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This paper proposes a modeling and analysis method for a Caputo–Fabrizio (C-F) definition-based fractional-order Boost converter with fractional-order inductive loads. The proposed method analyzes the system characteristics of a fractional-order circuit with three state variables. Firstly, this paper constructs a large signal model of a fractional-order Boost converter by taking advantage of the state space averaging method, providing accurate analytical solutions for the quiescent operating point and the ripple parameters of the circuit with three state variables. Secondly, this paper constructs a small signal model of the C-F definition-based fractional-order Boost converter by small signal linearization, providing the transfer function of the fractional-order system with three state variables. Finally, this paper conducts circuit-oriented simulation experiments where the steady-state parameters and the transfer function of the circuit are obtained, and then the effect of the order of capacitor, induced inductor, and load inductor on the quiescent operating point and ripple parameters is analyzed. The experimental results show that the simulation results are consistent with those obtained by the proposed mathematical model and that the three fractional orders in the fractional model with three state variables have a significant impact on the DC component and steady-state characteristics of the fractional-order Boost converter. In conclusion, the proposed mathematical model can more comprehensively analyze the system characteristics of the C-F definition-based fractional-order Boost converter with fractional-order inductive loads, benefiting the circuit design of Boost converters. [ABSTRACT FROM AUTHOR]

Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, we consider the generalized sequential boundary value problems of the Navier difference equations by using the post-quantum fractional derivatives of the Caputo-like type. We discuss on the existence theory for solutions of the mentioned (p ; q) -difference Navier problems in two single-valued and set-valued versions. We use the main properties of the (p ; q) -operators in this regard. Application of the fixed points of the ρ - θ -contractions along with the endpoints of the multi-valued functions play a fundamental role to prove the existence results. Finally in two examples, we validate our models and theoretical results by giving numerical models of the generalized sequential (p ; q) -difference Navier problems. [ABSTRACT FROM AUTHOR]

This paper is devoted to the research of the isomorphic multidimensional cyclic structure and multidimensional phase structure of the cyclic random process (CRP) and to its formation method, which enables a rigorous formalization of intuitive ideas concerning cyclic stochastic motion. The fundamental properties of the cyclic random process and analytical dependencies between the multidimensional cyclic structure, multidimensional phase structure and rhythm structure of the CRP have been established. This work shows that the CRP is able to take into account the cyclicity of multidimensional distribution functions of cyclic signals as well as the variability in the rhythm of the investigated signals. A subclass of the CRP is the periodic random process, which allows for the use of classical processing methods of cyclic signals with a regular rhythm. Based on a series of experiments, significant advantages of the CRP as a mathematical model of electrocardiographic signals (ECG) compared to the periodic random process are shown. [ABSTRACT FROM AUTHOR]

In this paper, we improved a mathematical model of monkeypox disease with a time delay to a crossover model by incorporating variable-order and fractional differential equations, along with stochastic fractional derivatives, in three different time intervals. The stability and positivity of the solutions for the proposed model are discussed. Two numerical methods are constructed to study the behavior of the proposed models. These methods are the nonstandard modified Euler Maruyama technique and the nonstandard Caputo proportional constant Adams-Bashfourth fifth step method. Many numerical experiments were conducted to verify the efficiency of the methods and support the theoretical results. This study's originality is the use of fresh data simulation techniques and different solution methodologies. [ABSTRACT FROM AUTHOR]

Fractional calculus is defined by expanded integer order integration and differentiation. In this paper, multiple mathematical models of a nonlinear dual-tank device are precisely formulated by fractional calculus. Using the accurate model, a multivariable fractional-order controller is designed for nonlinear devices. The merits of the fractional-order design include: (1) control of multivariable nonlinearities, (2) compensation of uncertainties, and (3) elimination of coupling effects. Simulations and experiments are conducted to verify the precision of the fractional order models and the effectiveness of the multivariable fractional-order control system design. [ABSTRACT FROM AUTHOR]

