Fractional Calculus Method Research Articles

Revolutionizing Heat Transfer and Fluid Flow Models: Fractional Calculus and Non-Newtonian Dynamics Meet Advanced Numerical Methods

Revolutionizing Heat Transfer and Fluid Flow Models: Fractional Calculus and Non-Newtonian Dynamics Meet Advanced Numerical Methods

A COMPUTATIONAL AND FRACTIONAL MATHEMATICAL MODEL FOR ASSESSING HEPATITIS B WITH OPTIMAL THERAPY

This paper presents a mathematical study applied to Hepatitis B disease utilizing fractional calculus and numerical methods. We begin by discussing mathematical fundamentals, focusing on fractional calculus concepts and relevant numerical aspects for disease modeling. We then formulate the problem, describing a nonlinear fractional differential system that models hepatitis B virus (HBV) infection and its response to drug therapy. A detailed theoretical analysis explores system properties and biological implications. We provide a qualitative analysis of the model, discussing important dynamic behaviors such as stability and parameter sensitivity. Subsequently, numerical analysis is performed to validate the model and estimate unknown parameters from patient data. Our findings include significant insights into biological parameters and their impact on therapeutic response. We conclude that the proposed fractional-order model is effective, offering optimal therapeutic strategies for hepatitis B.

Modeling time-dependent deformation in concrete: A fractional calculus method with finite element implementation

This study presents a novel numerical method to simulate the time-dependent behavior of concrete materials. The deformation response of both traditional and fractional calculus (FC) viscoelastic models is analyzed by the proposed method, in which models’ one-dimensional constitutive relationships are extended to the three-dimensional form. By applying numerical and finite difference methods based on Caputo-type fractional integration, the stress-strain behavior of the FC viscoelastic model is discretized over the time scale. This method exhibits broad application prospects, and can be employed to represent various forms of viscoelastic models. For the concrete material, suitable FC viscoelastic models are developed. To facilitate practical engineering applications of the proposed model, a numerical solution algorithm is implemented in the finite element (FE) analysis through the User-defined Material Mechanical Behavior (UMAT) interface of the commercial FE software ABAQUS. Finally, FE results are compared with the experimental results from several concrete creep tests. The consistency between the FE results and experimental data confirms the effectiveness of the proposed model in describing concrete behavior. The proposed method and model provide theoretical and numerical support for a more profound understanding and simulation of the time-dependent deformation of RC specimens in practical engineering applications.

Fourier–Gegenbauer Pseudospectral Method for Solving Periodic Higher-Order Fractional Optimal Control Problems

Optimal control (OC) problems involve determining the best control strategy for a system to achieve the desired outcome. Fractional-order OC models have emerged in recent decades to represent complex systems that exhibit memory effects and nonlocal interactions. These models offer advantages over the traditional integer-order models. However, a key challenge with existing fractional calculus methods to define fractional derivatives, such as Riemann–Liouville, Caputo, and Grünwald–Letnikov derivatives, is that they often fail to maintain the periodic nature of dynamical systems. This limitation hinders their application to real-world scenarios, in which periodicity is crucial. This study proposes a novel model for periodic higher-order fractional OC problems (PHFOCPs). We achieve this by employing recently developed periodic fractional derivatives that specifically preserve the periodicity within the model. This study outlines also a robust numerical solution method for this new class of problems. The numerical method is applicable for any positive fractional order [Formula: see text] and leverages several key techniques, including a useful change of variables, Fourier collocation, and Fourier and Gegenbauer quadratures, to retain numerical stability and high accuracy. We theoretically prove the exponential convergence of the proposed numerical method for sufficiently smooth periodic functions. The effectiveness of the proposed approach is further validated through various numerical simulations. In essence, this work presents a novel framework for the OC of periodic systems using fractional calculus, overcoming the limitations of classical methods and offering a powerful tool for various scientific and engineering applications.

