Non-local Kernel Research Articles

A comparison of heat and mass transfer on a Walter’s-B fluid via Caputo-Fabrizio versus Atangana-Baleanu fractional derivatives using the Fox-H function

In this research, a comparative study of modern differentiations based on singular versus non-singular and local versus non-local kernels have been analyzed for Walter’s-B liquid. In order to expose the efficiency of the two types of modern differentiations namely Caputo-Fabrizio and Atangana-Baleanu fractional differentiations, the partial differential equations governing Walter’s-B liquid are modeled through modern differentiations to study the free convection flow of Walter’s-B liquid. The critical focus is set on the combined heat and mass transfer. The fractional governing equations are solved by invoking the Laplace transform and general solutions are investigated for velocity, temperature and concentration analytically. The analytic solutions are transferred in terms of the Fox- H function for eliminating the gamma functions among the expressions of velocity, temperature and concentration. This comparative analysis indicates that the analytic results obtained via the Caputo-Fabrizio fractional differentiation have reciprocal trends in comparison with the Atangana-Baleanu fractional differentiation. Finally, graphical observations are also depicted for the check of influences of different pertinent parameters on the motion of Walter’s-B liquid.

NLPAR: Non-local smoothing for enhanced EBSD pattern indexing

Due to continued advances in phosphor sensitivity and camera technology, electron backscattered diffraction (EBSD) within a scanning electron microscope (SEM) has become an increasingly popular method for determining crystal orientations within a given microstructure. Concurrent advances in computational processing have also made it possible to store each individual diffraction pattern as it is collected, which has allowed more complex algorithms to be deployed for post-processing and indexing patterns. This paper proposes a new post-processing technique for pattern enhancement that aids in re-indexing by leveraging a non-local smoothing kernel whose weights are based on the exponential decay of the Euclidean distance between patterns. The advantage of this approach is its ability to utilize very large smoothing kernels without losing integrity near interface boundaries and while still operating on timescales comparable to traditional indexing approaches. Using an Inconel 600 nickel alloy sample, the capabilities and performance of the proposed approach are compared to other indexing schemes, including neighbor pattern averaging with re-indexing (NPAR) and a dictionary-based approach. The results demonstrate that the proposed method consistently produces a higher index success rate (ISR) than NPAR and comparable ISRs to the dictionary-based approach, a method with orders-of-magnitude greater computational demands. Source code for the NLPAR algorithm is available at https://github.com/USNavalResearchLaboratory/NLPAR

Strange attractor existence for non-local operators applied to four-dimensional chaotic systems with two equilibrium points.

Not every chaotic system has the particularity of displaying attractors with a fractal structure. That is why strange attractors remain enthralling not only for their fractal structure, but also for their amazing chaotic and multi-scroll dynamics. In this work, we apply the non-local and non-singular kernel operator to a four-dimensional chaotic system with two equilibrium points and show the existence of various types of attractors, including the butterfly type and strange type. Recently, there have been virulent communications related to the validity or not of the index law in fractional differentiation with non-local operators. These discussions resulted in pointing out many important features of the Mittag-Leffler function used as kernel and suitable to describe more complex real world problems. This paper follows the same momentum by pointing out another important feature of the non-local and non-singular kernel operator applied to chaotic models. We solve the model numerically and discuss the bifurcation and period doubling dynamics that eventually lead to chaos (in the form of butterfly attractor). Lastly, we provide related numerical simulations which prove the existence of a chaotic fractal structure (strange attractors).

Numerical solutions of the fractional Fisher's type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods.

The main objective of this paper is to investigate an accurate numerical method for solving a biological fractional model via Atangana-Baleanu fractional derivative. We focused our attention on linear and nonlinear Fisher's equations. We use the spectral collocation method based on the Chebyshev approximations. This method reduced the nonlinear equations to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them by using the finite difference method. This is the first time that this method is used to solve nonlinear equations in Atangana-Baleanu sense. We present the effectiveness and accuracy of the proposed method by computing the absolute error and the residual error functions. The results show that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels.

Accelerating propagation in a recursive system arising from seasonal population models with nonlocal dispersal

The system of interest is defined by iterations of the Poincaré map of a time periodic population model with symmetric nonlocal dispersal and periodic time delay. Under the KPP type setting, we show that the system admits the spreading speed c⁎∈(0,+∞] that coincides with the minimal speed of traveling waves, where c⁎<+∞ if the nonlocal dispersal kernel J is exponentially bounded and c⁎=+∞ if J is exponentially unbounded. Further, the latter case gives rises to an accelerating propagation that the level set of solutions with compactly supported initial values locate in between two exponential functions in time.