Breast cancer ranks among the most prevalent malignancies affecting the female population and is a prominent contributor to cancer-related mortality. Mathematical modeling is a significant tool that can be employed to comprehend the dynamics of breast cancer progression and dissemination and to formulate novel therapeutic approaches. This paper introduces a mathematical model of breast cancer that utilizes the Caputo–Fabrizio fractal-fractional derivative. The aim is to elucidate and comprehend the intricate dynamics governing breast cancer cells and cytotoxic T lymphocytes in the context of the fractional derivative. The derivative presented herein offers a broader perspective than the conventional derivative, as it incorporates the intricate fractal characteristics inherent in the process of tumor proliferation. The significance of this study lies in its contribution to a novel mathematical model for breast cancer, which incorporates the fractal characteristics of tumor development. The present model possesses the capability to investigate the impacts of diverse treatment strategies on the proliferation of breast cancer, as well as to formulate novel treatment strategies that exhibit enhanced efficacy. [ABSTRACT FROM AUTHOR]

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

In the literature, different approaches that are employed in designing automatic voltage regulators (AVRs) usually model the AVR as a single-input-single-output system, where the input is the generator reference voltage, and the output is the generator voltage. Alternately, it could be thought of as a double-input, single-output system, with the excitation voltage change serving as the additional input. In this paper, unlike in the existing literature, we designed the AVR system as a sextuple-input single-output (6ISO) system. The inputs in the model include the generator reference voltage, regulator signal change, exciter signal change, amplifier signal change, generator output signal change, and the sensor signal change. We also compared the generator voltage responses for various structural configurations and regulator parameter choices reported in the literature. The effectiveness of numerous controllers is investigated; the proportional, integral and differential (PID) controller, the PID with second-order derivative (PIDD2) controller, and the fractional order PID (FOPID) controller are the most prevalent types of controllers. The findings reveal that the regulator signal change and the generator output signal change significantly impact the generator voltage. Based on these findings, we propose a new approach to design the regulator parameter to enhance the response to generator reference voltage changes. This approach takes into consideration changes in the generator reference voltage as well as the regulator signal. We calculate the regulator settings using a cutting-edge hybrid technique called the Particle Swarm Optimization African Vultures Optimization algorithm (PSO–AVOA). The effectiveness of the regulator design technique and the proposed optimization algorithm are demonstrated. [ABSTRACT FROM AUTHOR]

The most commonly used model of solar cells is the single-diode model, with five unknown parameters. First, this paper proposes three variants of the single-diode model, which imply the voltage dependence of the series resistance, parallel resistance, and both resistors. Second, analytical relationships between the current and the voltage expressed were derived using the Lambert W function for each proposed model. Third, the paper presents a hybrid algorithm, Chaotic Snake Optimization (Chaotic SO), combining chaotic sequences with the snake optimization algorithm. The application of the proposed models and algorithm was justified on two well-known solar photovoltaic (PV) cells—RTC France solar cell and Photowatt-PWP201 module. The results showed that the root-mean-square-error (RMSE) values calculated by applying the proposed equivalent circuit with voltage dependence of both resistors are reduced by 20% for the RTC France solar cell and 40% for the Photowatt-PWP201 module compared to the standard single-diode equivalent circuit. Finally, an experimental investigation was conducted into the applicability of the proposed models to a solar laboratory module, and the results obtained proved the relevance and effectiveness of the proposed models. [ABSTRACT FROM AUTHOR]

The modeling of biological processes has increasingly been based on fractional calculus. In this paper, a novel fractional-order model is used to investigate the epidemiological impact of vaccination measures on the co-dynamics of viral hepatitis B and COVID-19. To investigate the existence and stability of the new model, we use some fixed point theory results. The COVID-19 and viral hepatitis B thresholds are estimated using the model fitting. The vaccine parameters are plotted against transmission coefficients. The effect of non-integer derivatives on the solution paths for each epidemiological state and the trajectory diagram for infected classes are also examined numerically. An infection-free steady state and an infection-present equilibrium are achieved when R 0 < 1 and R 0 > 1 , respectively. Similarly, phase portraits confirm the behaviour of the infected components, showing that, regardless of the order of the fractional derivative, the trajectories of the disease classes always converge toward infection-free steady states over time, no matter what initial conditions are assumed for the diseases. The model has been verified using real observations. [ABSTRACT FROM AUTHOR]

