Fractional Stochastic Model Research Articles

Novel Numerical Solutions to the Fractional-Stochastic Systems

Fractional-stochastic differential equations are widely used tools to simulate a wide range of engineering and scientific phenomena. In this paper, the applicability of the approach of indeterminate coefficients to various fractional-stochastic models is examined. These models have a fractional white noise term and are mostly produced by fractional-order derivative operators. We also look into the application of a polynomial chaos algorithm to stochastic Lotka-Volterra and Benney systems. Fractional-stochastic equations are entirely novel systems that have the potential to function as models for a wide range of scientific and engineering phenomena. It is noted that fractional-order systems with uncertainty or a noise term can benefit from the effective use of Galerkin type approaches used in this article. Keywords: Galerkin method, Numerical solutions, Stochastic-fractional systems

Leader-Following Formation Control for Discrete-Time Fractional Stochastic Multi-Agent Systems by Event-Triggered Strategy

Fractional differential equations, which are non-local and can better describe memory and genetic properties, are widely used to describe various physical, chemical, and biological phenomena. Therefore, the multi-agent systems based on discrete-time fractional stochastic models are established. First, some followers are selected for pinning control. In order to save resources and energy, an event-triggered based control mechanism is proposed. Second, under this control mechanism, sufficient conditions on the interaction graph and the fractional derivative order such that formation control can be achieved are given. Additionally, influenced by noise, the multi-agent system completes formation control in the mean square. In addition to that, these results are equally applicable to the discrete-time fractional formation problem without noise. Finally, the example of numerical simulation is given to prove the correctness of the results.

Some Fractional Stochastic Models for Neuronal Activity with Different Time-Scales and Correlated Inputs

In order to describe neuronal dynamics on different time-scales, we propose a stochastic model based on two coupled fractional stochastic differential equations, with different fractional orders. For the specified choice of involved functions and parameters, we provide three specific models, with/without leakage, with fractional/non-fractional correlated inputs. We give explicit expressions of the process representing the voltage variation in the neuronal membrane. Expectation values and covariances are given and compared. Numerical evaluations of the average behaviors of involved processes are presented and discussed.

Numerical treatment for a novel crossover mathematical model of the COVID-19 epidemic

<abstract><p>This paper extends a novel piecewise mathematical model of the COVID-19 epidemic using fractional and variable-order differential equations and fractional stochastic derivatives in three intervals of time. The deterministic models are augmented with hybrid fractional order and variable order operators, while the stochastic differential equations incorporate fractional Brownian motion. To probe the behavior of the proposed models, we introduce two numerical techniques: the nonstandard modified Euler Maruyama method for the fractional stochastic model, and the Caputo proportional constant-Grünwald-Letnikov nonstandard finite difference method for the fractional and variable-order deterministic models. Several numerical experiments corroborate the theoretical assertions and demonstrate the efficacy of the proposed approaches.</p></abstract>

A novel S-box design based on quantum tent maps and fractional stochastic models with an application in image encryption

A novel S-box design based on quantum tent maps and fractional stochastic models with an application in image encryption

Numerical simulation and analysis of the stochastic HIV/AIDS model in fractional order

The current work analyzes the HIV/AIDS model under stochastic fractional differential equations in the Caputo sense. The articulation of the model both in integer and arbitrary order with stochastic differential equation are presented. We provide the solution’s positivity and boundedness for the proposed system. The existence and uniqueness of the fractional stochastic order differential equations are given. We give a novel numerical approach based on Newton’s polynomial to numerically solve the fractional stochastic model. Also, we provide a numerical technique for the solution in the power law model. The results for the Caputo fractional case and stochastic fractional Caputo case are shown and discussed in detail. Sensitivity analysis are done and provide their respective results graphically. The numerical solution of the model based on their sensitive parameters is shown graphically in order to minimize the infection.

