Non-integer Order Research Articles

Witte’s conditions for uniqueness of solutions to a class of Fractal-Fractional ordinary differential equations

In this paper, Witte's conditions for the uniqueness solution of nonlinear differential equations with integer and non-integer order derivatives are investigated. We present a detailed analysis of the uniqueness solutions of four classes of nonlinear differential equations with nonlocal operators. These classes include classical and fractional ordinary differential equations in fractal calculus. For each case, theorems and lemmas and their proofs are presented in detail.

Approximate Analytic Frequency of Strong Nonlinear Oscillator

In this paper, a new analytic expression for the frequency of vibration of a strong nonlinear polynomial-type oscillator is introduced. The method for frequency calculation is based on the transformation of the nonlinear oscillators into linear ones using the equality of their amplitudes and periods of vibration. The frequency of the linear oscillator is assumed to be the sum of frequencies corresponding to each nonlinearity in the original oscillator separately, i.e., the sum of frequencies of truly nonlinear oscillators. The obtained frequency is a complex function of amplitude, coefficient and order of nonlinearity. For simplification, the frequencies of the truly nonlinear oscillators are modified as power order functions of the exact frequency of the cubic oscillator which is linearly dependent on the amplitude of vibration. In this paper, the approximate frequency expression is developed for the harmonic any-order nonlinear oscillator and oscillators with the sum of polynomial nonlinearities. The accuracy of the obtained frequencies is tested on the examples of non-integer order nonlinear oscillators and also on a quadratic-cubic oscillator. The difference between the analytical and exact, numerically obtained results is negligible. The suggested approximate frequency expression has a simple algebraic form and is suitable for application by engineers and technicians.

Predictive modeling of hepatitis B viral dynamics: a caputo derivative-based approach using artificial neural networks

A fractional model for the kinetics of hepatitis B transmission was developed. The hepatitis B virus significantly affects the world’s economic and health systems. Acute and chronic carrier phases play a crucial part in the spread of the HBV infection. The Hepatitis B infection can be spread by chronic carriers even though they show no symptoms. In this article, we looked into the Hepatitis B virus’s various stages of infection-related transmission and built a nonlinear epidemic. Then, a fractional hepatitis B virus model using a Caputo derivative and vaccine effects is created. First, we determined the proposed model’s essential reproductive value and equilibria. With the aid of Fixed Point Theory, a qualitative analysis of the problem’s approximative root has been produced. The Adams-Bashforth predictor-corrector scheme is used to aid in the iterative approximate technique’s evaluation of the fractional system under consideration that has the Caputo derivative. In the final section, a graphical representation compares various noninteger orders and displays the discovered scheme findings. In this study, we’ve utilized Artificial Neural Network (ANN) techniques to partition the dataset into three categories: training, testing, and validation. Our analysis delves deep into each category, comprehensively examining the dataset’s characteristics and behaviors within these divisions. The study comprehensively analyzes the fractional HBV transmission model, incorporating both mathematical and computational approaches. The findings contribute to a better understanding of the dynamics of HBV infection and can inform the development of effective public health interventions.

Fractional Derivative of Hyperbolic Function

Fractional derivative is a generalization of ordinary derivative with non-integer or fractional order. This research presented fractional derivative of hyperbolic function (hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, and hyperbolic cosecant) with order constraint . The hyperbolic function is presented in Maclaurin series form. Then, the fractional derivative can be determined by using definition of Riemann-Liouville fractional derivative. The result is simulated by using Matlab software

Online measurement of wear depth based on displacement signal of vertical tester

This paper proposes an accurate and applicable online measurement method of wear depth using displacement signal for vertical testers under mixed lubrication. Firstly, displacement signal is revealed to be composed of temperature drift and Brownian motion. Secondly, the first order Taylor expansion of the Brownian motion is revealed to be the unified form of wear models. Further, the online measurement method is established by introducing non-integer order and amplitude coefficient to eliminate systematic randomness. Finally, the method is verified by graphite-metal tribo-pairs on a standard tester and a high speed friction and wear testing bench, relative errors between online and offline measurement results are less than 10 %, extension errors by non-integer order within its application range are less than 2 %.