ФРАКТАЛЬНА РАДІОФІЗИКА ЧАСТИНА 3. ДРОБОВЕ ЧИСЛЕННЯ В ЕЛЕКТРОДИНАМІЦІ

Subject and Purpose. At the beginning of the 21st century, a fundamentally new scientific direction was formed, currently known as fractal radiophysics. The present work is an overview of the principal theoretical and practical ideas concerning "fractalization" in radio physics. The purpose is a systematic presentation of the main practical results suitable for application of the fractional calculus in modern theoretical radiophysics. Methods and Methodology. The basic theoretical principles of fractional calculus are outlined in a structured form. Results of applying fractional calculus methods in electrodynamics are systematized. Essential features, advantages and disadvantages of the technique are demonstrated and the problems still remaining discussed. Results. The basics of fractional (or fractal) calculus have been considered with emphasis on practical application to problems of radiophysics. A variety of approaches to constructing fractional integrals and Riemann–Liouville, etc. fractional derivatives have been presented. Using the Newton-Leibnitz formula and fundamental theorems of fractional calculus, principles of generalization of the classic vector calculus to fractal problems have been discussed, suggesting the examples of fractional vector-differential and vector-integral operators, Green’s and Stokes’ fractional formulas, etc. With the use of Gauss’s fractional formula the basics of fractal electrodynamics are expounded. Some different types of fractal Maxwellian equations has been induced and analyzed. Also, the main approaches to solving radio wave propagation problems in fractal media are discussed. Conclusions. As a practical example of applying fractals in modern theoretical radiophysics, results have been presented of the use of fractional calculus in electrodynamics. These results signify appearance of a fundamentally new direction in radiophysics, namely fractal electrodynamics.

Right fractional calculus to inverse-time chaotic maps and asymptotic stability analysis

Inverse or terminal value problems of fractional differential equations become popular recently. But memory effects or non-locality of fractional operators cause many difficulties for theoretical analysis. This study suggests a right fractional calculus method for inverse problem modelling and proposes a concept of inverse-time fractional chaotic maps. First, a simple right fractional linear differential equation's terminal value problem and the solutions are investigated. Then, some basics of the right discrete fractional calculus are introduced and the idea is extended to the discrete case. Right fractional sum equations are derived and numerical schemes are provided for dynamical analysis. Discrete chaos does exist in an inverse-time fractional logistic map and Hénon map, respectively. Their local stability conditions are given. It can be concluded that this right fractional calculus method is simpler but more efficient than the left one.

On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative

Abstract The fundamental objective of this article is to investigate about the boundary value problem with the uses of a generalized conformable fractional derivative introduced by Zarikaya et al. (On generalized the conformable calculus, TWMS J. App. Eng. Math. 9 (2019), no. 4, 792–799, http://jaem.isikun.edu.tr/web/images/articles/vol.9.no.4/11.pdf). In the development of the this article, by using classical methods of fractional calculus, we find a definition of the generalized fractional Wronskian according to the fractional differential operator defined by Zarikaya, a fractional version of the Sturm-Picone theorem, and in addition, the stability criterion given by the Hyers-Ulam theorem is studied with the use of the aforementioned fractional derivatives.

An Effective Spectral Approach to Solving Fractal Differential Equations of Variable Order Based on the Non-singular Kernel Derivative

A new differential operators class has been discovered utilising fractional and variable-order fractal Atangana-Baleanu derivatives that have inspired the development of differential equations' new class. Physical phenomena with variable memory and fractal variable dimension can be described using these operators. In addition, the primary goal of this study is to use the operation matrix based on shifted Legendre polynomials to obtain numerical solutions with respect to this new differential equations' class, which will aid us in solving the issue and transforming it into an algebraic equation system. This method is employed in solving two forms of fractal fractional differential equations: non-linear and linear. The suggested strategy is contrasted with the mixture of two-step Lagrange polynomials, the predictor-corrector algorithm, as well as the fractional calculus methods' fundamental theorem, using numerical examples to demonstrate its accuracy and simplicity. The estimation error was proposed to contrast the results of the suggested methods and the exact solution to the problems. The proposed approach could apply to a wider class of biological systems, such as mathematical modelling of infectious disease dynamics and other important areas of study, such as economics, finance, and engineering. We are confident that this paper will open many new avenues of investigation for modelling real-world system problems.

Analysis of Rheological Factors of Soft Rock Tunnel Based on Constitutive Model of Rock Parameters Attenuation with Equivalent Effect

Rock mass deformation is a time related process, especially for soft rock. Its deformation usually has a certain timeliness. Moreover, the deformation of surrounding rock is typically asymmetric. Therefore, it is of great significance to reasonably describe the time-dependent mechanical properties and behaviors of rock mass for practical engineering, especially when the actual engineering is a symmetric structure. Taking the chlorite schist section at the west end of the diversion tunnel of Jinping II Hydropower Station as the research object and introducing the characteristic that creep parameters attenuate with equal effect change into Burgers constitutive model used for numerical calculation, an improved Burgers model is proposed. Then, according to the actual situation of the project, the improved three-parameter H-K model, which is suitable for this study, is proposed by using the fractional calculus method. The effects of different factors on the rheological properties of soft rock tunnels are discussed. The results show that When K is equal to 1, creep is positively correlated with burial depth; When the burial depth H is 1500 m, the creep deformation is positively correlated with the horizontal geostress; The farther away from the working face, the greater the instantaneous elastic deformation release and the later creep displacement; After the tunnel excavation is stopped, the earlier the support is, the smaller the later creep deformation is; After excavation and support of the upper bench, the longer the stagnation time is, the more unfavorable the rheological deformation of the tunnel is.