A nonlinear mathematical model of COVID-19 containing asymptomatic as well as symptomatic classes of infected individuals is considered and examined in the current paper. The largest eigenvalue of the next-generation matrix known as the reproductive number is obtained for the model, and serves as an epidemic indicator. To better understand the dynamic behavior of the continuous model, the unconditionally stable nonstandard finite difference (NSFD) scheme is constructed. The aim of developing the NSFD scheme for differential equations is its dynamic reliability, which means discretizing the continuous model that retains important dynamic properties such as positivity of solutions and its convergence to equilibria of the continuous model for all finite step sizes. The Schur–Cohn criterion is used to address the local stability of disease-free and endemic equilibria for the NSFD scheme; however, global stability is determined by using Lyapunov function theory. We perform numerical simulations using various values of some key parameters to see more characteristics of the state variables and to support our theoretical findings. The numerical simulations confirm that the discrete NSFD scheme maintains all the dynamic features of the continuous model. [ABSTRACT FROM AUTHOR]

Fractals are geometric shapes and patterns that can describe the roughness (or irregularity) present in almost every object in nature. Many fractals may repeat their geometry at smaller or larger scales. This paper is the second (and last) part of a series of two papers dedicated to an eclectic survey of fractals describing the infinite complexity and amazing beauty of fractals from historical, theoretical, mathematical, aesthetical and technological aspects, including their diverse applications in various fields. In this article, our focus is on engineering, industrial, commercial and futuristic applications of fractals, whereas in the first part, we discussed the basics of fractals, mathematical description, fractal dimension and artistic applications. Among many different applications of fractals, fractal landscape generation (fractal landscapes that can simulate and describe natural terrains and landscapes more precisely by mathematical models of fractal geometry), fractal antennas (fractal-shaped antennas that are designed and used in devices which operate on multiple and wider frequency bands) and fractal image compression (a fractal-based lossy compression method for digital and natural images which uses inherent self-similarity present in an image) are the most creative, engineering-driven, industry-oriented, commercial and emerging applications. We consider each of these applications in detail along with some innovative and future ready applications. [ABSTRACT FROM AUTHOR]

This paper deals with the longitudinal movement control of an airplane (pitch angle) using fractional order adaptive controllers (FOACs). It shows the improvements achieved in the plane's behavior, in terms of the minimization of a given performance index. At the same time, less control effort is needed to accomplish the control objectives compared with the classic integer order adaptive controllers (IOACs). In this study, fractional order direct model reference adaptive control (FO-DMRAC) is implemented at the simulation level, and exhibits a better performance compared with the classic integer order (IO) version of the DMRAC (IO-DMRAC). It is also shown that the proposed control strategy for FO-DMRAC reduces the resultant system control structure down to a relative degree 2 system, for which the control implementation is simpler and avoids oscillations during the transient period. Moreover, it is interesting to note that this is the first time that an FOAC with fractional adaptive laws is applied to the longitudinal control of an airplane. A suitable model for the longitudinal movement of the airplane related to the pitch angle θ as the output variable with the lifter angle (δe) as the control variable, is first analyzed and discussed to obtain a reliable mathematical model of the plane. All of the other input variables acting on the plane are considered as perturbations. For certain operating conditions defined by the flight conditions, an FO-DMRAC is designed, simulated, and analyzed. Furthermore, a comparison with the implementation of the classical adaptive general direct control (relative degree ≥ 2) is presented. The boundedness and convergence of all of the signals are theoretically proven based on the new Lemma 3, assuring the boundedness of all internal signals ω(t). [ABSTRACT FROM AUTHOR]

Vaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model's dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine on coronavirus infection disease. We analyze the model under their equilibrium points. The model under the equilibrium points is discussed and proven that it is locally asymptotically stable if R v is lesser than unity. We present the backward bifurcation hypothesis of the model and show that there is a parameter β 2 that causes the backward bifurcation in the imperfect vaccine model. We show certain assumptions when ψ = 1 for the imperfect vaccine case; the model is both stable globally asymptotically at the disease-free ( R 0 ≤ 1 ) and endemic cases ( R 0 > 1 ). By using infected cases from the recent wave throughout Pakistan, we shall estimate the model parameters and calculate the numerical value of the basic reproductive number R 0 ≈ 1.2591 . We present the comprehensive graphical results for the realistic parameter values and show many useful suggestions regarding the elimination of the infection from society. The vaccination efficacy that provides an important role in disease elimination is discussed in detail. [ABSTRACT FROM AUTHOR]

