Caputo-type Fractional Derivatives Research Articles

A study on variable-order delay fractional differential equations: existence, uniqueness, and numerical simulation via a predictor corrector algorithm

In this study, we adapted a predictor-corrector technique to simulate delay differential equations incorporating variable-order Caputo-type fractional derivatives. We addressed the existence and uniqueness of solutions for the studied models. Then, we presented numerical simulation of some delay differential equations with variable-order fractional derivatives to demonstrate the efficiency of the used technique. Various periodic and chaotic characteristics of the studied models are observed for some variable-orders from the performed graphical simulations. The used technique can be modified and extended to handle different classes of initial value problems which involve variable-order fractional derivatives.

A study of nonlinear fractional-order biochemical reaction model and numerical simulations

This article depicts an approximate solution of systems of nonlinear fractional biochemical reactions for the Michaelis–Menten enzyme kinetic model arising from the enzymatic reaction process. This present work is concerned with fundamental enzyme kinetics, utilised to assess the efficacy of powerful mathematical approaches such as the homotopy perturbation method (HPM), homotopy analysis method (HAM), and homotopy analysis transform method (HATM) to get the approximate solutions of the biochemical reaction model with time-fractional derivatives. The Caputo-type fractional derivatives are explored. The proposed method is implemented to formulate a fractional differential biochemical reaction model to obtain approximate results subject to various settings of the fractional parameters with statistical validation at different stages. The comparison results reveal the complexity of the enzyme process and obtain approximate solutions to the nonlinear fractional differential biochemical reaction model.

Implementation of Caputo type fractional derivative chain rule on back propagation algorithm

Fractional gradient computation is a challenging task for neural networks. In this study, the brief history of fractional derivation is abstracted, and the core framework of the Faà di Bruno formula is implemented to the fractional gradient computation problem. As an analytical approach to solve the chain rule problem of fractional derivatives, the use of the Faà di Bruno formula for the Caputo-type fractional derivative is proposed. When the fractional gradient is calculated using the proposed approach, the problem of determining the bounds of the formula for calculating the Caputo derivative is addressed. As a consequence, the fundamental problem with the fractional back-propagation algorithm is solved analytically, paving the way for the use of any differentiable and integrable activation function in case they are stable. The development of the algorithm and its practical implementation on the photovoltaic fault detection data-set is investigated. The reliability of the neural network metrics is also investigated using analysis of variance (ANOVA), the results are then presented to decide which are the best metrics and the best fractional order. Ordinary back-propagation, fractional back-propagation with and without L2 regularization and momentum are presented in the results to show the advantages of using L2 regularization and momentum in fractional order gradient. Consequently, the test metrics are reliable but cannot be separated from each other. By changing the batch size from 2 to full batches, the proposed system performs better with bigger batches, but adaptive moment estimation (ADAM) performs better with small batches. The cross validation shows the performance of back propagate ion neural networks have better performance than the ordinary neural networks. It is interesting that the best order for a specific data-set cannot be determined because it depends on the batch size, number of epochs and the cross-validation folds .

Analytical solution of fuzzy heat problem in two-dimensional case under Caputo-type fractional derivative.

This work aims to investigate the analytical solution of a two-dimensional fuzzy fractional-ordered heat equation that includes an external diffusion source factor. We develop the Sawi homotopy perturbation transform scheme (SHPTS) by merging the Sawi transform and the homotopy perturbation scheme. The fractional derivatives are examined in Caputo sense. The novelty and innovation of this study originate from the fact that this technique has never been tested for two-dimensional fuzzy fractional ordered heat problems. We presented two distinguished examples to validate our scheme, and the solutions are in fuzzy form. We also exhibit contour and surface plots for the lower and upper bound solutions of two-dimensional fuzzy fractional-ordered heat problems. The results show that this approach works quite well for resolving fuzzy fractional situations.