A study of pattern forming systems with a fully nonlocal interaction kernel of possibly changing sign

A nonlocal model with a possibly sign changing kernel is proposed to study pattern formation problems in physical and biological systems with self-organization properties. One defines pre-solutions of a related linear problem and gives sufficient conditions for pre-solutions to be jump discontinuous solutions of the nonlocal model. Among a multitude of solutions a selection criterion is used to single out more significant critical solutions. These solutions are further classified into minimizing solutions, maximizing solutions and saddle solutions. Minimizing solutions are the most relevant for patterned states and they exist if the kernel changes sign. A non-negative kernel on the other hand behaves in some ways like the gradient term in the standard local model.

Pattern formation in the doubly-nonlocal Fisher-KPP equation

We study the existence, bifurcations, and stability of stationary solutions for the doubly-nonlocal Fisher-KPP equation. We prove using Lyapunov-Schmidt reduction that under suitable conditions on the parameters, a bifurcation from the non-trivial homogeneous state can occur. The kernel of the linearized operator at the bifurcation is two-dimensional and periodic stationary patterns are generated. Then we prove that these patterns are, again under suitable conditions, locally asymptotically stable. We also compare our results to previous work on the nonlocal Fisher-KPP equation containing a local diffusion term and a nonlocal reaction term. If the diffusion is approximated by a nonlocal kernel, we show that our results are consistent and reduce to the local ones in the local singular diffusion limit. Furthermore, we prove that there are parameter regimes, where no bifurcations can occur for the doubly-nonlocal Fisher-KPP equation. The results demonstrate that intricate different parameter regimes are possible. In summary, our results provide a very detailed classification of the multi-parameter dependence of the stationary solutions for the doubly-nonlocal Fisher-KPP equation.

New idea of Atangana and Baleanu fractional derivatives to human blood flow in nanofluids.

Applications of fractional derivatives are rare for blood flow problems, more exactly in nanofluids. The old definitions published in the literature for fractional derivatives, such as Riemann-Liouville definition, are rarely used by the researchers now; instead, they like to use the new definition introduced by Atangana and Baleanu quite recently. Therefore, in this article, a new idea of Atangana and Baleanu for fractional derivatives possessing a non-local and non-singular kernel has been applied to blood of nanofluids. Blood is considered as a base fluid, and single-wall carbon nanotubes are suspended in blood as nanoparticles in order to make a nanofluid. The magnetic effect with Lorentz force is also taken. The modelled problem is first written in the dimensionless form and later on solved by using an integral transform of Laplace. The effects of embedded parameters are shown in various plots on blood flow and temperature. The heart transfer rate is computed numerically in a tabular form. The results showed that Atangana and Baleanu fractional parameter slow down the blood motion, whereas increasing nanoparticles' volume fraction causes a significant increase in the heat transfer rate.

A fractional model for predator-prey with omnivore.

We consider the model of interaction of predator and prey with omnivore using three different waiting time distributions. The first waiting time is induced by the power law distribution which is the generator of Pareto statistics. The second waiting time is induced by exponential decay law with a particular property of Delta Dirac distribution when the fractional order tends to 1, this distribution is link to the Poison distribution. While the last waiting distribution, induced by the Mittag-Leffler distribution, presents a crossover from exponential to power law. For each model, we presented the conditions under which the existence of unique set of exact solutions is reached using the fixed-point Picard's method. Making use of a recent suggested numerical scheme, we solved each model numerically and some numerical simulations were generated for different values of fractional orders. We noticed a new attractor which can be considered as a combination of the Brownian motion and power law distribution in the model with the Atangana-Baleanu fractional derivative. With the aim to capture more attractors, we modified the model and presented also some numerical simulations. Our new model provides more attractors than the existing one even for fractional differential cases. We presented finally the Maximal Lyapunov exponent and the bifurcation diagrams. The comparative study shows that modeling with non-local and non-singular kernel fractional derivative leads to more attractors as this kernel is able to capture more physical problems.

Asymptotically Compatible SPH-Like Particle Discretizations of One Dimensional Linear Advection Models

Motivated by the smoothed particle hydrodynamics (SPH), we present nonlocal models for linear advection with a variable coefficient in one spatial dimension together with their particle based numerical discretizations. We establish that these numerical methods are robust in the sense that they are convergent as the particle spacing and the smoothing length shrink to zero independently of each other. We demonstrate the important role of nonlocal continuum models to ensure the stability of our numerical methods. The nonlocal models constructed here follow two different strategies: the first model relies on choosing an upwind kernel and the second on introducing a nonlocal viscous term. We study discrete numerical schemes for both models that are in essence particle-like quadrature based finite differences, yet the distinction is clearly drawn in the sense that the scheme for the first model is based on the first moment of the nonlocal kernel while the other is conceived on the basis of renormalized SPH.