Chemical graph theory (CGT) is a field of mathematical science that applies classical graph theory to chemical structures and processes. Chemical graphs are the principal data format used in cheminformatics to illustrate chemical interactions. Several researchers have addressed boundary-value problems using star graphs. Star graphs were used since their method requires a central point linked to other vertices but not to itself. Our objective is to expand the mechanism by introducing the idea of an isobutane graph that has the chemical formula C 4 H 10 and CAS number 75-28-5. By using the appropriate fixed point theory findings, this paper investigates the existence of solutions to fractional boundary value problems of Caputo type on such graphs. Additionally, two examples are provided to strengthen our important conclusions. [ABSTRACT FROM AUTHOR]

LFM signals are widely applied in radar, communication, sonar and many other fields. LFM signals received by these systems contain a lot of noise and outliers. In order to suppress the interference of strong impulse noise on target signals and realize the accurate estimation of LFM signal parameters, the impulse noise of echo signals need to be filtered. In this paper, to solve the problem of poor performance of LFM signal parameter estimation based on fractional Fourier transform in impulse noise, alpha stable distribution is used to establish the mathematical model of impulse noise. The proposed fastest tracking differentiator with an adaptive tracking factor is used to suppress the strong impulse noise, and fractional Fourier transform is used to estimate the parameter of the LFM signals. The experimental results show that the proposed fastest tracking differentiator with an adaptive tracking factor has a good filtering performance. It can effectively filter the impulse noise in the echo signal and allows the FrFT method to accurate estimate the parameters of the LFM signals in strong impulse noise. [ABSTRACT FROM AUTHOR]

This article deals with the random sequential adsorption (RSA) of 2D disks of the same size on fractal surfaces with a Hausdorff dimension 1 < d < 2. According to the literature and confirmed by numerical simulations in the paper, the high coverage regime exhibits fractional dynamics, i.e., dynamics in t1/d where d is the fractal dimension of the surface. The main contribution this paper is that it proposes to capture this behavior with a particular class of nonlinear model: a driftless control affine model. [ABSTRACT FROM AUTHOR]

Some researchers believe fractional differential operators should not have a non-singular kernel, while others strongly believe that due to the complexity of nature, fractional differential operators can have either singular or non-singular kernels. This contradiction in thoughts has led to the publication of a few papers that are against differential operators with non-singular kernels, causing some negative impacts. Thus, publishers and some Editors-in-Chief are concerned about the future of fractional calculus, which has generally brought confusion among the vibrant and innovative young researchers who desire to apply fractional calculus within their respective fields. Thus, the present work aims to develop a model based on a stochastic process that could be utilized to portray the effect of arbitrary-order derivatives. A nonlinear perturbation is used to study the proposed stochastic model with the help of white noises. The required condition(s) for the existence of an ergodic stationary distribution is obtained via Lyapunov functional theory. The finding of the study indicated that the proposed noises have a remarkable impact on the dynamics of the system. To reduce the spread of a disease, we imposed some control measures on the stochastic model, and the optimal system was achieved. The models both with and without control were coded in MATLAB, and at the conclusion of the research, numerical solutions are provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new Caputo-type modification of the Erdélyi-Kober fractional derivative. We pay attention to how to formulate representations of Erdélyi-Kober fractional integral and derivatives operators. Then, some properties of the new modification and relationships with other Erdélyi-Kober fractional derivatives are derived. In addition, a numerical method is presented to deal with fractional differential equations involving the proposed Caputotype Erdélyi-Kober fractional derivative. We hope the presented method will be widely applied to simulate such fractional models. [ABSTRACT FROM AUTHOR]

