Fractional Calculus Theory Research Articles

INNOVATIVE MATHEMATICAL MODELLING IN PREDATOR-PREY DYNAMICS: A SYSTEMATIC REVIEW

Advancements in mathematical ecology have greatly expanded predator-prey models beyond the classical Lotka-Volterra framework, incorporating complex factors such as role reversal, fractional calculus, and the Allee effect. This review highlights innovative models and advanced mathematical techniques that enhance our understanding of predator-prey dynamics and their applications across disciplines. It analysed recent studies introducing new ecological factors, including maturation delays, disease dynamics, and spatial heterogeneity. Techniques like fractional calculus and network theory were examined for their effectiveness in capturing complex behaviours. Models addressing role reversal between adult prey and juvenile predators or incorporating generalized fractional derivatives reveal significant impacts on population stability. Additionally, maturation delays, handling time, and gestation periods markedly influence oscillatory dynamics. The reviewed models demonstrate versatility in guiding pest control, understanding disease spread, and optimizing biotechnological processes. This review shows that modern predator-prey models, enriched by complex ecological factors and advanced mathematics, provide profound insights into system dynamics, with practical applications across various fields. As global challenges grow, these models offer crucial guidance for developing more resilient and sustainable systems, underscoring their potential to addressissues like food security, disease outbreaks, and ecosystem degradation. Future research should integrate emerging factors, such as anthropogenic noise and industrial pollutants, to further enhance the models’ real-world relevance.

Mild solutions to fourth-order parabolic equations modeling thin film growth with time fractional derivative

In this article, we study initial-boundary problems for fourth-order nonlinear parabolic equations modeling thin film growth with Caputo-type time fractional derivative. By means of the theory of abstract fractional calculus and \(L^p-L^q\) estimates, we establish the existence and uniqueness of local mild solutions in the spaces \(C([0,T]; L^{\frac{\beta N}{2-\beta}}(\Omega))\) with \(1<\beta<2\). Moreover, the local solutions can be extended globally if the initial data is sufficiently small. For more information see https://ejde.math.txstate.edu/Volumes/2024/58/abstr.html

Synchronization Analysis of Discrete-Time Fractional-Order Quaternion-Valued Uncertain Neural Networks.

This article studies synchronization issues for a class of discrete-time fractional-order quaternion-valued uncertain neural networks (DFQUNNs) using nonseparation method. First, based on the theory of discrete-time fractional calculus and quaternion properties, two equalities on the nabla Laplace transform and nabla sum are strictly proved, whereafter three Caputo difference inequalities are rigorously demonstrated. Next, based on our established inequalities and equalities, some simple and verifiable quasi-synchronization criteria are derived under the quaternion-valued nonlinear controller, and complete synchronization is achieved using quaternion-valued adaptive controller. Finally, numerical simulations are presented to substantiate the validity of derived results.

Topological degree method for a new class of Φ-Hilfer fractional differential Langevin equation

The aim of this article is to study the existence and uniqueness of the solution of a new class of fractional Langevin Φ-Hilfer differential equations. The proposed study is based on some fundamental definitions of fractional calculus and topological degree theory. We obtained the existence result by employing the topological degree method for condensing maps, and by making use of Banach’s fixed point theorem we deal with the uniqueness result. In order to illustrate our theoretical finding, we provide an example.

Dynamic analysis of fractional poroviscoelastic reinforced subgrade under moving loading

This paper conducts the dynamic analysis of fractional poroviscoelastic reinforced subgrade under moving loading. Based on the Biot theory and transversely isotropic (TI) parameter expression of the geogrid reinforced subgrade, the governing equations of the poroelastic reinforced subgrade are established in the wavenumber domain by the double Fourier transform. Considering the viscosity of the soil skeleton and the flow-dependent viscosity between the soil skeleton and pore water, the governing equations are extended to the fractional poroviscoelastic medium by introducing the Zener viscoelastic model, fractional calculus theory and the dynamic elastic-viscoelastic correspondence principle. Combining boundary conditions and interlayer continuity conditions, the extended precise integration method (PIM) and double Fourier integral transform are employed to obtain the solution of fractional poroviscoelastic reinforced subgrade in the spatial domain. After the numerical validation, a sensitivity analysis of the relaxation time, permeability, reinforcement ratio and the load velocity are conducted.