Fractional calculus & machine learning methods based rubber stress-strain relationship prediction

ABSTRACT Molecular dynamics simulation can be used to simulate the rubber stretching process and calculate the tensile strength of elastomer materials. However, molecular dynamics simulation process with low stretching rate that is comparable with the experimental case is time-consuming. A fractional long short-term memory (F-LSTM) neural network prediction model is proposed to obtain the tensile stress values under experimental strain rate in a short time and experiments show that the model can extrapolate the results of MD simulations well. Both long short-term memory (LSTM) neural network and fractional calculus have certain memory characteristics suitable for describing the stretching process of rubber. First, fractional calculus transformation is performed on four different tensile strain data sets corresponding to four different strain rates. Then, an LSTM network is established based on the transformed stress–strain data. Experiment results show that the introduction of fractional calculus improves the prediction accuracy of the LSTM model. The established F-LSTM model is further applied to the prediction of the stress value under the experimental strain rate of 1 × 106 s−1.

Mittag-Leffler Euler [formula omitted]-differences for Caputo fractional-order systems

Exponential Euler differences have got rapid development recently for integer-order differential equations. But there are few papers focusing on this difference to fractional differential equations. This paper establishes the basic structure of Mittag-Leffler implicit Euler scheme for fractional differential equations by using the constant variation methods in fractional calculus. More critically, since the acquired difference forms belong to the scope of implicit Euler differences and then the fractional PECE algorithms are proposed to solve these implicit differences in numerical calculations effectively. Besides, a novel nonlocal difference operator is proposed and some relative properties are presented. In the end, the existence and boundedness of a unique globally exponentially stable solution of some nonlocal difference system with short-term memory are investigated. Some calculative examples and numerical simulations are given to illustrate the primary research findings.

Impact of Magnetic Field on a Peristaltic Flow with Heat Transfer of a Fractional Maxwell Fluid in a Tube

Magnetic field and the fractional Maxwell fluids’ impacts on peristaltic flows within a circular cylinder tube with heat transfer was evaluated while assuming that they are preset with a low-Reynolds number and a long wavelength. Utilizing, the fractional calculus method, the problem was solved analytically. It was deduced for temperature, axial velocity, tangential stress, and heat transfer coefficient. Many emerging parameters and their effects on the aspects of the flow were illustrated, and the outcomes were expressed via graphs. A special focus was dedicated to some criteria, such as the wave amplitude's effect, Hartman and Grashof numbers, radius and relaxation–retardation ratios, and heat source, which were under discussions on the axial velocity, tangential stress, heat transfer, and temperature coefficients across one wavelength. Multiple graphs of physical interest were provided. The outcomes state that the effect of the criteria mentioned beforehand (the Hartman and Grashof numbers, wave amplitude, radius ratio, heat source, and relaxation–retardation ratio) were quite evident.

A ROBUST OPERATIONAL MATRIX OF NONSINGULAR DERIVATIVE TO SOLVE FRACTIONAL VARIABLE-ORDER DIFFERENTIAL EQUATIONS

Currently, a study has come out with a novel class of differential operators using fractional-order and variable-order fractal Atangana–Baleanu derivative, which in turn, became the source of inspiration for new class of differential equations. The aim of this paper is to apply the operation matrix to get numerical solutions to this new class of differential equations and help us to simplify the problem and transform it into a system of an algebraic equation. This method is applied to solve two types, linear and nonlinear of fractal differential equations. Some numerical examples are given to display the simplicity and accuracy of the proposed technique and compare it with the predictor–corrector and mixture two-step Lagrange polynomial and the fundamental theorem of fractional calculus methods.

