Fractional Order Derivative Research Articles

CMMSE: New Properties of Auto‐Wave Solutions in Activator‐Inhibitor Reaction‐Diffusion Systems With Fractional Derivatives

ABSTRACTIn this article, we analyze new properties of auto‐wave solutions in fractional reaction‐diffusion systems. These new properties arise due to a change in fractional derivative order and do not occur in systems with classical derivatives. It is shown that the stability of steady‐state solutions and their evolution are mainly determined by the eigenvalue spectrum of a linearized system and the fractional derivative order. It is also demonstrated that the basic properties of auto‐wave solutions in fractional‐order systems can essentially differ from those in standard systems. The results of the linear stability analysis are confirmed by computer simulations of the generalized fractional van der Pol–FitzHugh–Nagumo mathematical model. A common picture of possible instabilities and auto‐wave solutions in time‐fractional two‐component activator‐inhibitor systems is presented.

Higher-order approximations for space-fractional diffusion equation

Second-order and third-order finite difference approximations for fractional derivatives are derived from a recently proposed unified explicit form. The Crank-Nicholson schemes based on these approximations are applied to discretize the space-fractional diffusion equation. We theoretically analyse the convergence and stability of the Crank-Nicholson schemes, proving that they are unconditionally stable. These schemes exhibit unconditional stability and convergence for fractional derivatives of order in the range . Numerical examples further confirm the convergence order and unconditional stability of the approximations, demonstrating their effectiveness in practice.

Numerical analysis of double-fractional PDEs in MHD hybrid nanofluid blood flow with slip velocity, heat source, and radiation effects

Abstract The study of blood flow in cylindrical geometries resembling small arteries is crucial for advancing drug delivery systems, cardiovascular health, and treatment methods. However, Conventional models have failed to capture the complex memory effects and non-local behavior inherent in blood flow dynamics, which hinders their accuracy in predicting critical flow and heat transfer properties for medical applications. To overcome these limitations, this research introduces a novel fractional-order magnetohydrodynamic model for blood flow, incorporating a $ZnO$ and $Fe_3O_4$ hybrid nanofluid. The model uniquely integrates boundary slip velocity effects within the double fractional Maxwell model (DFMM) rheology framework and utilizes the dual fractional phase lag bioheat model (DFPLM) applied to a porous cylindrical structure. Fractional-order time derivatives in the thermal and momentum equations are formulated using the Caputo approach, with numerical solutions derived via finite difference methods leveraging L1 and L2 approximations for Caputo fractional derivatives. The study examines the effects of fractional orders, relaxation time, and phase lags for heat and temperature, along with parameters such as thermal radiation, wall slip velocity, and porosity. These factors are analyzed for their impact on velocity, temperature, skin friction, and the Nusselt number. Results indicate that the hybrid nanofluid enhances heat transfer compared to blood or mono-hybrid nanofluids, while also reducing skin friction. Furthermore, fractional-order models provide more reliable and realistic predictions under varying flow conditions. The DFMM shows smoother transitions in velocity and friction, while the DFPLM predicts higher temperatures and greater heat transfer enhancement compared to classical and single-phase lag models. By integrating fractional calculus, this model offers improved simulation of complex transport phenomena in small arteries, contributing to the development of more effective cardiovascular treatments.

Numerical Solutions of Time fractional Klein Gordon Equation using Crank-Nicolson Finite Difference Method

Finite difference methods are widely used numerical techniques used to solve partial differential equations observed in many fields, such as science and engineering. This research presents a study on the numerical solutions of the Klein-Gordon equation, which describes anomalous diffusion and wave propagation in quantum fields and possesses a fractional derivative in the Caputo sense. The content of the paper begins by discretizing the region of the problem while taking into account the fundamental characteristics of finite difference methods. Subsequently, the time derivative algorithm, and the other terms, are discretized using the Crank-Nicolson finite difference approach, resulting in a system of algebraic equations. Solving this algebraic equation system yields numerical solutions. The numerical results are calculated for various values of the parameters associated with the equation and fractional order derivatives , leading to the computation of error norms. Graphical findings illustrate the physical behavior of approximation solutions for a variety of fraction order values. Additionally, the stability analysis of the numerical scheme is investigated using von-Neumann stability analysis. The results of this paper will help other researchers studying in the field to apply the presented method to other problems modelling the natural phenomena.