Analysis of a class of fractal hybrid fractional differential equation with application to a biological model

Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. Different problems of real world processes have been investigated by using the concepts of fractional calculus and important and applicable outcomes were obtained. Because, there has been a lot of interest in fractional differential equations. It is brought on by both the extensive development of fractional calculus theory and its applications. The use of linear and quadratic perturbations of nonlinear differential equations in mathematical models of a variety of real-world problems has received a lot of interest. Therefore, motivated by the mentioned importance, this research work is devoted to analyze in detailed, a class of fractal hybrid fractional differential equation under Atangana- Baleanu- Caputo ABC derivative. The qualitative theory of the problem is examined by using tools of non-linear functional analysis. The Ulam-Hyer’s (U-H) type stability criteria is also applied to the consider problem. Further, the numerical solution of the model is developed by using powerful numerical technique. Lastly, the Wazewska-Czyzewska and Lasota Model, a well-known biological model, verifies the results. Several graphical representations by using different fractals fractional orders values are presented. The detailed discussion and explanations are given at the end.

Effects of Lévy noise and impulsive action on the averaging principle of Atangana–Baleanu fractional stochastic delay differential equations

As delays are common, persistent, and ingrained in daily life, it is imperative to take them into account. In this work, we explore the averaging principle for impulsive Atangana–Baleanu fractional stochastic delay differential equations driven by Lévy noise. The link between the averaged equation solutions and the equivalent solutions of the original equations is shown in the sense of mean square. To achieve the intended outcomes, fractional calculus, semigroup properties, and stochastic analysis theory are used. We also provide an example to demonstrate the practicality and relevance of our research.

Stability and stabilization of fractional-order singular interconnected delay systems

An analytical approach based on fractional calculus and singular value theory to finite-time stability and stabilization of fractional-order singular interconnected delay systems is proposed. Particularly, we study fractional singular equations with interval time-varying delays. We first give new sufficient conditions for finite-time stability of such equations. Then, the feedback stabilizing controllers are designed via solving a tractable linear matrix inequality (LMI) and Mittag-Leffler function. Finally, numerical examples with simulations are given to illustrate the feasibility and effectiveness of the proposed results.

Fuzzy discrete fractional calculus and fuzzy fractional discrete equations

This paper aims to highlight certain limitations in the study of fuzzy fractional discrete equations (FFDEs) based on the generalized Hukuhara difference (gH-difference) in the previous papers. In general, the equivalence between FFDEs and the associated fuzzy discrete fractional sum equations (FDFSEs) is not achieved, requiring the introduction of an appropriate hypothesis to establish this equivalence. Furthermore, this paper introduces the fundamental theory of fuzzy fractional discrete calculus through granular arithmetic operations between fuzzy intervals to address restrictions in the formerly mentioned approaches involving the generalized Hukuhara difference. These operations are constructed based on the concept of the horizontal membership function (HMF) utilized in multidimensional fuzzy arithmetic (MFA). Additionally, the paper proposes the application of fractional discrete calculus to two types of time-discretization diffusion equations with non-zero right-hand sides. Finally, several numerical examples are provided to validate the main results.

Fractional-Order Modeling and Nonlinear Dynamic Analysis of Forward Converter

To accurately investigate the nonlinear dynamic characteristics of a forward converter, a fractional-order state-space averaged model of a forward converter in continuous conduction mode (CCM) is established based on the fractional calculus theory. And nonlinear dynamical bifurcation maps which use PI controller parameters and a reference current as bifurcation parameters are obtained. The nonlinear dynamic behavior is analyzed and compared with that of an integral-order forward converter. The results show that under certain operating conditions, the fractional-order forward converter exhibits bifurcations characterized by low-frequency oscillations and period-doubling as certain circuit and control parameters change. Under the same circuit conditions, there is a difference in the stable parameter region between the fractional and integral-order models of the forward converter. The stable zone of the fractional-order forward converter is larger than that of the integral-order one. Therefore, the circuit struggles to enter states of bifurcation and chaos. The stability domain for low-frequency oscillations and period-doubling bifurcations can be accurately predicted by using a small signal model and a predictive correction model of the fractional-order forward converter, respectively. Finally, by performing circuit simulations and hardware-in-the-loop experiments, the rationality and correctness of the theoretical analysis are verified.

A rational multiscale nonlinear constitutive model for freeze–thaw rocks under triaxial compression

The stability and durability of rocks in cold regions are significantly impacted by the degradation of mechanical properties caused by freeze–thaw (F–T) environment. In this work, we shall propose a rational multiscale nonlinear constitutive model based on thermodynamics, micromechanics, and fractional calculus theory to describe the complete deformation and failure process of F–T rocks under triaxial compression. The F–T rocks at the mesoscale are regarded as consisting of porous matrix and cracks, while porous matrix is composed of the micropores and elastic solid grains at the microscale. According to experimental observations, we assume the F–T action mainly causes micropores growth and cracks opening, and mechanical damage is resulted from the initiation and propagation of cracks. In this context, the effects of F–T and mechanical damage on effective elastic properties of rocks can be quantitatively analyzed by using the two-step Mori–Tanaka (M-T) homogenization method. After subtly deriving the total free energy function of F–T rocks under compression, we systematically develop specific criteria for describing open cracks closure deformation, mechanical damage evolution and frictional sliding induced plastic distortion. Note that to correctly capture the plastic deformation characteristics, the non-orthogonal flow rule based on fractional differential calculations is employed. Following that, analytical analyses and numerical implementation of the proposed model are conducted. The performance of the model is evaluated by the simulations with experimental data on two kinds of F–T rocks, and discussions on parameters sensitivity and effects of fractional order are followed.

