Fractional Differential Model Research Articles

Existence and uniqueness of well-posed fractional boundary value problem.

In this paper, the existence and uniqueness of solution for a fractional differential model involving well-posed boundary conditions and implicit fractional differential equation is considered. The desired goals are achieved by using Banach contraction principle and Scheafer's fixed point theorem. To show the results applicability some examples are presented. The basic mathematical concept of well-posed fractional boundary value issues is investigated in this study. It dives into the existence and uniqueness of these difficulties, offering light on the conditions that allow for both practical and singular solutions. This study contributes to a better knowledge of fractional calculus and its applications in a variety of scientific and technical areas, giving significant insights for both scholars and practitioners.

A study of nonlinear fractional-order biochemical reaction model and numerical simulations

This article depicts an approximate solution of systems of nonlinear fractional biochemical reactions for the Michaelis–Menten enzyme kinetic model arising from the enzymatic reaction process. This present work is concerned with fundamental enzyme kinetics, utilised to assess the efficacy of powerful mathematical approaches such as the homotopy perturbation method (HPM), homotopy analysis method (HAM), and homotopy analysis transform method (HATM) to get the approximate solutions of the biochemical reaction model with time-fractional derivatives. The Caputo-type fractional derivatives are explored. The proposed method is implemented to formulate a fractional differential biochemical reaction model to obtain approximate results subject to various settings of the fractional parameters with statistical validation at different stages. The comparison results reveal the complexity of the enzyme process and obtain approximate solutions to the nonlinear fractional differential biochemical reaction model.

A Caputo fractional derivative dynamic model of hepatitis E with optimal control based on particle swarm optimization

Hepatitis E, as a zoonotic disease, has been a great challenge to global public health. Therefore, it has important research value and practical significance for the transmission and control of hepatitis E virus (HEV). In the exploration of infectious disease transmission dynamics and optimal control, mathematical models are often applied. Among them, the fractional differential model has become an important and practical tool because of its good memory and genetic characteristics. In this paper, an HEV propagation dynamic model is constructed by the Caputo fractional derivative. First, the properties of the model are analyzed, including the existence, non-negativity, boundedness, and stability of the equilibrium points. Then, from the perspective of fractional optimal control (FOC), control measures were proposed, including improving the awareness and prevention of hepatitis E among susceptible people, strengthening the treatment of infected people, and improving environmental hygiene. Then, an FOC model of HEV was constructed. After analyzing the necessary conditions for optimality, the particle swarm optimization is introduced to optimize the control function. In addition, four control strategies are applied. Finally, the numerical simulation is completed by the fractional Adams–Bashforth–Moulton prediction correction algorithm. The four strategies and no control were compared and analyzed. The numerical simulation results of different fractional orders are also compared and analyzed. The results illustrate that the optimal strategy, compared with no control, reduces the HEV control time by nearly 60 days. Therefore, this method would contribute to the study of HEV transmission dynamics and control mechanisms, thus contributing to the development of global public health.

Convergence analysis of the effects of antiviral drug treatment in the fractional differential model of HIV-1 infection of CD4+ T-cells

ABSTRACT The time fractional-order nonlinear model of HIV-1 infection of T-cells with and without the rate of virus-producing cells’ death is very important in mathematical modeling. This paper presents proofs of the existence and uniqueness theorems using the Banach fixed point theorem, along with new approximate analytical solutions to two nonlinear fractional Equationuations characterizing HIV-1 infection in CD 4 + T-cells. We use a combination of two methods, namely the natural transform method to deal with the linear terms and the Adomian decomposition method to deal with the nonlinear terms. There are two nonlinear numerical examples presented in this work, namely, we find approximate solutions to the time fractional-order nonlinear model of HIV-1 infection of T cells with and without the rate of virus-producing cells’ death using an effective method called the Adomian natural decomposition method (ANDM). The present approach, which has numerous applications in the science and engineering fields, is a great alternative to the many existing methods for solving systems of differential Equationuations. It also holds great promise for additional real-world applications. The Mathematica 12 package is used to carry out some of the nonlinear term calculations and the graphs presented in this work.