Fractional stochastic modelling of monkeypox dynamics

Monkeypox virus is primarily transferred to humans via wild animals including rodents and more often the transmission ensues between humans to humans. This disease has been neglected in the past and little effort has been made by research to study the dynamics of the disease. This paper aims to use a fractional stochastic modeling approach to investigate the Monkeypox model dynamics. In this work, a fractionalized mathematical model is constructed with an emphasis on the unstigmatized and stigmatized in the community. First, we carried out a deterministic model using fractional-order operator with Atangana–Baleanu–Caputo derivative. The basic properties and the dynamical behavior of the deterministic model are investigated. Next, the existence and uniqueness of the solutions of the model based on the fractional stochastic approach are examined. Numerical simulations of deterministic and stochastic fractional model are conducted to support the analytical results which depict that fractional order derivative has a serious effect on the dynamics of the monkeypox. This work presented more detailed the characteristic of the monkeypox dynamics than deterministic fractional operators. The infection rate in humans has an impact on the spread of the disease in the human compartment.

Analytical and Computational Analysis of Fractional Stochastic Models Using Iterated Itô Integrals

Biological and financial models are examples of dynamical systems where both stochastic and historical behavior are important to be considered. The fractional Brownian motion (fBM) is commonly used, sometimes with fractional-order derivatives, to model the combined stochastic and fractional effects. Recently, spectral techniques are used to analyze models with fBM using, e.g., iterated Itô fractional integrals such as the fractional Wiener-Hermite (FWHE). In the current work, FWHE is generalized and adapted to be consistent with the Malliavin calculus approach. The conditions for existence and uniqueness are outlined in addition to the proof of convergence. The solution algorithm is described in detail. Using FWHE, the stochastic fractional model is replaced by a deterministic fractional-order system that can be handled using well-known mathematical tools to evaluate the solution statistics. Analytical solutions can be obtained for many important models such as the fractional stochastic Black–Scholes model. The convergence is studied and compared with the exact solution and high convergence is noticed compared with other techniques. A general numerical algorithm is described to analyze the resultant deterministic system in the case of no feasible analytical solutions. The algorithm is applied to study and simulate the population model with nonlinear losses for different values of the Hurst parameter. The results show the efficiency of FWHE in analyzing practical linear and nonlinear models.

On mathematical modeling of fractional-order stochastic for tuberculosis transmission dynamics

Tuberculosis remains one of the most dangerous diseases globally and has affected many people in Sub-Saharan Africa. In this paper, a fractional stochastic model of tuberculosis disease was formulated and analyzed. The existence and uniqueness of solutions are presented in the new approach far from the deterministic fractional operators. We carry out numerical simulations of the model using three fractional operators. Our numerical results suggest that the Caputo–Fabrizio operator has a more random effect among the three different operators than the Atangana–Baleanu and the Caputo operators. Further, it is envisaged that the fractional-order derivatives significantly impact the dynamics of the disease.

Robust H∞ sliding mode control scheme for uncertain fractional stochastic systems: Nonlinear analysis and design

AbstractThis paper investigates the sliding mode control (SMC) design for fractional stochastic systems. We study a very general category of stochastic systems that are nonlinear and driven by fractional Brownian motion (fBm). A robust SMC scheme is presented for a fractional stochastic model with external disturbance, state‐ and disturbance‐dependent noise, and uncertainties, which ensures that the closed‐loop system is stochastically stable. We propose a novel sliding surface and then prove its reachability in the state space. Furthermore, the conditions for the stochastic stability of the sliding motion are derived via nonlinear Hamilton–Jacobi (HJ)‐type inequalities. In addition, an SMC method is developed for a special class of fractional stochastic models, and two sets of linear matrix inequality (LMI) conditions are obtained, which are sufficient for stochastic stability. Eventually, the validity of the results is validated via a simulation example.

New Coronavirus (2019-nCov) Mathematical Model Using Piecewise Hybrid Fractional Order Derivatives; Numerical Treatments

A new mathematical model of Coronavirus (2019-nCov) using piecewise hybrid fractional order derivatives is given in this paper. Moreover, in order to be consistent with the physical model problem, a new parameter μ is presented. The boundedness, existence, and positivity of the solutions for the proposed model are discussed. Two improved numerical methods are presented in this paper. The Caputo proportional constant nonstandard modified Euler–Maruyama method is introduced to study the fractional stochastic model, and the Grünwald–Letnikov nonstandard finite difference method is presented to study the hybrid fractional order deterministic model. Comparative studies with real data from Spain and Wuhan are presented.