An improved creep model for unsaturated reticulated red clay

Establishment of a creep model is an important method to analyze the relationship between soil creep deformation and time, and the element model is widely used for studying soil creep. However, the element creep model is employed for fitting saturated soil, and the mechanical element model is generally linear, which cannot well fit the nonlinear deformation of the soil with time in practice. The creep process of the soil is not only time-dependent, but also related to the deviatoric stress level. Therefore, the fractional calculus theory and a parameter n reflecting the effect of deviatoric stress level on the creep properties of the soil were introduced into the element model, and the fractional qBurgers creep model was established by using the fractional Koeller dashpot and Caputo fractional calculus. The proposed model was used to fit the triaxial test data of reticulated red clay under different net confining pressures and matric suctions by unsaturated triaxial apparatus. The proposed model can well describe the nonlinearity of unsaturated reticulated red clay, has memory and global correlation to the creep development process of unsaturated reticulated red clay, and has clear physical meaning. The functional relationships of the model parameters with the matric suction, net confining pressure and deviatoric stress level were deduced, so that the creep curves of unsaturated reticulated red clay can be obtained for any conditions, which is of great value for the study of unsaturated soils.

Editorial Conclusion for the Special Issue “New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization”

Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, variational analysis, convex analysis, dynamical system theory, mathematical economics, data mining, signal processing, control theory, and many more [...]

Fractional stochastic Schrödinger evolution system with complex potential and poisson jumps: Qualitative behavior and T-controllability

Establishing the fractional stochastic Schrödinger equations (FSSEs) in Hilbert space with complex potential and Poisson jumps is the primary goal of this work. Stochastic analysis, Mönch fixed point theorem, fractional calculus, and semigroup theory all support the possibility of a moderate solution for the suggested system. We construct and prove the criteria for the mild solution’s exponential fall. The trajectory (T-) controllability for the fractional Schrödinger equation is established in the latter portion, extending the findings of Durga and Muthukumar (2023); Grecksch and Lisei (2011); Keller (2013); Wang et al. (2012). An example is given to verify the theoretical findings gained. We discuss the use of fractional Schrödinger’s equation in the Oil and Gas Industry.

The Duality Theory of Fractional Calculus and a New Fractional Calculus of Variations Involving Left Operators Only

Through duality, it is possible to transform left fractional operators into right fractional operators and vice versa. In contrast to existing literature, we establish integration by parts formulas that exclusively involve either left or right operators. The emergence of these novel fractional integration by parts formulas inspires the introduction of a new calculus of variations, where only one type of fractional derivative (left or right) is present. This applies to both the problem formulation and the corresponding necessary optimality conditions. As a practical application, we present a new Lagrangian that relies solely on left-hand side fractional derivatives. The fractional variational principle derived from this Lagrangian leads us to the equation of motion for a dissipative/damped system.

State of Charge Estimation for Lithium Battery Based on Fractional Order Square Root Cubature Kalman Filter and Adaptive Multi-innovation Unscented Kalman Filter

Accurate state of charge (SOC) estimation of batteries is of great significance for electric vehicles. A SOC estimation method based on a fractional order square root cubature Kalman filter (FOSRCKF) and an adaptive multi-innovation unscented Kalman filter (AMIUKF) is proposed. The battery is modelled using fractional order calculus theory and the model parameters are identified by adaptive genetic algorithm. The FOSRCKF estimates the battery SOC, while the AMIUKF online updates the internal resistance in the model, and there exchanges information between two filters. The experimental results under the Urban Dynamometer Driving Schedule (UDDS) and the US06 Highway Driving Schedule show that the proposed method has lower SOC estimation error and lower terminal prediction error compared with the traditional SRCKF method based on integer order models, which demonstrates the effectiveness, accuracy and robustness of the proposed method.