Parameter Identification of Fractional Order Partial Differential Equation Model Based on Polynomial–Fourier Method

Parameter Identification of Fractional Order Partial Differential Equation Model Based on Polynomial–Fourier Method

A FRACTIONAL DIFFERENTIAL EQUATION MODEL FOR THE SPREAD OF POTATO LEAF ROLL VIRUS (PLRV) ON POTATOES

Potatoes infected with the PLRV virus will experience a decrease in production up to 90%. In this paper, The PLRV distribution fractional differential equation model with potato and vector populations is reformulated by adding one new parameter, namely the rate of vector death due to predators. The model is divided into susceptible and infected classes. The PLRV dispersion model was developed and converted to a fractional order form for 0<σ ≤ 1. Next, the invariant region, positive solutions, basic reproduction number, equilibrium point, and stability were determined. Based on the stability analysis, it is shown that the stability of the disease-free equilibrium point is locally stable and globally stable if the basic reproduction number (R0)<1, and the stability of the endemic equilibrium point is globally stable if the basic reproduction number (R0)>1. Numerical solutions were also carried out to determine the effect of several parameters on the PLRV distribution model on potatoes. The numerical solution results show that the elimination rate of infected potatoes and the infection rate of potatoes have a significant role in controlling the spread of PLRV in potatoes.

Finite difference discretization for one-dimensional higher-order integral fractional Laplacian and its application

A simple and easy-to-implement discrete approximation is proposed for one-dimensional higher-order integral fractional Laplacian (IFL), and our method is applied to discrete the fractional biharmonic equation, multi-term fractional differential model and fractal KdV equation. Based on the generating function, a fractional analogue of the central difference scheme to higher-order IFL is provided, the convergence of the discrete approximation is proved. Extensive numerical experiments are provided to confirm our analytical results. Moreover, some new observations are discovered from our numerical results.

Deterministic-stochastic analysis of fractional differential equations malnutrition model with random perturbations and crossover effects

To boost the handful of nutrient-dense individuals in the societal structure, adequate health care documentation and comprehension are permitted. This will strengthen and optimize the well-being of the community, particularly the girls and women of the community that are welcoming the new generation. In this article, we extensively explored a deterministic-stochastic malnutrition model involving nonlinear perturbation via piecewise fractional operators techniques. This novel concept leads us to analyze and predict the process from the beginning to the end of the well-being growth, as it offers the possibility to observe many behaviors from cross over to stochastic processes. Moreover, the piecewise differential operators, which can be constructed with operators such as classical, Caputo, Caputo-Fabrizio, Atangana-Baleanu and stochastic derivative. The threshold parameter is developed and the role of malnutrition in society is examined. Through a rigorous analysis, we first demonstrated that the stochastic model’s solution is positive and global. Then, using appropriate stochastic Lyapunov candidates, we examined whether the stochastic system acknowledges a unique ergodic stationary distribution. The objective of this investigation is to design a nutritional deficiency in pregnant women using a piecewise fractional differential equation scheme. We examined multiple options and outlined numerical methods of coping with problems. To exemplify the effectiveness of the suggested concept, graphical conclusions, including chaotic and random perturbation patterns, are supplied. Consequently, fractional calculus’ innovative aspects provide more powerful and flexible layouts, enabling us to more effectively adapt to the system dynamics tendencies of real-world representations. This has opened new doors to readers in different disciplines and enabled them to capture different behaviors at different time intervals.

Plants and age-structured pest dynamics with natural enemy as control strategy: A fractional differential equations model

In recent years, plant pests have drawn much attention from researchers. Biological control proved an efficient replacement for chemical and other pest control techniques. The present study proposes the cognition of an immature pest population unaffected by the natural enemy and grows exponentially by the harvesting effort of the mature pest population to plants. The fractional calculus environment proposes A four-compartment: plant, immature and mature pest, and natural enemy model. Equilibrium points and the system’s stability are studied theoretically and numerically. Simulation study for different values of fractional order generates the optimum solution. The suggested method is much more benignant than the existing method of ordinary differential equations as we can figure out the infrequent changes in the variable using a fractional derivative. Creating the memory effect in the system using fractional derivatives makes it more realistic.