Simulation of near-field pulse-like ground motions using a correlated bimodal fractional stochastic model

This paper introduces a two-part stochastic method to simulate completely nonstationary pulse-like records and capture several peaks in the power spectrum. Based on the evolutionary power spectral density compatibility of the acceleration ground motion in different stages, the model is fitted to the target records. The proposed model is flexible enough to consider the decay rate variability of the high-and low-frequency content of the ground motions. Also, using the circular shift and IFFT techniques, the discretization of the model is performed such that the simulation cost is significantly reduced. The horizontal components of 178 near-filed pulse-like ground motions with a rupture distance of less than 10 km, with strike-slip and dip-slip mechanisms, are simulated using the proposed method. The model's efficiency is investigated using different criteria such as the elastic response spectra, Arias intensity, zero-crossing, Fourier spectrum, and the 2DOF system response. Moreover, the simulated records' pulse-like characteristics are investigated, and the model's capability to generate an ensemble of desirable ground motions is studied. Comparing the results with the previous models reveals that the model can generate more compatible records with the seed ones according to various aspects.

Development of a fractional Wiener-Hermite expansion for analyzing the fractional stochastic models

The fractional Brownian motion (FBM) is a common model for long and short-range dependent phenomena that appears in different fields, including physics, biology, and finance. In the current work, a new spectral technique named the fractional Wiener Hermite Expansion (FWHE) is developed to analyze stochastic models with FBM. The technique has a theoretical background in the literature and proof of convergence. A new complete orthogonal Hermite basis set is developed. Calculus derivations and statistical analysis are performed to handle the mixed multi-dimensional fractional and/or integer-order integrals that appear in the analysis. Formulas for the mean and variance are deduced and are found to be based on fractional integrals. Using the developed expansion with the statistical properties of the basis functionals will help to reduce the stochastic model to equivalent deterministic fractional models that can be analyzed numerically or analytically with the well-known techniques. A numerical algorithm is developed to be used in case there is no available analytical solution. The numerical algorithm is compared with the fractional Euler-Maruyama (EM) technique to verify the results. In comparison to sampling based techniques, FWHE provides an efficient analytical or numerical alternative. The applicability of FWHE is demonstrated by solving different examples with additive and multiplicative FBM.

A novel approach for solving decision-making problems with stochastic linear-fractional models

Stochastic chance-constrained optimization has a wide range of real-world applications. In some real-world applications, the decision-maker has to formulate the problem as a fractional model where some or all of the coefficients are random variables with joint probability distribution. Therefore, these types of problems can deal with bi-objective problems and reflect system efficiency. In this paper, we present a novel approach to formulate and solve stochastic chance-constrained linear fractional programming models. This approach is an extension of the deterministic fractional model. The proposed approach, for solving these types of stochastic decision-making problems with the fractional objective function, is constructed using the following two-step procedure. In the first stage, we transform the stochastic linear fractional model into two stochastic linear models using the goal programming approach, where the first goal represents the numerator and the second goal represents the denominator for the stochastic fractional model. The resulting stochastic goal programming problem is formulated. The second stage implies solving stochastic goal programming problem, by replacing the stochastic parameters of the model with their expectations. The resulting deterministic goal programming problem is built and solved using Win QSB solver. Then, using the optimal value for the first and second goals, the optimal solution for the fractional model is obtained. An example is presented to illustrate our approach, where we assume the stochastic parameters have a uniform distribution. Hence, the proposed approach for solving the stochastic linear fractional model is efficient and easy to implement. The advantage of the proposed approach is the ability to use it for formulating and solving any decision-making problems with the stochastic linear fractional model based on transforming the stochastic linear model to a deterministic linear model, by replacing the stochastic parameters with their corresponding expectations and transforming the deterministic linear fractional model to a deterministic linear model using the goal programming approach

