elimination of dengue infection.

Mathematical modeling and analysis of the novel Coronavirus using Atangana–Baleanu derivative

The novel Coronavirus infection disease is becoming more complex for the humans society by giving death and infected cases throughout the world. Due to this infection, many countries of the world suffers from great economic loss. The researchers around the world are very active to make a plan and policy for its early eradication. The government officials have taken full action for the eradication of this virus using different possible control strategies. It is the first priority of the researchers to develop safe vaccine against this deadly disease to minimize the infection. Different approaches have been made in this regards for its elimination. In this study, we formulate a mathematical epidemic model to analyze the dynamical behavior and transmission patterns of this new pandemic. We consider the environmental viral concentration in the model to better study the disease incidence in a community. Initially, the model is constructed with the derivative of integer-order. The classical epidemic model is then reconstructed with the fractional order operator in the form of Atangana–Baleanu derivative with the nonsingular and nonlocal kernel in order to analyze the dynamics of Coronavirus infection in a better way. A well-known estimation approach is used to estimate model parameters from the COVID-19 cases reported in Saudi Arabia from March 1 till August 20, 2020. After the procedure of parameters estimation, we explore some basic mathematical analysis of the fractional model. The stability results are provided for the disease free case using fractional stability concepts. Further, the uniqueness and existence results will be shown using the Picard–Lendelof approach. Moreover, an efficient numerical scheme has been proposed to obtain the solution of the model numerically. Finally, using the real fitted parameters, we depict many simulation results in order to demonstrate the importance of various model parameters and the memory index on disease dynamics and possible eradication.

Solvability for a New Class of Moore-Gibson-Thompson Equation with Viscoelastic Memory, Source Terms, and Integral Condition

This paper deals with the existence and uniqueness of solutions of a new class of Moore-Gibson-Thompson equation with respect to the nonlocal mixed boundary value problem, source term, and nonnegative memory kernel. Galerkin’s method was the main used tool for proving our result. This work is a generalization of recent homogenous work.

Fractional diffusion maps

In this paper, we extend the diffusion maps algorithm on a family of heat kernels that are either local (having exponential decay) or nonlocal (having polynomial decay), arising in various applications. For example, these kernels have been used as a regularizer in various supervised learning tasks for denoising images. Importantly, these heat kernels give rise to operators that include (but are not restricted to) the generators of the classical Laplacian associated to Brownian processes as well as the fractional Laplacian associated with β -stable Lévy processes. For local kernels, while the method is a version of the diffusion maps algorithm, we show that the applications with non-Gaussian local heat kernels approximate temporally rescaled Laplace-Beltrami operators. For the non-local heat kernels, we modify the diffusion maps algorithm to estimate fractional Laplacian operators. Here, the graph distance is used to approximate the geodesic distance with appropriate error bounds. While this approximation becomes numerically expensive as the number of data points increases, it produces an accurate operator estimation that is robust to the choice of the kernel bandwidth parameter value. In contrast, the local kernels are numerically more efficient but more sensitive to the choice of kernel bandwidth parameter value. In an application to estimate non-smooth regression functions, we find that using the nonlocal kernel as a regularizer produces a more robust and accurate estimate than using local kernels. For manifolds with boundary, we find that the proposed fractional diffusion maps framework implemented with non-local kernels approximates the regional fractional Laplacian.

ACHIEVING MORE PRECISE BOUNDS BASED ON DOUBLE AND TRIPLE INTEGRAL AS PROPOSED BY GENERALIZED PROPORTIONAL FRACTIONAL OPERATORS IN THE HILFER SENSE

A user-friendly approach depending on nonlocal kernel has been constituted in this study to model nonlocal behaviors of fractional differential and difference equations, which is known as a generalized proportional fractional operator in the Hilfer sense. It is deemed, for differentiable functions, by a fractional integral operator applied to the derivative of a function having an exponential function in the kernel. This operator generalizes a novel version of Čebyšev-type inequality in two and three variables sense and furthers the result of existing literature as a particular case of the Čebyšev inequality is discussed. Some novel special cases are also apprehended and compared with existing results. The outcome obtained by this study is very broad in nature and fits in terms of yielding an enormous number of relating results simply by practicing the proportionality indices included therein. Furthermore, the outcome of our study demonstrates that the proposed plans are of significant importance and computationally appealing to deal with comparable sorts of differential equations. Taken together, the results can serve as efficient and robust means for the purpose of investigating specific classes of integrodifferential equations.