Thermal–viscoelastic analysis of polymethyl methacrylate using a fractional differential viscoelastic model

Vacuum forming is used to manufacture large molded parts. As forming conditions have a significant effect on the dimensional accuracy, these should be determined accordingly. In this study, a geometric nonlinear creep analysis of polymethyl methacrylate (PMMA), which is a common thermoplastic resin, was carried out at the target temperature of 393.15[Formula: see text]K and target strain of approximately 50% for vacuum forming. The proposed fractional differential viscoelastic model was extended to a three-element model, consisting of a single hyperelastic spring and two fractional differential (FD) models. It was further extended by time–temperature superposition (TTS) for thermo-viscoelastic analysis. The model determined all material constants by measuring the temperature/frequency sweeps at small strain amplitudes of 0.01% using dynamic mechanical analysis (DMA). Numerical analysis confirmed the validity of the proposed method through creep and stress-relaxation tests by DMA at the target temperature/strain. The results demonstrated that the finite element analysis constructed using the proposed method could predict the mechanical properties during vacuum-forming-oriented creep tests. These results are expected to provide important insights into the complex mechanical behavior of PMMA, which varies with the temperature and strain rate.

Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with p-Laplacian

The fractional order p-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary value problem (BVP) for a class of nonlinear Hadamard fractional differential equation with p-Laplacian operator. By virtue of an important fixed point theorem on a complete metric space with two distances, we study the solvability and approximation of this BVP. Based on nonlinear analysis methods, we further discuss the generalized Ulam-Hyers (GUH) stability of this problem. Eventually, we supply two example and simulations to verify the correctness and availability of our main results. Compared to many previous studies, our approach enables the solution of the system to exist in metric space rather than normed space. In summary, we obtain some sufficient conditions for the existence, uniqueness, and stability of solutions in the metric space.

Viscoelastic behavior of jute fiber reinforced composite filled with ceramic particles after water aging

AbstractDue to the special multi‐layered structure of plant fiber, jute fiber reinforced composite is sensitive to water. Its mechanical properties will be significantly decreased after water aging. In this paper, alumina ceramic particles are added into jute fiber reinforced composite to improve its water aging resistance and viscoelastic properties. Effects of temperature, vibration frequency and loading time on the viscoelastic properties of the composite before and after water aging are measured. Reversible and irreversible performances degradation caused by water aging is investigated. The results show that ceramic particles can effectively improve the viscoelastic properties of the composite. The storage modulus and relaxation modulus of the composite increase with the particle content, but the strength decreases due to the stress concentration caused by the particles. Therefore, filler content between 10%–30% is more suitable. It is also noticed that its mechanical properties partially recover after redry. Plasticization of the matrix can be reversed after water is removed, but hydrolysis of the matrix and interface damage are irreversible. In addition, a practical Fractional Differential Kelvin Model is established to predict the viscoelastic behavior of jute fiber reinforced composite filled with alumina ceramic particles, and the fitting results agree well with test data.

Fractional order delay differential model of a tumor-immune system with vaccine efficacy: Stability, bifurcation and control

The aim of this paper is to present a mathematical model of tumor vaccine efficacy in the context of immunotherapy treatment. The model is governed by fractional-order delay differential equations with a control variable. Transforming growth factor beta (TGF-β) inhibition alone gives limited clinical advantages, but when combined with a tumor vaccine, it can dramatically increase the immune system’s anti-tumor response. The mathematical model takes into account tumor growth dynamics, TGF-β concentrations, cytotoxic effector cells, and regulatory T-cells. The non-negativity of the solutions to such a model has been examined. The steady-state stability and Hopf bifurcation of tumor time delays τ are investigated. An optimal fractional-order control problem is derived and analyzed in the presence of immunotherapy treatments. Using Adams–Bashforth–Moulton predictor–corrector algorithms, we illustrate numerical examples that confirm the analytical findings. The optimal control treatment method reduces the load of tumor cells while increasing the effector cells.

Fractional Differential Equation in the Cost Control System of Cross-Border E-Commerce

Abstract This paper establishes a cost management model of cross-border e-commerce based on a fractional differential equation. Firstly, this paper constructs a fractional differential equation game model of the platform’s evolution and consumer and government regulation. Then this paper studies each participant’s interests and cost control in the supervision and management of international e-commerce. Finally, this paper makes a dynamic simulation of cross-border e-commerce. It discusses the decision-making process of cross-border e-commerce enterprises in the operation factors of the alliance. Through the numerical simulation of MATLAB, it can be seen that standardized management can reduce the cost sensitivity of enterprises in a sense. The cost control system model of cross-border e-commerce based on fractional differential equation has high accuracy and strong applicability.