Discrete fractional stochastic Grönwall inequalities arising in the numerical analysis of multi-term fractional order stochastic differential equations

This paper is devoted to the rigorous derivation of some discrete versions of stochastic Grönwall inequalities involving a martingale, which are commonly used in the numerical analysis of multi-term stochastic time-fractional diffusion equations. A Grönwall lemma is also established to deal with the numerical analysis of multi-term stochastic fractional diffusion equations with delay. The proofs of the established inequalities are based on a corresponding deterministic version of the discrete fractional Grönwall lemma in case of smooth solutions and an inequality bounding the supremum in terms of the infimum for discrete time martingales. A numerical application is introduced finally in which the constructed inequalities are handled to derive a priori estimates for a discrete fractional stochastic model.

Fractional stochastic models for COVID-19: Case study of Egypt

In this paper, COVID-19 dynamics are modelled with three mathematical dynamic models, fractional order modified SEIRF model, stochastic modified SEIRF model, and fractional stochastic modified SEIRF model, to characterize and predict virus behavior. By using Euler method and Euler-Murayama method, the numerical solutions for the considered models are obtained. The considered models are applied to the case study of Egypt to forecast COVID-19 behavior for the second virus wave which is assumed to be started on 15 November 2020. Finally, comparisons between actual and predicted daily infections are presented.

Why Do Big Data and Machine Learning Entail the Fractional Dynamics?

Fractional-order calculus is about the differentiation and integration of non-integer orders. Fractional calculus (FC) is based on fractional-order thinking (FOT) and has been shown to help us to understand complex systems better, improve the processing of complex signals, enhance the control of complex systems, increase the performance of optimization, and even extend the enabling of the potential for creativity. In this article, the authors discuss the fractional dynamics, FOT and rich fractional stochastic models. First, the use of fractional dynamics in big data analytics for quantifying big data variability stemming from the generation of complex systems is justified. Second, we show why fractional dynamics is needed in machine learning and optimal randomness when asking: “is there a more optimal way to optimize?”. Third, an optimal randomness case study for a stochastic configuration network (SCN) machine-learning method with heavy-tailed distributions is discussed. Finally, views on big data and (physics-informed) machine learning with fractional dynamics for future research are presented with concluding remarks.

Statistical Methods for Parameter Identification of Temperature Dependent Viscoelastic Models

AbstractA fully fractional stochastic model for linear viscoelastic materials is proposed. Time dependent stress‐strain response for creep tests is described through Adaptive Links theory as a random process of successive micro‐stretching events due to disentanglements between groups of material fibers. The process is modeled by a Continuous Time Random Walk (CTRW) that tends in average to power‐law function, which corresponds to the creep kernel of the usual fractional model for viscoelasticity. Thermal effects are explained through changes in waiting time and jump compliance distribution parameters. Simulated CTRWs are compared with experimental results of specimens of epoxy resin. It was observed that simulated mean function approaches to measurements in long term during the creep phase. Additionally, parameter averages decrease for higher temperatures, representing more frequent jumps of fiber creep strain and faster development of creep process.

Option Pricing in Stochastic Volatility Models Driven by Fractional Jump-Diffusion Processes

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

Developed mathematical technique for fractional stochastic point kinetics model in nuclear reactor dynamics

Fractional stochastic kinetics equations have proven to be valuable tools for the point reactor kinetics model, where the nuclear reactions are not fully described by deterministic relations. A fractional stochastic model for the point kinetics system with multi-group of precursors, including the effect of temperature feedback, has been developed and analyzed. A major mathematical and inflexible scheme to the point kinetics model is obtained by merging the fractional and stochastic technique. A novel split-step method including mathematical tools of the Laplace transforms, Mittage–Leffler function, eigenvalues of the coefficient matrix, and its corresponding eigenvectors have been used for the fractional stochastic matrix differential equation. The validity of the proposed technique has been demonstrated via calculations of the mean and standard deviation of neutrons and precursor populations for various reactivities: step, ramp, sinusoidal, and temperature reactivity feedback. The results of the proposed method agree well with the conventional one of the deterministic point kinetics equations.