A spatio-temporal infection epidemic model with fractional order, general incidence, and vaccination analysis

The paper investigates a spatio-temporal SEIR epidemic model on a global level with fractional order. It employs four partial differential equations incorporating fractional derivatives and account for diffusion to characterize infection dynamics. By applying fixed-point theorem results, the paper establishes the existence, uniqueness, and boundedness of the solution. Equilibrium points are determined based on vaccination values and the basic reproduction number R0. Global stability of each equilibrium is confirmed using the Lyapunov direct method. Through simulations with a predictor–corrector algorithm, key insights into epidemiological dynamics are provided, elucidating the impact of vaccination on reducing disease transmission and altering R0. Trajectories of various compartments closely align with theoretical equilibrium points, affirming the model’s predictive precision. Furthermore, simulations indicate the potential for attaining disease-free equilibria with heightened vaccination rates, underscoring the pivotal role of vaccination strategies in epidemic control and disease eradication. It was shown that the fractional derivative order has no effect on the equilibrium stability but rather only on the convergence speed towards the equilibrium points.

Elastoinertial stability analysis and structure formation in viscoelastic subdiffusive pipe flow

The modal temporal stability analysis of viscoelastic, subdiffusive, pressure driven, axisymmetric pipe flow, representing thick polymer solutions, exhibits the presence of temporally stable regions at high fluid inertia. The stress constitutive equation, previously derived for channel flows [Chauhan et al., Phys. Fluids 35(12), 123121 (2023)] is the fractional variant of the upper convected Maxwell equation. The parameters governing the stability are the Reynolds number, Re=ρU0R0η0, the elasticity number, El=λαη0ρR02, and the ratio of the solvent to the polymer solution viscosity, ν=ηsη0, where R0,U0,ρ,λ,α are the pipe radius, the maximum mean flow velocity, density, the polymer relaxation time, and the fractional order of the time derivative, respectively. The neutral curves indicate, in the limit of small elasticity numbers or in the limit when the viscosity ratio approaches unity, El(1−ν)≪1, that the critical Reynold number, Rec diverges as Rec∼[(1−ν)El]−3α/2, while the critical wavenumber, kc increases as kc∼[(1−ν)El]−α/2. Using a novel fractional variant of the pressure correction method as well as a metric in the Riemannian manifold of symmetric positive definite conformation tensors, the direct numerical simulations quantify the formation of spatiotemporally stable macrostructures (or the non-homogeneous regions of high viscosity) at moderate inertia, thereby corroborating the qualitative features of the experimentally observed flow-instability transition of subdiffusive axisymmetric pipe flows.

Analytical approximate solutions of some fractional nonlinear evolution equations through AFVI method

In this article, we construct and investigate a new fractional variational iteration technique which is named as AFVI method. After that, we formulate some IVPs corresponding to the fractional nonlinear evolution equations NLTFFWE, mNLTFFWE, TFmCHE and TFmDPE. The Caputo fractional order derivative has been used to fractionalize the considered NLEEs. Then, we solve the considered IVPs by applying the formulated AFVI method. Lastly, we check the accuracy of the attained AASs by making some graphical and numerical comparisons with their corresponding exact solutions and other existing equivalent AASs. The result of this article confirm that the efficiency, appropriateness and time spent capability of the newly constructed AFVI method are better than that of the other existing analogous fractional analytical approximation methods. Here, we apply the Maple 2021 programming software to obtain the AASs of the formulated IVPs and drawing the 3D graphs of the attained AASs. Finally, in this article we validate the applicability of the Caputo fractional order derivative to form a novel analytical approximation method as well as to fractionalize some essential NLEEs of Mathematical physics.

CMINNs: Compartment model informed neural networks — Unlocking drug dynamics

In the field of pharmacokinetics and pharmacodynamics (PKPD) modeling, which plays a pivotal role in the drug development process, traditional models frequently encounter difficulties in fully encapsulating the complexities of drug absorption, distribution, and their impact on targets. Although multi-compartment models are frequently utilized to elucidate intricate drug dynamics, they can also be overly complex. To generalize modeling while maintaining simplicity, we propose an innovative approach that enhances PK and integrated PK–PD modeling by incorporating fractional calculus or time-varying parameter(s), combined with constant or piecewise constant parameters. These approaches effectively model anomalous diffusion, thereby capturing drug trapping and escape rates in heterogeneous tissues, which is a prevalent phenomenon in drug dynamics. Furthermore, this method provides insight into the dynamics of drug in cancer in multi-dose administrations. Our methodology employs a Physics-Informed Neural Network (PINN) and fractional Physics-Informed Neural Networks (fPINNs), integrating ordinary differential equations (ODEs) with integer/fractional derivative order from compartmental modeling with neural networks. This integration optimizes parameter estimation for variables that are time-variant, constant, piecewise constant, or related to the fractional derivative order. The results demonstrate that this methodology offers a robust framework that not only markedly enhances the model’s depiction of drug absorption rates and distributed delayed responses but also unlocks different drug–effect dynamics, providing new insights into absorption rates, anomalous diffusion, drug resistance, persistence, and pharmacokinetic tolerance, all within a system of just two (fractional) ODEs with explainable results. These findings have the potential to streamline drug development by improving the prediction of drug behavior in complex biological systems and shedding light on cancer cell death mechanisms, ultimately aiding in the design of more effective therapeutic strategies.