Locality estimates for Fresnel-wave-propagation and stability of x-ray phase contrast imaging with finite detectors

Coherent wave-propagation in the near-field Fresnel-regime is the underlying contrast-mechanism to (propagation-based) x-ray phase contrast imaging (XPCI), an emerging lensless technique that enables 2D- and 3D-imaging of biological soft tissues and other light-element samples down to nanometer-resolutions. Mathematically, propagation is described by the Fresnel-propagator, a convolution with an arbitrarily non-local kernel. As real-world detectors may only capture a finite field-of-view, this non-locality implies that the recorded diffraction-patterns are necessarily incomplete. This raises the question of stability of image-reconstruction from the truncated data—even if the complex-valued wave-field, and not just its modulus, could be measured. Contrary to the latter restriction of the acquisition, known as the phase-problem, the finite-detector-problem has not received much attention in literature. The present work therefore analyzes locality of Fresnel-propagation in order to establish stability of XPCI with finite detectors. Image-reconstruction is shown to be severely ill-posed in this setting—even without a phase-problem. However, quantitative estimates of the leaked wave-field reveal that Lipschitz-stability holds down to a sharp resolution limit that depends on the detector-size and varies within the field-of-view. The smallest resolvable lengthscale is found to be ≈ times the detector’s aspect length, where is the Fresnel number associated with the latter scale. The stability results are extended to phaseless imaging in the linear contrast-transfer-function regime.

Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana–Baleanu fractional operator

In this article, by popularization of the reproducing kernel Hilbert space method in the sense of the Atangana–Baleanu fractional operator; set of first-order integrodifferential equations are solved with respect to Fredholm operator and initial conditions of optimality. The solvability approach based on use of the generalized Mittag–Leffler function in order to avoid nonsingular and nonlocal kernel functions appears in the classical fractional operator's. The procedure of solution is studied and described in details under some hypotheses, which provides the theoretical structure behind the utilized numerical method. Indeed, error analysis and convergence of numerical solution for the identification of the method is introduced in Hilbert space. In this analysis, some computational results and graphical representations are presented to demonstrated the suitability and portability of the utilized new fractional operator. Finally, the gained results reach to that; the utilized method is simple, direct, and powerful tool in finding numerical solutions for considered fractional equations.

The Nonlocal Kernel in van der Waals Density Functionals as an Additive Correction: An Extensive Analysis with Special Emphasis on the B97M-V and ωB97M-V Approaches.

The development of van der Waals density functional approximations (vdW-DFAs) has gained considerable interest over the past decade. While in a strictest sense, energy calculations with vdW-DFAs should be carried out fully self-consistently, we demonstrate conclusively for a total of 11 methods that such a strategy only increases the computational time effort without having any significant effect on energetic properties, electron densities, or orbital-energy differences. The strategy to apply a nonlocal vdW-DFA kernel as an additive correction to a fully converged conventional DFA result is therefore justified and more efficient. As part of our study, we utilize the extensive GMTKN55 database for general main-group thermochemistry, kinetics, and noncovalent interactions [ Phys. Chem. Chem. Phys. 2017, 19, 32184], which allows us to analyze the very promising B97M-V [ J. Chem. Phys. 2015, 142, 074111] and ωB97M-V [ J. Chem. Phys. 2016, 144, 214110] DFAs. We also present new DFT-D3(BJ) based counterparts of these two methods and of ωB97X-V [ J. Chem. Theory Comput 2013, 9, 263], which are faster variants with similar accuracy. Our study concludes with updated recommendations for the general method user, based on our current overview of 325 dispersion-corrected and -uncorrected DFA variants analyzed for GMTKN55. vdW-DFAs are the best representatives of the three highest rungs of Jacob's Ladder, namely, B97M-V, ωB97M-V, and DSD-PBEP86-NL.

A fractional model of vertical transmission and cure of vector-borne diseases pertaining to the Atangana–Baleanu fractional derivatives

The model of transmission dynamics of vector-borne diseases with vertical transmission and cure within a targeted population is extended to the concept of fractional differentiation and integration with non-local and non-singular fading memory introduced. The effect of vertical transmission and cure rate on the basic reproduction number is shown. The Atangana–Baleanu fractional operator in caputo sense (ABC) with non-singular and non-local kernels is used to study the model. The existence and uniqueness of solutions are investigated using the Picard–Lindel method. Ultimately, for illustrating the acquired results, we perform some numerical simulations and show graphically to observe the impact of the arbitrary order derivative. It is expected that the proposed model will show better approximation than the classical model established before.

