Power-law Kernel Research Articles

Nonequilibrium fractional correlation functions and fluctuation–dissipation in linear viscoelasticity

A simple model for time retarded fractional density and velocity hydrodynamic fluctuations correlations of a linear viscoelastic fluid under an external gradient is proposed. A fractional generalized Langevin equation with a Caputo’s fractional time derivative is formulated for a power-law memory kernel with long correlation noise. The fractional density and velocity fluctuations correlation functions, as well as the fractional light scattering spectra, and the fractional intermediate scattering function (FISF) are calculated analytically for power-law viscoelasticity. Their relation to the longitudinal modulus of the viscoelastic fluid is also obtained. We find that for different frequency intervals appropriate to perform isothermal light scattering experiments using Fabry–Perot interferometry and photon correlation, the non-equilibrium fractional structure factor may be smaller or larger than its fractional equilibrium values. We also discuss the validity of a fractional fluctuation–dissipation theorem for this model in terms of the asymptotic behavior of the equilibrium fractional correlation functions of the model.

Integral decomposition for the solutions of the generalized Cattaneo equation

We present the integral decomposition for the fundamental solution of the generalized Cattaneo equation with both time derivatives smeared through convoluting them with some memory kernels. For power-law kernels t^{-α},α∈(0,1] this equation becomes the time fractional one governed by the Caputo derivatives in which the highest order is 2. To invert the solutions from the Fourier-Laplace domain to the space-time domain we use analytic methods based on the Efross theorem and find out that solutions looked for are represented by integral decompositions which tangle the fundamental solution of the standard Cattaneo equation with nonnegative and normalizable functions being uniquely dependent on the memory kernels. Furthermore, the use of methodology arising from the theory of complete Bernstein functions allows us to assign such constructed integral decompositions the interpretation of subordination. This fact is preserved in two limit cases built into the generalized Cattaneo equations, i.e., either the diffusion or the wave equations. We point out that applying the Efross theorem enables us to go beyond the standard approach which usually leads to the integral decompositions involving the Gaussian distribution describing the Brownian motion. Our approach clarifies puzzling situation which takes place for the power-law kernels t^{-α} for which the subordination based on the Brownian motion does not work if α∈(1/2,1].

A Nonlocal Fractional Peridynamic Diffusion Model

This paper proposes a nonlocal fractional peridynamic (FPD) model to characterize the nonlocality of physical processes or systems, based on analysis with the fractional derivative model (FDM) and the peridynamic (PD) model. The main idea is to use the fractional Euler–Lagrange formula to establish a peridynamic anomalous diffusion model, in which the classical exponential kernel function is replaced by using a power-law kernel function. Fractional Taylor series expansion was used to construct a fractional peridynamic differential operator method to complete the above model. To explore the properties of the FPD model, the FDM, the PD model and the FPD model are dissected via numerical analysis on a diffusion process in complex media. The FPD model provides a generalized model connecting a local model and a nonlocal model for physical systems. The fractional peridynamic differential operator (FPDDO) method provides a simple and efficient numerical method for solving fractional derivative equations.

Hawkes-based models for high frequency financial data

Compared with low frequency data, high frequency data exhibit distinct empirical properties, including, for instance, essentially discontinuous evolution paths, time-varying intensities, and self-exciting features. All these make it more challenging to model appropriately the dynamics associated with high frequency data such as order arrival and price formation. To capture more accurately the microscopic structures and properties pertaining to the limit order books, this paper focuses on modeling high frequency data using Hawkes processes. Two models, one with exponential kernels and the other with power-law kernels, are introduced systematically, algorithmized precisely, and compared with each other extensively from various perspectives, including the goodness of fit to the original data and the computational time in searching for the maximum likelihood estimator, with search algorithm being taken into consideration as well. To measure the goodness of fit, a number of quantities are proposed. Studies based on both multiple-trading-day data of one stock and multiple-stock data on one trading day indicate that Hawkes processes with slowly-decaying kernels are able to reproduce the intensity of jumps in the price processes more accurately. The results suggest that Hawkes processes with power-law kernels and their implied long memory nature of self-excitation phenomena could, on the level of microstructure, serve as a realistic model for high frequency data.