A deterministic design approach of tilt integral derivative controller for integer and fractional-order system with time delay

A tilt integral derivative (TID) controller modifies the proportional integral derivative (PID) controller in the fractional domain. It converts the proportional gain as a function of frequency and is thereby capable of achieving optimal system response. The usual practice for the parameter estimation of the TID controller is by minimization of the error-based objective functions using optimization techniques. Although precise results can be achieved, these nature-inspired algorithms are stochastic and hence produce different solutions during different iterations. Therefore, a comparative statistical study is usually necessary to validate the best possible result. This study shows a deterministic analytical procedure for the paramssseter estimation of TID controllers. The magnitude and phase angle criteria, along with the frequency-domain loop shaping specifications, are used for the explicit evaluation of the TID parameters. Because of its model-independent nature, this tuning strategy can be used for a variety of integral and nonintegral order systems with different plant structures. In this article, the authenticity of the applied procedure is demonstrated through suitable numerical examples. The complexity of the design problem is enhanced by using it for both integer and non-integer (fractional) order plus time-delay systems. Further, the robustness of the control system in the presence of a TID controller was examined under the influence of external parameters and input reference changes. Simulation studies validate the supremacy of TID controllers over PID controllers in terms of reference tracking and disturbance rejection capabilities.

Modeling the Dynamics of Supercapacitors by Means of Riemann–Liouville Integral Definition

The application of fractional calculus to obtain dynamic models for supercapacitors represents an alternative approach to obtaining simpler and more accurate models. This paper presents a model for the supercapacitor in the time domain, based on the use of the fractional or non-integer order integral. This fractional model is compared with the conventional simple model, which is typically used in industrial applications. This fractional integral-based model provides satisfactory fits in relation to the number of parameters used in the model. Furthermore, an interpretation of the effect of the application of fractional integration is presented for constant current charging and discharging processes at constant current, using the Riemann–Liouville definition for the non-integer order integral. Supercapacitors are devices that exhibit non-linear behavior, with a distinct charging and discharging operation. There are several methods of dynamic analysis for the characterization of supercapacitors. The information extracted from these methods is essential for understanding the behavior of supercapacitors and, thus, ensuring that processes involving supercapacitors are as efficient as possible. This paper presents a dynamic analysis based on charge and discharge operations with constant currents. The conclusion is that the fractional model provides fairly accurate fits.

Fractional Calculus Meets Neural Networks for Computer Vision: A Survey

Traditional computer vision techniques aim to extract meaningful information from images but often depend on manual feature engineering, making it difficult to handle complex real-world scenarios. Fractional calculus (FC), which extends derivatives to non-integer orders, provides a flexible way to model systems with memory effects and long-term dependencies, making it a powerful tool for capturing fractional rates of variation. Recently, neural networks (NNs) have demonstrated remarkable capabilities in learning complex patterns directly from raw data, automating computer vision tasks and enhancing performance. Therefore, the use of fractional calculus in neural network-based computer vision is a powerful method to address existing challenges by effectively capturing complex spatial and temporal relationships in images and videos. This paper presents a survey of fractional calculus neural network-based (FC NN-based) computer vision techniques for denoising, enhancement, object detection, segmentation, restoration, and NN compression. This survey compiles existing FFC NN-based approaches, elucidates underlying concepts, and identifies open questions and research directions. By leveraging FC’s properties, FC NN-based approaches offer a novel way to improve the robustness and efficiency of computer vision systems.

Mathematical analysis using fractional operator to study the dynamics of dengue fever

Researchers and analysts are intensively studying modeling contagious diseases using non-integer order derivatives to enhance understanding and prediction. Taking this idea forward, in this study, we consider the fractional model for dengue fever disease. The Hilfer fractional model was initially formulated to address epidemic dynamics. This study employed the numerical technique, the Laplace homotopy analysis transform method (LHATM), to examine the fractional dengue fever model for analysis. We employed homotopy analysis and Laplace transform to formulate the proposed technique. There is also a consideration of the uniqueness and convergence of the solution. Utilizing MATLAB21a, numerical simulation for different integer and non-integer orders within the interval (0, 1) has been drawn.

Fractional neutron point kinetics model for reactivity transients of the NuScale and comparison with the classical kinetics approach

In this work, the behavior of the neutron and natural circulation parameters for Reactivity Transients in the PWR reactor called NuScale are analyzed through the novel system equations. It model includes non-integer and ordinary order differentials coupling the neutron point kinetic with the thermos-hydraulic core of the reactor. Transient processes of increase and decrease of the feed flow and the coolant inlet temperature are analyzed to identify the feedback of the thermohydraulic effects with the neutron phenomena in a non-integer order scheme. In the results, it is observed that for short times the power and reactivity have the same behavior with the classical model and the non-integer order model for all values of the anomalous diffusion coefficient simulated. However, for long times of simulation, there is more reactivity and less power, in contrast to the classical model, when reducing the value of the anomalous diffusion coefficient these counter-intuitive effects are increased. Additionally, a delay is observed between the change in reactivity and power. In conclusion, it is observed that sub-diffusive processes are relevant when there are slow changes in reactivity, which can be analyzed through the Fractional Order Neutron Density Equation.