Numerical analysis of dissipative system with noise model with the Atangana–Baleanu fractional derivative

The model of dissipative system with noise has not been so far a center of interest of many researchers, while this model is very interesting physical problem that deserves to capture attentions. Not to mention that no study has been done for this model using the newly established concept of fractional differential operators based on non-singular kernels. In this paper, we aimed to provide an analysis mainly numerical analysis of the model with the non-local non-singular kernel differential operators. We made use of a recent powerful numerical scheme and provided some numerical simulations for different values of fractional order.

Physically consistent nonlocal kernels for predicting vibrational characteristics of single walled carbon nanotubes

In the present work, at first, the drawbacks of the continuum mechanics (CM) based method in predicting the vibrational and wave dispersion characteristics of carbon nanotubes (CNTs) are illustrated and discussed. For this purpose, the response of the Euler-Bernoulli and Timoshenko beam theories together with the differential form of the Eringen’s nonlocal elasticity theory under different boundary conditions are examined. The same problems are simulated via molecular dynamics (MD) approach. By considering the results of MD simulation as the benchmark solutions, it is shown that the optimum value of the length scale parameter of the nonlocal elasticity theory of Eringen depends on the nanotube boundary conditions, aspect ratio and vibrational mode number of the CNTs. To overcome these drawbacks, a kernel with unique length scale variable is introduced by combing different Gaussian kernels and specifying two related new scale variables. These newly defined scale parameters resolve former problem in shifting between different scales. Accordingly, an improved nonlocal constitutive relation is suggested. It is shown that the proposed model can accurately predict the vibrational and flexural vibrational dispersion relation (FVDR) characteristics of CNTs even when using Euler-Bernoulli theory (EBT).

Application of Atangana–Baleanu fractional derivative to MHD channel flow of CMC-based-CNT's nanofluid through a porous medium

The objective of this article is to apply the Atangana–Baleanu derivative fractional in Caputo sense to convective flow of Carboxy–Methyl–Cellulose (CMC) based Carbon nanotubes (CNT's) nanofluid in a vertical microchannel. The magnetohydrodynamic (MHD) flow through a porous medium together with heat transfer is considered. The Atangana–Baleanu fractional derivative without singular and the non-local kernel is used in the mathematical formulation to get the time fractional governing equations subject to physical initial and boundary conditions. The Laplace transform technique is used to obtain the exact analytical solutions for velocity and temperature distributions. Finally, the influence of parameters of interest is studied through plots and discussed physically.

Reduction of Thermal Conductive Flux by Non-local Effects in the Presence of Turbulent Scattering.

The heat flux in a plasma is determined by the degree of anisotropy in the particle distribution function, which is in turn driven by gradients in the ambient density and temperature. When the mean free path at the thermal speed is substantially smaller than the scale length associated with the temperature variation, the heat flux simply depends on the local value of the temperature gradient. However, when the temperature scale length and mean free path are comparable, heat conduction becomes substantially non-local in character: the magnitude of the heat flux now depends on the overall temperature profile and is generally smaller than the locally determined value. In the presence of angular scattering associated with turbulence, the mean free path (and its velocity dependence) can be significantly smaller than its collisional value; this makes the expression for the heat flux more local in character, but also results in a heat flux that is lower than that obtained through a purely collisional analysis. Therefore, whether or not turbulence is present, the heat flux is generally smaller than the value obtained from a local collisional analysis. We here present an analytic expression for the conductive heat flux in terms of a convolution of the local heat flux with a non-local kernel function that incorporates both Coulomb collisions and turbulent scattering. We comment on the need to include both non-local and turbulent scattering effects in the modeling of quasi-static active region loops and in the conductive cooling of post-flare loops.

Analysis and numerical simulation of multicomponent system with Atangana–Baleanu fractional derivative

In this paper, we consider the mathematical analysis and numerical simulation of time-fractional multicomponent systems. Here, the classical time derivatives in such systems are replace with the Atangana–Baleanu fractional derivative in the sense of Caputo. This derivative is found useful in the sense that it combines both the non-local and nonsingular kernels in its formulation. A two-step family of Adams–Bashforth method is derived for the approximation of the Atangana–Baleanu derivative. Numerical experiments presented for different instances of α, 0 < α ≤ 1 correspond to our theoretical findings.

A novel method for a fractional derivative with non-local and non-singular kernel

Atangana and Baleanu gave a derivative with fractional order to search the valuable inquiries. We present a novel technique for investigating fractional differential equations including the Atangana-Baleanu fractional derivative in this paper. We give some examples for this method for equations of fractional initial value problems.