EEG signal artefact removal using flower pollination fractional calculus optimisation

PurposeThe main aim of this paper is to design a technique for improving the quality of EEG signal by removing artefacts which is obtained during acquisition. Initially, pre-processing is done on EEG signal for quality improvement. Then, by using wavelet transform (WT) feature extraction is done. The artefacts present in the EEG are removed using deep convLSTM. This deep convLSTM is trained by proposed fractional calculus based flower pollination optimisation algorithm.Design/methodology/approachNowadays' EEG signals play vital role in the field of neurophysiologic research. Brain activities of human can be analysed by using EEG signals. These signals are frequently affected by noise during acquisition and other external disturbances, which lead to degrade the signal quality. Denoising of EEG signals is necessary for the effective usage of signals in any application. This paper proposes a new technique named as flower pollination fractional calculus optimisation (FPFCO) algorithm for the removal of artefacts from EEG signal through deep learning scheme. FPFCO algorithm is the integration of flower pollination optimisation and fractional calculus which takes the advantages of both the flower pollination optimisation and fractional calculus which is used to train the deep convLSTM. The existed FPO algorithm is used for solution update through global and local pollinations. In this case, the fractional calculus (FC) method attempts to include the past solution by including the second order derivative. As a result, the suggested FPFCO algorithm approaches the best solution faster than the existing flower pollination optimization (FPO) method. Initially, 5 EEG signals are contaminated by artefacts such as EMG, EOG, EEG and random noise. These contaminated EEG signals are pre-processed to remove baseline and power line noises. Further, feature extraction is done by using WT and extracted features are applied to deep convLSTM, which is trained by proposed fractional calculus based flower pollination optimisation algorithm. FPFCO is used for the effective removal of artefacts from EEG signal. The proposed technique is compared with existing techniques in terms of SNR and MSE.FindingsThe proposed technique is compared with existing techniques in terms of SNR, RMSE and MSE.Originality/value100%.

Numerical solution of Abel's general fuzzy linear integral equations by fractional calculus method

The aim of this article is to give a numerical method for solving Abel's general fuzzy linear integral equations with arbitrary kernel. The method is based on approximations of fractional integrals and Caputo derivatives. The convergence analysis for the proposed method is also given and the applicability of the proposed method is illustrated by solving some numerical examples. The results show the utility and the greater potential of the fractional calculus method to solve fuzzy integral equations.

MODELING OF TELECOMMUNICATION REVENUE AS A PERCENTAGE OF GROSS DOMESTIC PRODUCT’S FOR COUNTRIES WITH FRACTIONAL CALCULUS

This study explores the modeling of the share of telecommunication revenues in gross domestic product from the year 2000 to 2018 for 5 countries including France, Germany, Italy, Turkey, the UK, and the OECD average. First, a new mathematical model based on Fractional Calculus and Least Square Method is proposed. Later, the telecommunication revenues in GDP dataset is modeled. Further, we compare the new Fractional approach to the classical Polynomial approach in three different settings. The results show that employing Fractional Calculus yields better modeling performance when compared to the classical Polynomial Approach in terms of Mean Absolute Percentage Error (MAPE). The Fractional approach outperforms the Polynomial approach by 0.1329 % MAPE on average. The largest MAPE is found for Turkey while the smallest MAPE is obtained for Italy in all settings.

Time glass: A fractional calculus approach

Out of equilibrium states in glasses and crystals have been a major topic of research in condensed-matter physics for many years, and the idea of time crystals has triggered a flurry of new research. Here, we provide the first description for the recently conjectured Time Glasses using fractional calculus methods. An exactly solvable effective theory is introduced, with a continuous parameter describing the transition from liquid through normal glass, Time Glass, into the Gardner phase. The phenomenological description with a fractional Langevin equation is connected to a microscopic model of a particle in a sub-Ohmic bath in the framework of a generalized Caldeira-Leggett model.

Preface to a thematic special issue of methods of mathematics and fractional calculus

Preface to a thematic special issue of methods of mathematics and fractional calculus

RenoRmalization Group and Fractional Calculus Methods in a Complex World: A Review

Abstract The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory. In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking.

Application of fractional calculus methods to viscoelastic behaviours of solid propellants.

A three-branch viscoelastic model based on fractional derivatives is proposed for the viscoelastic behaviours of solid propellants. The simulation results show a satisfactory agreement with the stress relaxation modulus and complex modulus of solid propellants. As a comparison, the static modulus is also characterized by traditional viscoelastic model with integer-order derivatives. Results show that the application of the fractional derivatives to the viscoelastic constitutive model can effectively reduce the number of the required parameters while giving an accurate prediction of viscoelastic behaviours of solid propellants. Moreover, a simple and effective direct search method based on simulated annealing and Powell's mothed is proposed for the data fitting. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.