Analysis and achievement for fractional optimal control of Hepatitis B with Caputo operator

Deciphering the effect of long memory on dynamic behavior is important for controlling disease transmission. Employing fractional order calculus, here, we propose a model with Caputo operator to emphasize the regulation effect of the long memory on the optimal control of Hepatitis B. Considering four different health classes of populations, susceptible, acute infections, chronic infections, and recovered, we try to elucidate the existence of the optimal solution of the fractional optimal control problems (FOCP) by applying a strategy of isolation, treatment, and vaccination. Initially, we discuss the locally asymptotically stability of the equilibrium solutions of the fractional differential model. Sensitivity analysis and numerical simulations are also carried out to verify how parameter changes affect the system dynamics, that is, the threshold effect depending on the basic reproduction number, which provides inspiration for constructing the controlled system. After that, the necessary conditions for the optimality of the Hepatitis B system are derived to focus on the Caputo fractional operator with a singular kernel. An effective numerical simulation algorithm based on trapezoidal approximation is used to validate the control effect in the sense of the transient response by comparing the difference between the fractional-order and integer-order derivatives, and the numeric results can vividly show the attenuation effect of memory on control. This study may provide a powerful rationale for uncovering the importance of the long-memory effects in disease control.

Study on a new [formula omitted]-Laplacian fractional differential model generated by instantaneous and non-instantaneous impulsive effects

In this paper, we discuss a new p-Laplacian fractional differential equation involving instantaneous and non-instantaneous impulses, supplemented with Sturm–Liouville boundary conditions. To study the stated problem, we introduce a new norm for fractional space, denoted by Eα,p. Under this framework, using the critical point theorem proposed by Bonanno and Marano, we prove the existence result of multiple solutions. Finally, an illustrative example is presented.

Application of Fractional Differential Model in Image Enhancement of Strong Reflection Surface

Combined with advanced fractional differential mask operation, this paper used a fractional differential to normalize the 5 × 5 mask and conducted experiments to select fractional v = 0.7 to determine the equation. The position of the center of the light band was obtained by the gray centroid method, and the center of the light band was enhanced by the fractional differential algorithm. Three samples of hard disk substrate, roller, and printed circuit board were selected. The traditional processing was compared to the fractional differential algorithm in this paper, and several advanced algorithms were compared with the algorithm in this paper. Experimental data showed that fractional differential enhancement can effectively improve the accuracy of extracting the center of light fringes. It can be found that the average error of extracting the center by fractional differential processing was relatively small, and the image outline was clearer.

Clinical effects of 2-DG drug restraining SARS-CoV-2 infection: A fractional order optimal control study.

Fractional calculus is very convenient tool in modeling of an emergent infectious disease system comprising previous disease states, memory of disease patterns, profile of genetic variation etc. Significant complex behaviors of a disease system could be calibrated in a proficient manner through fractional order derivatives making the disease system more realistic than integer order model. In this study, a fractional order differential equation model is developed in micro level to gain perceptions regarding the effects of host immunological memory in dynamics of SARS-CoV-2 infection. Additionally, the possible optimal control of the infection with the help of an antiviral drug, viz. 2-DG, has been exemplified here. The fractional order optimal control would enable to employ the proper administration of the drug minimizing its systematic cost which will assist the health policy makers in generating better therapeutic measures against SARS-CoV-2 infection. Numerical simulations have advantages to visualize the dynamical effects of the immunological memory and optimal control inputs in the epidemic system.

Modeling consolidation of soft clay by developing a fractional differential constitutive model in conjunction with an intelligent displacement inversion method.

Studying the constitutive relation of soft clays is of critical importance for fundamentally understanding their complex consolidation behavior. This study proposes a fractional differential constitutive model in conjunction with an intelligent displacement inversion method based on the classic particle swarm optimization for modeling the deformation behavior of soft clay. The model considered the rheological properties of soft clay at different consolidation stages. In addition, statistical adaptive dynamic particle swarm optimization-least squares support vector machines were implemented to identify the model parameters efficiently. The accuracy and effectiveness of the model were validated using available experimental results. Finally, the application results showed that the proposed model could efficiently simulate coupling properties of soft clay’s primary and secondary consolidations.

Multiplicity of solutions for the Dirichlet boundary value problem to a fractional quasilinear differential model with impulses

This paper aims to consider the multiplicity of solutions for a kind of boundary value problem to a fractional quasilinear differential model with impulsive effects. By establishing a new variational structure and overcoming the difficulties brought by the influence of impulsive effects, some new results are acquired via the symmetry mountain-pass theorem, which extend and enrich some previous results.