Applications of Fractional-Order Calculus in Robotics

Fractional calculus, a branch of mathematical analysis, extends traditional calculus that encompasses integrals and derivatives of non-integer orders [...]

On the stability of a Caputo fractional order predator-prey framework including Holling type-II functional response along with nonlinear harvesting in predator

In this study, we look at a fractional-order two-species predator-prey framework that uses Caputo Sense's fractional derivative to try to understand its dynamics, which include a simplified Holling type II functional response and nonlinear harvesting in the predator. Initially, we establish certain mathematical facts, such as the solutions of fractional order dynamical systems being unique, non-negative, bounded, and existing. The dissipativeness of the FDE system solution is addressed. Our investigation also includes a look at the local stability requirements for every possible equilibrium point. As a bifurcation constraint, the order of derivatives has been considered in our discussion of the existence condition of Hopf bifurcation. Our analysis has been strengthened by the incorporation of fractional Hopf bifurcation, and the dynamics of the proposed model are influenced by selective non-linear harvesting. Furthermore, the investigation focuses on the impact of the predator's pace of harvesting about the complexities of the prey-predator relationship. Through the use of certain setting values, empirical evidence indicates as the non-integer order framework displays stable to unstable, when the non-integer order is increased. The analytical findings are shown by various instances presented in the numerical part.

Experimentally validated analytical single parametric FOTID control for time-delayed fractional order processes

In current studies, non-integer order (NIO) plants with dead time have gained the significant attention of many researchers in the area of control theory. However, the design methods for fractional order controller tuning suffer from a lack of direct systematic tuning strategies due to a higher number of tuneable parameters. Most of the works on such design are optimisation-based whereas the analytical solutions are achieved with assumptions and approximations leading to compromised closed-loop robustness. This paper proposes a simple analytical design of a fractional-order tilt-integral-derivative (FOTID) controller for time-delayed NIO processes with a single tuneable parameter. Explicit tuning rules in terms of plant parameters are derived with user-defined stability margins. To prove the efficiency of the proposed control law, simulation comparisons with recently developed tuning rules are included. Lastly, the proposed control law is tested through experiments in a two-tank level loop for real-time validation.

The use of Hermite wavelet collocation method for fractional cancer dynamical system

Tumour is a very serious and dangerous health risk, which is defined as a mass or lump of tissue formalized by the aggregation of abnormal cells. In this paper, we have presented a comparative and chaotic investigation of tumour and effector cells using non-integer order tumour immune dynamical model. Also, we have looked at the interactions between different tumour cell populations and immunological composition using a model of a real-world medical research problem. Additionally, stability analysis of the proposed model is established with the help of fixed-point iteration. Further bifurcation diagrams, as well as phase portraits, have been used to study the proposed system numerically and to analyse its behaviour. For numerical simulations, we have used the Hermite wavelet operational matrix and Taufik-Atangana methods using Caputo fractional derivatives. models. The main objective of the paper is to convert nonlinear fractional systems into algebraic equations. Our findings will be valuable to biologists in cancer treatment.

An innovative approach to approximating solutions of fractional partial differential equations

The RPS-M (residual power series method) is a valuable technique for solving F-PDEs (fractional partial differential equations). However, the derivative of the residual function to obtain the coefficients of the series is required in RPS-M. This makes the application of the classical RPS-M limited to a certain extent due to the complexity of the derivation of the residual function for higher iterations. To overcome this obstacle, in this study, we present a simplified version of this approach with the help of Laplace transform that requires less computation and offers higher accuracy. This modified method does not require derivation as well as limit of the residual function to estimate the unknown coefficients of the series solution. To demonstrate its effectiveness, we apply the proposed method to nonlinear F-PDEs to obtain their semi-analytical solution. The obtained solutions exhibit excellent agreement when compared to results obtained using other established approaches. We have also provided the convergence analysis of the obtained solution. Furthermore, by comparing the outcomes for various values of the non-integer order σ, we observe that as the value approaches an integer order, the solution converges towards the exact solution.