A numerical study on the nonlinear fractional Klein–Gordon equation

This article helps to develop a numerical approach based on Fibonacci wavelets for solving fractional Klein-Gordan equations (FKGEs). The FKGEs are solved with Caputo-type fractional derivative. Using the definition of Fibonacci wavelets, we constructed the operational matrices of integration. These operational matrices of integration led to the development of the collocation method called the Fibonacci wavelet collocation method (FWCM). This method transforms the given nonlinear partial differential equation into a system of nonlinear algebraic equations using collocation points to determine the unknown coefficients. By substituting the unknown coefficients in the method, we obtained the numerical solution of the present approach. We furnish the different error norms for the present technique. The obtained results are compared with the Clique polynomial method. These findings demonstrate the computational attractiveness, efficiency, effectiveness, reliability, and robustness of the proposed method for addressing a variety of physical models in science and engineering.

Chaotic dynamics of fractional viscoelastic PET membranes subjected to combined harmonic and variable axial loads

Nonlinear chaotic vibrations of fractional viscoelastic PET (polyethylene terephthalate) membranes subjected to combined harmonic and variable axial loads is investigated in this paper. Axial tension variations arise from the machine disturbances of the processing line of roll-to-roll manufacturing. The viscoelasticity of PET membrane is characterized by the fractional Kelvin-Voigt model. Based on the Hamilton principle, the equation of motion of the membrane is established with the consideration of geometric nonlinearity, and the Galerkin procedure is employed to discretize the resulting governing equation. For the solution, the finite difference method is utilized in conjunction with the Caputo-type fractional derivative to reliably estimate the nonlinear response of fractional viscoelastic PET membrane. The reliability of this numerical strategy is proved by the available results of the fractional system and comparison examples. The influence of system parameters on chaotic behaviors is described by the bifurcation diagram and the detailed responses at the set bifurcation parameters. The fractional model together with the analysis provides a fundamental framework for the control of viscoelastic substrates in flexible manufacturing.

Nonlinear dynamics of fractional viscoelastic PET membranes with linearly varying density

In this paper, the nonlinear dynamics of fractional viscoelastic polyethylene terephthalate (PET) membranes with linearly varying density are elaborated. The viscoelasticities of the PET membranes are characterized with the fractional Kelvin–Voigt model, and the density distribution is considered a linear fluctuation in the lateral direction. The geometrically nonlinear formulation is established with the von-Karman theory and the Hamilton principle. The Bubnov–Galerkin method is used to solve the resulting fractional differential equations, and a refined numerical scheme combining the finite difference with a discrete Caputo-type fractional derivative approximation is developed. Comparison examples are provided to substantiate the accuracy and effectiveness of the present method. A detailed parametric analysis is performed to illuminate the effects of the fractional order, density coefficient and various system parameters on the time response of the viscoelastic PET membranes. This study paves the way for fractional modelling and vibrational analysis of viscoelastic substrates in flexible electronics manufacturing.

Fractional mathematical modeling of the Stuxnet virus along with an optimal control problem

In this digital, internet-based world, it is not new to face cyber attacks from time to time. A number of heavy viruses have been made by hackers, and they have successfully given big losses to our systems. In the family of these viruses, the Stuxnet virus is a well-known name. Stuxnet is a very dangerous virus that probably targets the control systems of our industry. The main source of this virus can be an infected USB drive or flash drive. In this research paper, we study a mathematical model to define the dynamical structure or the effects of the Stuxnet virus on our computer systems. To study the given dynamics, we use a modified version of the Caputo-type fractional derivative, which can be used as an old Caputo derivative by fixing some slight changes, which is an advantage of this study. We demonstrate that the given fractional Caputo-type dynamical model has a unique solution using fixed point theory. We derive the solution of the proposed non-linear non-classical model with the application of a recent version of the Predictor–Corrector scheme. We analyze various graphs at different values of the arrival rate of new computers, damage rate, virus transmission rate, and natural removal rate. In the graphical interpretations, we verify the values of fractional orders and simulate 2-D and 3-D graphics to understand the dynamics clearly. The major novelty of this study is that we formulate the optimal control problem and its important consequences both theoretically and mathematically, which can be further extended graphically. The main contribution of this research work is to provide some novel results on the Stuxnet virus dynamics and explore the uses of fractional derivatives in computer science. The given methodology is effective, fully novel, and very easy to understand.