Modeling and Predicting Passenger Load Factor in Air Transportation: A Deep Assessment Methodology with Fractional Calculus Approach Utilizing Reservation Data

This study addresses the challenge of predicting the passenger load factor (PLF) in air transportation to optimize capacity management and revenue maximization. Leveraging historical reservation data from 19 Turkish Airlines market routes and sample flights, we propose a novel approach combining deep assessment methodology (DAM) with fractional calculus theory. By modeling the relationship between PLF and the number of days remaining until a flight, our method yields minimal errors compared to traditional techniques. Through a continuous curve constructed using the least-squares approach, we enable the anticipation of future flight values. Our analysis demonstrates that the DAM model with a first-order derivative outperforms linear techniques and the Fractional Model-3 in both modeling capabilities and prediction accuracy. The proposed approach offers a data-driven solution for efficiently managing air transport capacity, with implications for revenue optimization. Specifically, our modeling findings indicate that the DAM wd model improves prediction accuracy by approximately 0.67 times compared to the DAM model, surpassing the fractional model and regression analysis. For the DAM wd modeling method, the lowest average mean absolute percentage error (AMAPE) value achieved is 0.571, showcasing its effectiveness in forecasting flight outcomes.

Investigating the Constitutive Model of Frozen Supersulfate Saline Soil: Insights from Fractional Calculus

The study of the mechanical properties of frozen saline soil is one of the key issues in addressing the design of infrastructure in cold regions. This research focuses on the supersulfated saline soil of the Ningxia Yellow River Irrigation Area in China, conducting triaxial tests under negative temperatures (−5, −10, −15 °C, and − 20 °C) with varying water contents (12%, 16%, 20%). Based on fractional calculus theory and incorporating an exponential decay factor, this study proposes a novel fractional-order constitutive model for a unified description of the softening and hardening behaviors in frozen saline soil. The model treats frozen saline soil as a composite blend of ideal solids and ideal fluids in varying proportions, taking into account the material's inherent time-dependency and non-linear stress-strain relationships. Finally, the validity of the model is verified by the calculated values of the model and the triaxial tests. The results indicate that, based on preliminary judgment, due to the presence of salt solutes, a large amount of liquid water remains in the supersulfated saline soil at temperatures ranging from 0 to −10 °C, forming an unstable state called “warm frozen saline soil”. The mechanical properties of frozen saline soil depend on the relative content of unfrozen water, ice crystals, and salt crystals and the formation of ice and salt crystals significantly enhances the strength of frozen saline soil. The computational results of the improved fractional constitutive model align well with experimental results, effectively describing the stress-strain relationship of frozen supersulfate saline soil. In the model, the parameter κ functions analogously to an elastic modulus and exhibits a linear relationship with temperature, and the parameter α characterizes the strain hardening of saline soil, while β describes its softening behavior. The proposed fractional constitutive model, with only three parameters having clear physical significance, is convenient for practical engineering applications.

Cluster synchronization of fractional‐order complex networks via variable‐time impulsive control

This paper investigates the cluster synchronization of fractional‐order complex networks. Considering that impulsive control can reduce the update of controller, and the appearance of impulse is always dependent on each node in the networks instead of appearing at fixed instant, thus we design a variable‐time impulsive controller to control the considered networks. Foremost, several assumptions are proposed to guarantee the every solution of coupled error networks intersect each discontinuous impulsive surface exactly once. In addition, by utilizing the B‐equivalence method and the theory of fractional calculus, the variable‐time impulsive fractional‐order system is reduced to a fixed‐time impulsive fractional‐order system, which can be regarded as the comparison system of the former. Next, under the framework of 1‐norm, some sufficient conditions are presented to ensure that fractional‐order system and target trajectory ultimately achieve cluster synchronization. In the end, a numerical example is designed to illustrate the validity and feasibility of theoretical results.

Novel Controller Design for Finite-Time Synchronization of Fractional-Order Nonidentical Complex Dynamical Networks under Uncertain Parameters

The synchronization of complex networks, as an important and captivating dynamic phenomenon, has been investigated across diverse domains ranging from social activities to ecosystems and power systems. Furthermore, the synchronization of networks proves instrumental in solving engineering quandaries, such as cryptography and image encryption. And finite-time synchronization (FTS) controls exhibit substantial resistance to interference, accelerating network convergence speed and heightening control efficiency. In this paper, finite-time synchronization (FTS) is investigated for a class of fractional-order nonidentical complex networks under uncertain parameters (FONCNUPs). Firstly, some new FTS criteria for FONCNUPs are proposed based on Lyapunov theory and fractional calculus theory. Then, the new controller is designed based on inequality theory. Compared to the general controller, it controls all nodes and adds additional control to some of them. When compared to other controllers, it has lower control costs and higher efficiency. Finally, a numerical example is presented to validate the effectiveness and rationality of the obtained results.