Global Dynamics of a Social Hierarchy-Stratified Malaria Model: Insight from Fractional Calculus

In this study, a mathematical model for the transmission dynamics of malaria among different socioeconomic groups in the human population interacting with a susceptible-infectious vector population is presented and analysed using a fractional-order derivative of the Caputo type. The total human population is stratified into two distinguished classes of lower and higher income individuals, with each class further subdivided into susceptible, infectious, and recovered populations. The socio hierachy-structured fractional-order malaria model is analyzed through the application of different dynamical system tools. The theory of positivity and boundedness based on the generalized mean value theorem is employed to investigate the basic properties of solutions of the model, while the Banach fixed point theory approach is used to prove the existence and uniqueness of the solution. Furthermore, unlike the existing related studies, comprehensive global asymptotic dynamics of the fractional-order malaria model around both disease-free and endemic equilibria are explored by generalizing the usual classical methods for establishing global asymptotic stability of the steady states. The asymptotic behavior of the trajectories of the system are graphically illustrated at different values of the fractional (noninteger) order.

Numerical computation of fractional Bloch equation by using Jacobi operational matrix

In this work, we present a numerical scheme based on the operational matrix of fractional Caputo-Fabrizio (CF) integration for handling fractional Bloch equation (FBE) in nuclear magnetic resonance (NMR). The understanding of Bloch equation provides us a fundamental framework for describing magnetic resonance phenomena, facilitating breakthrough in diverse fields such as medical diagnostics, quantum computing and materials characterization. The non-integer order derivative and integration are presented in the Caputo-Fabrizio sense. To construct the operational matrix, Jacobi polynomial is used as a basis. The fractional Bloch equation is transformed into a set of algebraic equations by using the operational matrix. In order to examine the fractional order problem, we obtain an approximate solution for FBE and present the numerical results in graphical and tabular forms.

Investigating the Dynamic Behavior of Integer and Noninteger Order System of Predation with Holling’s Response

The paper’s primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor’s theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system’s behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter.

Fractional order modelling and optimal control of dual active bridge converters

As the construction of new power systems centred around renewable energy gains traction, the installed capacity of new energy sources has experienced explosive growth. Research has shown that the actual external characteristics of inductors and capacitors in circuits exhibit fractional order characteristics. As a core device in dual active bridge converters, the non-integer order of inductors and capacitors has an important impact on their dynamic performance, frequency domain characteristics, and control system design. This paper conducts modelling and control research on dual active bridge converters based on fractional-order calculus theory. First, based on the definition of fractional-order calculus, according to the operating mode of the fractional-order dual active bridge converter and the characteristics of fractional-order components, The fractional-order state space average model was established, and the influence of the order of inductance and capacitance on the amplitude frequency characteristics, phase frequency characteristics and dynamic performance of the converter was analyzed. Secondly, the average model and circuit model were built on the MATLAB/Simulink simulation platform, and the simulation results under different fractional orders were compared. Finally, in order to further improve the control performance of the converter, a fractional-order P I λ control strategy is designed based on the established state space average model of the fractional-order dual active bridge converter and based on the transfer function from the control to the output of the fractional-order dual active bridge converter. The research results show that the fractional-order model can more accurately describe the actual characteristics of the converter. In addition, the fractional-order controller P I λ enables the fractional-order dual active bridge converter to obtain better robustness and improve the dynamic performance of the converter.

Analysis and Applications of the Chaotic Hyperbolic Memristor Model with Fractional Order Derivative

This work presents a new five-dimensional fractional-order chaotic system that includes a feedback memristor. A Lorenz-Stenflo-based fractional-order chaotic system with five dimensions that contains feedback memristor dynamics is used to do this. This work focuses on chaotic models through the ABC fractional derivative, a concept we rigorously examine. We designed and usedsophisticated numerical algorithms to model fractional-order dynamics with the utmost precision to understand this system’s complex characteristics. These numerical approaches excel at non-integer order differentiation, which standard numerical methods struggle with. Our research uses fractional calculus to increase the system’s complexity and robustness, revealing its hyperchaotic nature. We demonstrate that the system is chaotic and stable over fractional orders using eigenvalues, Lyapunov exponents, Kaplan-Yorke dimensions, maximal exponents, phase portraits, and equilibrium points. Our conclusions are strengthened by using these numerical systems, intended for this work, and analytical tools. We improve our understanding of fractional-order chaotic systems with our research. It illuminates how to improve electrical integrations and create more sophisticated nonlinear dynamical systems. This work establishes the approach and insights for future research, guiding the development of new systems with enhanced dynamical features