Unsteady flow of fractional Burgers’ fluid in a rotating annulus region with power law kernel

Keeping in view of the complex fluid mechanics in bio-medicine and engineering, the Burgers’ fluid with a fractional derivatives model analyzed through a rotating annulus. The governing partial differential equation solved for velocity field and shear stress by using integral transformation method and using Bessel equations. The transformed equation inverted numerically by using Gaver-Stehfest’s algorithm. The approximate analytical solution for rotational velocity, and shear stress are presented. The influence of various parameters like fractional parameters, relaxation and retardation time parameters material constants, time and viscosity parameters are drawn numerically. It is found that the relaxation time and time helps the flow pattern, on the other hand other material constants resist the fluid rotation. Fractional parameters effect on fluid flow is opposite to each other. Finally, to check the validity of the solution there are comparisons for velocity field and shear stress for obtained results with an other numerical algorithm named Tzou’s algorithm.

First Solution of Fractional Bioconvection with Power Law Kernel for a Vertical Surface

The present study provides the heat transfer analysis of a viscous fluid in the presence of bioconvection with a Caputo fractional derivative. The unsteady governing equations are solved by Laplace after using a dimensional analysis approach subject to the given constraints on the boundary. The impact of physical parameters can be seen through a graphical illustration. It is observed that the maximum decline in bioconvection and velocity can be attained for smaller values of the fractional parameter. The fractional approach can be very helpful in controlling the boundary layers of the fluid properties for different values of time. Additionally, it is observed that the model obtained with generalized constitutive laws predicts better memory than the model obtained with artificial replacement. Further, these results are compared with the existing literature to verify the validity of the present results.

Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

Dynamical study on three-species population eco-epidemiological model with fractional order derivatives

The essential target of this work is to analyse the dynamical behaviour of the arbitrary order eco - epidemiological model by adopting the new numerical method. We applied the power law kernel, exponential decay kernel, and generalized Mittag–Leffler kernel functions for treatment of arbitrary order eco-epidemiological model where the considered eco-epidemiological model is a non-linear dynamical system with three population species. Further, we calculated the uniqueness and existence of the solutions by adopting the fixed point hypothesize. Further, we studied the stability analysis for aforesaid model. We examine the possibility for finding new dynamical phase portraits with singular and non-singular arbitrary order operator and demonstrate the dynamical phase portraits at various values of arbitrary order. Furthermore, we have found out the maximal bifurcation graph of the eco-epidemiological system and this eco-epidemiological model is numerically solved by adopting Atangana-Seda (AS) numerical method. We have used Newton’s polynomial.

Stochastic fractional model of Alzheimer disease

Differential operators based on convolution definitions have been recognized as powerful mathematics tools to help model real world problems due to the properties associated to their different kernels. In particular the power law kernel helps include into mathematical formulation the effect of long range, while the exponential decay helps with fading memory, also with Poisson distribution properties that lead to a transitive behavior from Gaussian to non-Gaussian phases respectively. In this paper, we presented the well-poseness of the models for different differential operators that were presented in detail.

A data-driven deep learning approach for options market making

We develop a data-driven approach for options market making. Using stock options data from CBOE, we find that both buy and sell orders exhibit strong self-excitation but insignificant cross-excitation. We show that a Hawkes process with a time-varying baseline intensity and the power law kernel provides a good fit to the data of market order flows for stock options. To solve the optimal market making problem for a single option, we approximate the market making strategy at each decision time by a neural network and train them to optimize the expected utility of the market maker. We study feature selection for the neural networks and compare the out-of-sample performance of the optimal neural network strategy trained from data generated by the Hawkes process and the Poisson process. We find that using the more realistic Hawkes model improves the out-of-sample performance significantly. Furthermore, utilizing the Hawkes process intensity or the expected number of market order arrivals computed under the Hawkes model as an additional input feature can boost the performance. We also show how to solve the market making problem for option portfolios with Greeks and inventory constraints using neural network approximation.

A thermal optimization throughan innovative mechanism of free convection flow of Jeffrey fluid using non-local kernel

Optimality for operating temperature can improve the reliability of thermodynamics of convection flow of Jeffrey fluid through mathematical techniques; this is because thermal convection is invoked in several engineering problems for knowingan accurate assessment and stable or unstable heat transference characteristics. In this context, the free convection flow of Jeffrey fluid among two vertical plates is characterized by stable or unstable heat transfer. The heat transference mechanism is designed by employing a generalized and a fractional form of Fourier's law that delivers damping for thermal flux. In this process, we make use of the Caputo time-fractional derivative (CTFD) having a power-law singular kernel. The free convection is induced by the temperature gradient. The analytical solutions have been obtained through a method of Laplace coupled with finite sine-Fourier transform and have been embedded with regards to the Mittag-Leffler function. The behavior of velocity and temperature profiles is analyzed through numerical computations and graphical representations for different embedded parameters with Mathcad. Certainly, this article presents a comprehensive discussion as well as a graphical interpretation of the achieved results.