This paper proposes a nonlocal fractional peridynamic (FPD) model to characterize the nonlocality of physical processes or systems, based on analysis with the fractional derivative model (FDM) and the peridynamic (PD) model. The main idea is to use the fractional Euler-Lagrange formula to establish a peridynamic anomalous diffusion model, in which the classical exponential kernel function is replaced by using a power-law kernel function. Fractional Taylor series expansion was used to construct a fractional peridynamic differential operator method to complete the above model. To explore the properties of the FPD model, the FDM, the PD model and the FPD model are dissected via numerical analysis on a diffusion process in complex media. The FPD model provides a generalized model connecting a local model and a nonlocal model for physical systems. The fractional peridynamic differential operator (FPDDO) method provides a simple and efficient numerical method for solving fractional derivative equations. [ABSTRACT FROM AUTHOR]

This paper proposes a fractional-order economic growth model with time delay based on the Solow model to describe the economic growth path and explore the underlying growth factors. It effectively captures memory characteristics in economic operations by adding a time lag to the capital stock. The proposed model is presented in the form of a fractional differential equations system, and the sufficient conditions for the local stability are obtained. In the simulation, the theoretical results are verified and the sensitivity analysis is performed on individual parameters. Based on the proposed model, we predict China's GDP in the next thirty years through optimization and find medium-to-high-speed growth in the short term. Furthermore, the application results indicate that China is facing the disappearance of demographic dividend and the deceleration of capital accumulation. Therefore, it is urgent for China to increase the total factor productivity (TFP) and transform its economic growth into a trajectory dependent on TFP growth. [ABSTRACT FROM AUTHOR]

In this paper, the new mathematical correlation of two quantum systems that were initially allowed to interact and then separated is being formulated and analyzed. These correlations are illustrated by many examples and are also connected with fractals at a certain level. The main idea of the paper arises from the EPR paradox, the paradox of Einstein, Podolsky, and Rosen that occurs when the measurement of a physical observable performed on one system has an immediate effect on the other separate system being entangled with it. That is a physical phenomenon, especially when the particles are separated by a large distance. In this paper, we define distant correlations as the advanced method for the exact interpretation of strong connection and influence among those particles even when they are widely separated. On the given topological space (X , τ) , we define a notion of τ-metric such that the set X is a τ-metric space and we prove some properties of these spaces. By using this new proposed model, we nullify the contradiction that appears in the EPR paradox. An illustrative example involving fractals is given. This innovative mathematical approach to this physical phenomenon may be attractive for future research in the field of quantum physics. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of reduced-order H ∞ filter design for singular fractional-order systems with order 0 < α < 1 . It provides necessary and sufficient conditions for designs of both reduced-order H ∞ filters and zeroth-order H ∞ filters. When reduced to special cases, the present results are shown to include those in recent works as special cases. Illustrative examples are presented to demonstrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

A single-species population model exhibiting a symmetric slow variation for the carrying capacity and intrinsic growth rate is evaluated explicitly. However, it is unrealistic to eliminate the possibility of a clear separation in the evolution of the biotic environmental elements; thus, this paper considers the situation where these elements have a hierarchical variation on the time scales. Accordingly, two particular situations are recognized, which are the carrying capacity varies faster than the growth rate and vice versa. Applying the multi-time scaling technique in such a system provides a small parameter, which leads us to construct analytical approximate expressions for the population behavior, using the perturbation approach. Such approximations display very good agreement with the numerical simulations. [ABSTRACT FROM AUTHOR]

Fractional-order mathematical modelling of physical phenomena is a hot topic among various researchers due to its many advantages over positive integer mathematical modelling. In this context, the appropriate solutions of such fractional-order physical modelling become a challenging task among scientists. This paper presents a study of unsteady free convection fluid flow and heat transfer of Maxwell fluids with the presence of Clay nanoparticle modelling using fractional calculus. The obtained model was transformed into a set of linear nondimensional, partial differential equations (PDEs). The finite difference scheme is proposed to discretize the obtained set of nondimensional PDEs. The Maple code was developed and executed against the physical parameters and fractional-order parameter to explain the behavior of the velocity and temperature profiles. Some limiting solutions were obtained and compared with the latest existing ones in literature. The comparative study witnesses that the proposed scheme is a very efficient tool to handle such a physical model and can be extended to other diversified problems of a complex nature. [ABSTRACT FROM AUTHOR]