Synchronization of the Circadian Rhythms with Memory: A Simple Fractional-Order Dynamical Model Based on Tow Coupled Oscillators

Disruptions of the circadian rhythm are associated with internal desynchronization. It affects some internal functions of our body and behavior that are important to our health. Our modern lifestyle has contributed to millions of people developing some circadian rhythm disruptions, making the subject very important clinically as well as economically. Motivated by studying a simple mathematical model that can reveal some features of internal synchronization desynchronization, in this contribution, we extend the coupling oscillator phase model proposed by Strogatz [14] in the sense that memory is considered in the modeling. Such memory is a result of the introduction of Caputo-type fractional derivatives in the coupling oscillators’ phase model dynamics, resulting in a fractional phase model. We show that the proposed fractional coupling oscillator phase model is well-posed. Furthermore, we analyzethe synchronization phenomena. We obtain the synchronized solutions explicitly when the memory is equally distributed between the oscillators. In contrast, when distinct levels of memory are imposed in the modeling, we obtain lower and upper bounds because any existing synchronized solution must be confined in between. We present numerical realizations that support the theoretical findings in great detail.

Stochastic bifurcations of nonlinear vibroimpact system with time delay and fractional derivative excited by Gaussian white noise

This paper investigates the stochastic dynamics in the nonlinear vibroimpact system with time delay and fractional derivative under Gaussian white noise excitation. Firstly, based on the definition of Caputo-type fractional derivative and the method of non-smooth transformation, the original system is transformed into an equivalent delayed stochastic vibroimpact system without fractional derivative. Then, by using the stochastic averaging method, the stationary density function of the stochastic Itoˆ equation is obtained. The effectiveness of the proposed method was validated by comparing the consistency between the original system and the optimized system without fractional derivative or the term of time delay. At last, we also explore the stochastic P-bifurcation induced by the power spectral density of two uncorrelated noises, time delay, fractional order, and restitution coefficient of the system.

Estimating China's CO2 emissions under the influence of COVID-19 epidemic using a novel fractional multivariate nonlinear grey model.

The COVID-19 pandemic has posed a severe challenge to the global economic recovery and China's economic growth. Although China has achieved stage victory against the epidemic, the impact and influence of the epidemic on China's economy will linger. On the positive side, the epidemic has led to a dramatic reduction in air pollution levels and a sharp slowdown in greenhouse gas emissions such as CO2. Forecasting China's CO2 emissions under the impact of the COVID-19 epidemic is crucial for formulating policies that contribute to smooth economic development and rational energy consumption. To this end, this paper improves on the traditional grey model, GM(1, N), by developing a novel fractional multivariate nonlinear grey model, FMNGM(q, N). These improvements include two key points. First, different power exponents are assigned to each relevant factor variable to explain their nonlinear, uncertain, and complicated relationships with the system characteristic variables. Second, the integer accumulation is changed to fractional accumulation to preprocess the data, and the Caputo-type fractional derivative is used to characterize the endogenous relationships among the system characteristic data. In addition, the quantum particle swarm optimization (QPSO) algorithm is used to optimize the above parameters with the goal of minimizing MAPE. The collected data on China's CO2 emission from 2001 to 2018 were divided into two parts according to different stages of economic development: 2001-2009 and 2010-2018. Both sets of data were modelled and analysed separately and compared with other models. Results showed that the proposed model had better prediction accuracy than other models. Finally, the model built using the 2010-2018 data was used to forecast China's CO2 emissions. Results show that China's CO2 emissions in 2020 under the impact of COVID-19 will decrease by approximately 3.15% from 2019 to 9893 million tons. The results bear important policy implications for planners in investment in clean energy and infrastructure to achieving the goal of a low-carbon transition.