How Does Tempering Affect the Local and Global Properties of Fractional Brownian Motion?

The present paper investigates the effects of tempering the power law kernel of the moving average representation of a fractional Brownian motion (fBm) on some local and global properties of this Gaussian stochastic process. Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) are the processes that are considered in order to investigate the role of tempering. Tempering does not change the local properties of fBm including the sample paths and p-variation, but it has a strong impact on the Breuer–Major theorem, asymptotic behavior of the third and fourth cumulants of fBm and the optimal fourth moment theorem.

Computational dynamics of predator-prey model with the power-law kernel

Evolution system which contains fractional derivatives can give rise to useful mathematical model for describing some important real-life or physical scenarios. Here, we suggest some numerical techniques for solving fractional-in-time reaction-diffusion models in the sense of Caputo operator. The suggested schemes are formulated with difference scheme and Fourier-spectral algorithm. In the simulation framework, it was observed that the spectral method retains the advantage of spectral accuracy over its finite difference counterpart. Both techniques are easy to adapt and extend to high-dimensions in space and time. The existence of solution, uniqueness of solution, linear stability analysis as well as the calculation of the Lyapunov exponent of the main system, are well established. Suitability of the suggested numerical techniques are tested on non-diffusive model, one-dimensional diffusive example and two-dimensional diffusive experiments.

Fractional B-spline collection method for solving fractal-differential equations

This study used the fractional B-spline collocation technique to obtain the numerical solution of fractal-fractional differential equations. The technique was considered to solve the fractal-fractional differential equations (FFDEs) with ($0 <gamma_i < 1,, i = 1, 2, cdots, N$). In this suggested technique, the B-spline of fractional order was utilised in the collocation technique. The scheme was easily attained, efficient, and relatively precise with reduced computational work numerical findings. Via the proposed technique, FFDEs can be reduced for solving a system of linear algebraic equations using an appropriate numerical approach. The verified numerical illustrative experiments were presented will show the effectiveness of the technique proposed in this study in solving FFDEs in three cases of nonlocal integral and differential operators namely power law kernel, when the kernels are exponential and the generalization of Mittag-Leffler kernel. The approximate solution is very good and accurate to the exact solution.

A Fractal-Fractional Model for the MHD Flow of Casson Fluid in a Channel

An emerging definition of the fractal-fractional operator has been used in this study for the modeling of Casson fluid flow. The magnetohydrodynamics flow of Casson fluid has cogent in a channel where the motion of the upper plate generates the flow while the lower plate is at a static position. The proposed model is non-dimensionalized using the Pi-Buckingham theorem to reduce the complexity in solving the model and computation time. The non-dimensional fractal-fractional model with the power-law kernel has been solved through the Laplace transform technique. The Mathcad software has been used for illustration of the influence of various parameters, i.e., Hartman number, fractal, fractional, and Casson fluid parameters on the velocity of fluid flow. Through graphs and tables, the results have been implemented and it is shown that the boundary conditions are fully satisfied. The results reveal that the flow velocity is decreasing with the increasing values of the Hartman number and is increasing with the increasing values of the Casson fluid parameter. The findings of the fractal-fractional model have elucidated that the memory effect of the flow model has higher quality than the simple fractional and classical models. Furthermore, to show the validity of the obtained closed-form solutions, special cases have been obtained which are in agreement with the already published solutions.