Computer-Aided Diagnosis of Graphomotor Difficulties Utilizing Direction-Based Fractional Order Derivatives

Children who do not sufficiently develop graphomotor skills essential for handwriting often develop graphomotor disabilities (GD), impacting the self-esteem and academic performance of the individual. Current examination methods of GD consist of scales and questionaries, which lack objectivity, rely on the perceptual abilities of the examiner, and may lead to inadequately targeted remediation. Nowadays, one way to address the factor of subjectivity is to incorporate supportive machine learning (ML) based assessment. However, even with the increasing popularity of decision-support systems facilitating the diagnosis and assessment of GD, this field still lacks an understanding of deficient kinematics concerning the direction of pen movement. This study aims to explore the impact of movement direction on the manifestations of graphomotor difficulties in school-aged. We introduced a new fractional-order derivative-based approach enabling quantification of kinematic aspects of handwriting concerning the direction of movement using polar plot representation. We validated the novel features in a barrage of machine learning scenarios, testing various training methods based on extreme gradient boosting trees (XGBboost), Bayesian, and random search hyperparameter tuning methods. Results show that our novel features outperformed the baseline and provided a balanced accuracy of 87 % (sensitivity = 82 %, specificity = 92 %), performing binary classification (children with/without graphomotor difficulties). The final model peaked when using only 43 out of 250 novel features, showing that XGBoost can benefit from feature selection methods. Proposed features provide additional information to an automated classifier with the potential of human interpretability thanks to the possibility of easy visualization using polar plots.

A Comprehensive Approach to Load Frequency Control in Hybrid Power Systems Incorporating Renewable and Conventional Sources with Electric Vehicles and Superconducting Magnetic Energy Storage

This study addresses the load frequency control (LFC) within a multiarea power system characterized by diverse generation sources across three distinct power system areas. area 1 comprises thermal, geothermal, and electric vehicle (EV) generation with superconducting magnetic energy storage (SMES) support; area 2 encompasses thermal and EV generation; and area 3 includes hydro, gas, and EV generation. The objective is to minimize the area control error (ACE) under various scenarios, including parameter variations and random load changes, using different control strategies: proportional-integral-derivative (PID), two-degree-of-freedom PID (PID-2DF), fractional-order PID (FOPID), fractional-order integral (FOPID-FOI), and fractional-order integral and derivative (FOPID-FOID) controllers. The result analysis under various conditions (normal, random, and parameter variations) evidences the superior performance of the FOPID-FOID control scheme over the others in terms of time-domain specifications like oscillations and settling time. The FOPID-FOID control scheme provides advantages like adaptability/flexibility to system parameter changes and better response time for the current power system. This research is novel because it shows that the FOPID-FOID is an excellent control scheme that can integrate these diverse/renewable sources with modern systems.

Fractional Sturm-Liouville operators on compact star graphs

Abstract In this article, we examine two problems: a fractional Sturm-Liouville boundary value problem on a compact star graph and a fractional Sturm-Liouville transmission problem on a compact metric graph, where the orders α i {\alpha }_{i} of the fractional derivatives on the ith edge lie in ( 0 , 1 ) (0,1) . Our main objective is to introduce quantum graph Hamiltonians incorporating fractional-order derivatives. To this end, we construct a fractional Sturm-Liouville operator on a compact star graph. We impose boundary conditions that reduce to well-known Neumann-Kirchhoff conditions and separated conditions at the central vertex and pendant vertices, respectively, when α i → 1 {\alpha }_{i}\to 1 . We show that the corresponding operator is self-adjoint. Moreover, we investigate a discontinuous boundary value problem involving a fractional Sturm-Liouville operator on a compact metric graph containing a common edge between the central vertices of two star graphs. We construct a new Hilbert space to show that the operator corresponding to this fractional-order transmission problem is self-adjoint. Furthermore, we explain the relations between the self-adjointness of the corresponding operator in the new Hilbert space and in the classical L 2 {L}^{2} space.