An Error Analysis of Implicit Finite Difference Method with Mamadu-Njoseh Basis Functions for Time Fractional Telegraph Equation

In this paper, we proposed and analyzed the error estimate of an implicit finite difference method with Mamadu-Njoseh as basis functions for time fractional telegraph equation. To enhance the efficacy of the method we first transform the Caputo type fractional derivative into Riemann-Liouville derivatives. The error analysis of the method is stated and proven. Also, the optimal results for scalars unknown in norm were derived for the two-dimensional case. Numerical illustrations are presented to test the reliability of the analytical and computed results. The resulting numerical evidence shows that the proposed method convergences more rapidly than the standard finite difference method. MAPLE 18 is used for all mathematical procedures in this paper.

Post-Pandemic Sector-Based Investment Model Using Generalized Liouville–Caputo Type

In this article, Euler’s technique was employed to solve the novel post-pandemic sector-based investment mathematical model. The solution was established within the framework of the new generalized Caputo-type fractional derivative for the system under consideration that serves as an example of the investment model. The mathematical investment model consists of a system of four fractional-order nonlinear differential equations of the generalized Liouville–Caputo type. Moreover, the existence and uniqueness of solutions for the above fractional order model under pandemic situations were investigated using the well-known Schauder and Banach fixed-point theorem technique. The stability analysis in the context of Ulam—Hyers and generalized Ulam—Hyers criteria was also discussed. Using the investment model under consideration, a new analysis was conducted. Figures that depict the behavior of the classes of the projected model were used to discuss the obtained results. The demonstrated results of the employed technique are extremely emphatic and simple to apply to the system of non-linear equations. When a generalized Liouville–Caputo fractional derivative parameter (ρ) is changed, the results are asymmetric. The current work can attest to the novel generalized Caputo-type fractional operator’s suitability for use in mathematical epidemiology and real-world problems towards the future pandemic circumstances.

Probabilistic evolution analysis and first passage analysis of a class of stochastic dynamic systems with fractional derivative based on Complex Fractional Moment method

In this paper, the application of complex fractional moment(CFM) method in the stochastic dynamic systems with Caputo-type fractional derivative under the excitation of Gaussian white noise are investigated. By the approximation method, the FPK equation governing the amplitude is derived, and the semi-analytic solution of FPK equation is obtained by CFM method. The results are verified by numerical simulation. Meanwhile, the probability evolution of transient Probability Density Function(PDF) is analyzed when the stochastic bifurcation induced by fractional coefficient for the first time. In addition, the relation of equivalence between first passage time(FPT) and CFM is established by a novel method, the accuracy of this method is verified, the influences of system parameters change on reliability function and the FPT are discussed by the novel method.

Solution of a dengue fever model via fractional natural decomposition and modified predictor–corrector methods

In this paper, we solved a model of a well-known infectious disease called dengue fever via fractional natural decomposition and modified Predictor–Corrector (PC) methods. A study of the dengue epidemic in the Cape Verde Islands off the coast of West Africa in 2009 has been resumed here for a better understanding of the results. The results are obtained using Liouville–Caputo and new generalized Caputo-type fractional derivatives. The numerical simulations are presented for various orders of given derivatives. Existence and uniqueness analysis of the given problem are also performed in the new generalized Caputo sense. The explored results are verified using figures. The main target of this paper is to explore the different dynamics of the given dengue fever model via two types of fractional numerical algorithms.

Numerical solutions of fractional epidemic models with generalized Caputo-type derivatives

Recently, a general framework of fractional operators, that includes the Caputo model as a particular case, has been introduced and some applications in the area of fractional calculus have been presented. In this paper, novel fractional epidemic models with generalized Caputo-type derivatives were proposed. The universal predictor-corrector method was modified here to deal with the considered epidemic models for the purposes of simulation. The behavior and complex dynamic of these hybrid fractional epidemic models were studied using the modified method. The dynamics of the generalized Caputo-type fractional SIR, HIV and SEIR models were investigated by numerical simulation. Basically, the effect of generalized Caputo-type fractional derivative operator parameters on the dynamic behavior of the proposed epidemic models was discussed.