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

The harvesting management is developed to protect the biological resources from over-exploitation such as harvesting and trapping. In this article, we consider a predator–prey interaction that follows the fractional-order Rosenzweig–MacArthur model where the predator is harvested obeying a threshold harvesting policy (THP). The THP is applied to maintain the existence of the population in the prey–predator mechanism. We first consider the Rosenzweig–MacArthur model using the Caputo fractional-order derivative (that is, the operator with the power-law kernel) and perform some dynamical analysis such as the existence and uniqueness, non-negativity, boundedness, local stability, global stability, and the existence of Hopf bifurcation. We then reconsider the same model involving the Atangana–Baleanu fractional derivative with the Mittag–Leffler kernel in the Caputo sense (ABC). The existence and uniqueness of the solution of the model with ABC operator are established. We also explore the dynamics of the model with both fractional derivative operators numerically and confirm the theoretical findings. In particular, it is shown that models with both Caputo operator and ABC operator undergo a Hopf bifurcation that can be controlled by the conversion rate of consumed prey into the predator birth rate or by the order of fractional derivative. However, the bifurcation point of the model with the Caputo operator is different from that of the model with the ABC operator.

Analysis and new applications of fractal fractional differential equations with power law kernel

&lt;p style='text-indent:20px;'&gt;We obtain the solutions of fractal fractional differential equations with the power law kernel by reproducing kernel Hilbert space method in this paper. We also apply the Laplace transform to get the exact solutions of the problems. We compare the exact solutions with the approximate solutions. We demonstrate our results by some tables and figures. We prove the efficiency of the proposed technique for fractal fractional differential equations.&lt;/p&gt;

ATANGANA–SEDA NUMERICAL SCHEME FOR LABYRINTH ATTRACTOR WITH NEW DIFFERENTIAL AND INTEGRAL OPERATORS

In this paper, we present a new numerical scheme for a model involving new mathematical concepts that are of great importance for interpreting and examining real world problems. Firstly, we handle a Labyrinth chaotic problem with fractional operators which include exponential decay, power-law and Mittag-Leffler kernel. Moreover, this problem is solved via Atangana-Seda numerical scheme which is based on Newton polynomial. The accuracy and efficiency of the method can be easily seen with numerical simulations.

Role of fractal–fractional derivative on ferromagnetic fluid via fractal Laplace transform: A first problem via fractal–fractional differential operator

A deformity of magnetic field discloses that different particle of ferrofluid exhibits different pattern to accumulate more closely with strong magnetic properties; this is because of the existence of a homogeneous ferromagnetic liquid. In this context, the Newtonian fluid (ordinary liquid) is magnetized by dispersing into porous media. A mathematical model for ferromagnetic fluid is developed by means of fractal–fractionalderivative based on the power law. A powerful technique of Laplace transform is invoked on the fractal-fractionalized MHD Newtonian fluid model embedded in porous medium based within power law kernel. The solution is investigated for velocity field by particularizing the format of mathematical special function so called Fox-H function Hα,β+11,αF. The Fox-H function Hα,β+11,αF is employed among solution for the elimination of gamma functions. In order to generate the physical insight of fractal-fractionalized ferrofluid, a novel behaviors of velocity field based on the power law has disclosed several similarities and differences through the embedded rheological parameters. In short, It is worth noting that there is no previous paper related to ferrofluid involved with fractal–fractional​ derivative based on the power law.

Power law memory of natural convection flow of hybrid nanofluids with constant proportional Caputo fractional derivative due to pressure gradient

In this work, influence of hybrid nanofluids on heat transfer flow of a viscous fluid due to pressure gradient is discussed with innovative constant proportional Caputo fractional derivative. For this purpose, we consider an infinite vertical wall which is exponentially moving in the x-direction with variable temperature. Nanosized particles of Cu and $$\hbox {Al}_{2}\hbox {O}_{3}$$ are suspended in water, the base fluid. The governing equations of the problem are converted into dimensionless form. Further, we develop the constant proportional Caputo fractional model with a new operator with power law kernel which can be used to study the fluid behaviour for different values of fractional parameter at the present time. We applied the Laplace transform method to obtain the solutions and to see the impact of hybrid nanofluids and fractional parameter $$\alpha $$ respectively. We compared the present results with the recently published work (Nehad et al, Adv. Mech. Eng. 11(7): 1 (2019)) with Caputo fractional derivative. As a result, we have found that the present solutions are best to describe the memory concept of temperature and velocity. For small values of fractional parameter, temperature and velocity have maximum values and for larger values of fractional parameter, temperature and velocity have minimum values. Further, rate of heat transfer and skin friction are also computed in tabular forms and it is found that Nusselt number with CPC is much less than that is computed with Caputo fractional derivative for greater values of fractional parameter $$\alpha $$ .