Study of two-dimensional nonlinear coupled time-space fractional order reaction advection diffusion equations using shifted Legendre-Gauss-Lobatto collocation method

In this article, the nonlinear coupled two-dimensional space-time fractional order reaction-advection–diffusion equations (2D-STFRADEs) with initial and boundary conditions is solved by using Shifted Legendre-Gauss-Lobatto Collocation method (SLGLCM) with fractional derivative defined in Caputo sense. The SLGLC scheme is used to discretize the coupled nonlinear 2D-STFRADEs into the shifted Legendre polynomial roots to convert it to a system of algebraic equations. The efficiency and efficacy of the scheme are confirmed through error analysis while applying the scheme on two existing problems having exact solutions. The impact of advection and reaction terms on the solution profiles for various space and time fractional order derivatives are shown graphically for different particular cases. A drive has been made to study the convergence of the proposed scheme, which has been applied on the proposed mathematical model.

Exact solutions of time-delay integer- and fractional-order advection equations

Transport phenomena play a vital role in various fields of science and engineering. In this work, exact solutions are derived for advection equations with integer- and fractional-order time derivatives and a constant time-delay in the spatial derivative. Solutions are obtained, for arbitrary separable initial conditions, by incorporating recently introduced delay functions in a separation of variables approach. Examples are provided showing oscillatory and translatory behaviours that are fundamentally different to standard propagating wave solutions.

Dynamic response of thermoelasticity based on Green-Lindsay theory and Caputo-Fabrizio fractional-order derivative

To extend the applicability and accuracy of the generalized thermoelasticity theory of thermoelasticity theory for one-dimensional problems involving a moving heat source, this study proposes a fractional-order thermoelasticity coupling theoretical model based on the Green-Lindsay theory and the Caputo-Fabrizio fractional-order derivative. The model's uniqueness and reciprocity are well established. To show its application, we analyzed the thermoelastic coupled dynamic response of a fixed-end rod subjected to a moving heat source. Using Laplace transforms and its numerical inverse method, the distribution patterns of non-dimensional displacement, temperature, and stress were obtained. A comprehensive analysis was conducted to investigate the effects of the fractional coefficient, two thermal relaxation time factors, and moving heat source speed on non-dimensional displacement, temperature, and stress. The findings reveal that the fractional coefficient and the speed of the moving heat source significantly influence all non-dimensional physical variables. While the two distinct thermal relaxation time factors have a minimal influence on the non-dimensional temperature, they exert a more pronounced effect on non-dimensional displacement and stress.

Normalized acceleration based online tuning of variable-order fractional derivatives: a case study on quadcopter position control

In recent years, the use of Variable-Order (VO) fractional operators in control system design has been gaining popularity due to their adaptive nature, enabled by their dynamically adjustable fractional derivative and integral orders. This paper presents an online tuning method for adjusting the VO fractional derivatives in fractional controllers. The method is formulated to strategically accelerate or decelerate the system response's rate of change to enhance reference tracking and disturbance rejection performance while preserving closed-loop stability. It uses the normalised acceleration of the system response, a metric that provides insights into the ‘fastness’ or ‘slowness’ of the system response. The effectiveness of the proposed online tuning method is validated through a case study on quadcopter position control. Our research includes both simulation and real-time testing of Variable-Order Fractional PD (VOFPD) controllers, which utilise our online tuning method to adjust their fractional derivatives in real-time. Stability analysis via the D-decomposition method confirms that the quadcopter's closed-loop stability is preserved. Comparative results show significant improvements in reference tracking and disturbance rejection in terms of time-domain criteria.

New crossover lumpy skin disease: Numerical treatments

This work expands on a novel piecewise mathematical model of Lumpy Skin Disease (LSD) in three time intervals by utilizing fractional stochastic derivatives and variable-order differential equations. The LSD model is developed into two crossover hybrid variable-order derivatives. This study's piecewise mathematical model representation of lumpy skin disease has revealed a property that has never been taken into account or seen in previous research employing mathematical models based on classical, different fractional derivatives and variable order fractional derivatives. The Caputo derivative and the Riemann-Liouville integral are merged linearly to produce the hybrid fractional order derivative. The variable-order fractional and hybrid fractional operators are approximated using the Grünwald-Letnikov approximation. We introduce the hybrid variable-order operator combined with the non-standard finite difference method. The stability, boundedness, positivity, and existence of the suggested model are examined. The effectiveness of the techniques and the validity of the theoretical results were verified through a number of numerical experiments.