An Application of the Homotopy Analysis Method for the Time- or Space-Fractional Heat Equation

This paper focuses on the usage of the homotopy analysis method (HAM) to solve the fractional heat conduction equation. In the presented mathematical model, Caputo-type fractional derivatives over time or space are considered. In the HAM, it is not necessary to discretize the considered domain, which is its great advantage. As a result of the method, a continuous function is obtained, which can be used for further analysis. For the first time, for the considered equations, we proved that if the series created in the method converges, then the sum of the series is a solution of the equation. A sufficient condition for this convergence is provided, as well as an estimation of the error of the approximate solution. This paper also presents examples illustrating the accuracy and stability of the proposed algorithm.

A modification of Newton’s cooling law with correlation to fractional derivative

Newton’s cooling law (NCL) provides a linear differential equation governing the rate of heat loss of a heated body using the temperature difference of the body with the environment. However, the prediction of NCL still did not fit with the data under the laboratory framework. Previous works have modified NCL by incorporating fractional derivative as a basis of their models, and in particular cases of convective fluid, higher correlations with the experimental data can be observed. In this study, to model the empirical value obtained from the experiment, the conventional model is enhanced by appending a new parameter as the exponent of time in the differential equation. The comparison is shown between the conventional NCL and the modified model, along with its adjusted R-Squared value from the regression of the experimental data, which results in a significant enhancement from the conventional model. In a numerical relationship between the two parameters, correlation of this newly proposed model is found with an established model of NCL using Caputo type fractional derivative, thereby providing some support to the theoretical basis of the model. Moreover, the performance of this modified model is to a certain degree higher than the fractional derivative model.

THE DYNAMICS OF HIV/AIDS MODEL WITH FRACTAL-FRACTIONAL CAPUTO DERIVATIVE

The human immunodeficiency virus (HIV) is a major global public health issue and causes millions of deaths around the globe. The most severe phase of HIV infection is known as AIDS. In recent years, a number of mathematical models based on classical integer-order derivative have been developed to analyze the insight dynamics of HIV/AIDS. This paper presents the transmission dynamics of HIV/AIDS using fractional order (FO) and a fractal-fractional order compartmental model with the power-law kernel. In the first phase, the proposed model is formulated using the Caputo-type fractional derivative. The basic properties such as the solution positivity and existence as well as uniqueness of the fractional model are presented. The equilibria and the basic reproductive number [Formula: see text] are evaluated. Further, using fractional stability concepts the stability of the model (both local and global) around the equilibrium is presented in the disease-free case. In addition, the fractional model is solved numerically, and the graphical results with many values of [Formula: see text] are shown. In the second phase, the concept of a fractal-fractional (FF) operator is applied to obtain a more generalized model that addresses the dynamics of HIV/AIDS. The uniqueness and existence of the solutions of the FF-based model are shown via the Picard–Lindelof approach while the modified Adams–Bashforth method is utilized to present the numerical solution. Detailed numerical simulations are presented for various values fractional as well as the fractal orders, [Formula: see text] and [Formula: see text] respectively. The graphical results reveal that the FF-based model provides biologically more feasible results than the models in fractional and classical integer-order cases.

NUMERICAL METHODS FOR THE CAPUTO-TYPE FRACTIONAL DERIVATIVE WITH AN EXPONENTIAL KERNEL

In the present article, several typical numerical discrete formulas for the Caputo-type fractional derivative with an exponential kernel (call "exponential Caputo derivative" for brevity) with order $ \alpha\in(0,1) $ and $ \alpha\in(1,2) $ are constructed, which are L1, L1-2, L2-1$ _\sigma $ formulas for $ \alpha\in(0,1) $, and H2N2 and L2$ _{1} $ formulas for $ \alpha\in(1,2) $, respectively. And the estimates of the truncation errors are determined. Meanwhile, the properties of the coefficients in these formulas are studied. Finally, some numerical examples are displayed which support the theoretical analysis.