A spline-based framework for solving the space–time fractional convection–diffusion problem

In this study we consider a spline-based collocation method to approximate the solution of fractional convection–diffusion equations which include fractional derivatives in both space and time. This kind of fractional differential equations are valuable for modeling various real-world phenomena across different scientific disciplines such as finance, physics, biology and engineering.The model includes the fractional derivatives of order between 0 and 1 in space and time, considered in the Caputo sense and the spatial fractional diffusion, represented by the Riesz–Caputo derivative (fractional order between 1 and 2). We propose and analyze a collocation method that employs a B-spline representation of the solution. This method exploits the symmetry properties of both the spline basis functions and the Riesz–Caputo operator, leading to an efficient approach for solving the fractional differential problem. We discuss the advantages of using Greville Abscissae as collocation points, and compare this choice with other possible distributions of points. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Monitoring soil salinity in coastal wetlands with Sentinel-2 MSI data: Combining fractional-order derivatives and stacked machine learning models

Monitoring soil salinity is essential for understanding the behavior of coastal wetland ecosystems and implementing effective management strategies. Despite the advantages of the Multi-Spectral Instrument (MSI) data for large-scale, high-frequency soil salinity monitoring, challenges remain in data preprocessing and model construction. We combined fractional-order derivative (FOD) technology with stacked machine learning models to monitor and map soil salinity using Sentinel-2 MSI data. The base models included Elastic Net Regression, Support Vector Regression, Artificial Neural Network, Extreme Gradient Boosting, and Random Forest, with Non-Negative Least Squares as the meta-learner. The results showed that low-order FOD enhanced image gradients and maintained a high peak signal-to-noise ratio, thereby improving the correlation with soil salinity. Notably, the 0.25-order FOD showed the best performance, increasing the correlation coefficient with soil salinity by up to 13 %. The stacked machine learning models effectively combined the strengths of different base models, enhancing prediction accuracy by more than 8 % compared to single models. Furthermore, combining stacked models with FOD further improved prediction accuracy, with an increase in R² of up to 9 %. The combination of 0.25-order FOD and the stacked machine learning model achieved the best performance (R² = 0.82, RMSE = 10.19 ppt, RPD = 2.38, RPIQ = 4.69). This approach provides a reference for rapid and effective large-scale digital mapping of soil salinity in coastal wetlands.

A novel fast tempered algorithm with high-accuracy scheme for 2D tempered fractional reaction-advection-subdiffusion equation

In this article, we propose a novel and computationally efficient difference scheme for evaluating the Caputo tempered fractional derivative (TFD). Our approach introduces a new fast tempered FλL2−1σ difference method, achieving a higher convergence rate of order O(Δt3−α). Specifically, we apply this method to a class of two-dimensional tempered-fractional reaction-advection-subdiffusion equations (2D-TF-RASDE) characterized by the tempered parameter λ and a tempered fractional derivative of order α, (where 0<α<1). The novel fast tempered scheme is based on the sum of exponents (SOE) technique. Additionally, we develop a novel high order compact finite difference (CFD) scheme using an alternating direction implicit (ADI) formulation for the 2D-TF-RASDE. We investigate the stability and convergence of this FLλ2−1σ-ADI-CD scheme. Numerical simulations reveal a convergence order of O(Δt1+α+hϰ4+hy4+ϵ) under robust regularity assumptions underlining the scheme's superior computational efficiency. Furthermore, our computational results align well with theoretical analysis, demonstrating both accuracy and reduced computational complexity and storage requirements as compared to the standard tempered Lλ2−1σ-ADI-CD scheme. Notably, the fast tempered FLλ2−1σ-ADI-CD scheme exhibits competitive performance and reduced CPU time relative to the standard Lλ2−1σ-ADI-CD scheme, thereby offering a distinct advantage for efficiently solving complex fractional differential equations.

A Special Note According to Possible Applications of Fractional-Order Calculus for Various Special Functions

The main aim of this special study is to recall certain information about fractional (arbitrary) order calculus, which has wide and fruitful applications in science and engineering. Then, it aims to consider various essential definitions related to fractional order integrals and derivatives for stating and proving some results, as well as to present some of their possible applications to the attention of related researchers.