Fractional Differential Equation Model Research Articles

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.

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.

Numerical simulation of time fractional Benjamin-Bona-Mahony-Burger equation describing propagation of long waves on the water surface

The unidirectional propagation of long waves in a certain nonlinear dispersive system is explained in the Benjamin-Bona-Mahony-Burger equation. The motivation of this manuscript is to examine the Benjamin-Bona-Mahony-Burger equation numerically and analytically by employing reproducing kernel functions. The implementation of reproducing kernel Hilbert space method produces an analytical solution in the form of a series representation, and after truncating the infinite series, an approximate solution is attained. With the support of numerical examples, the accuracy of this method has been shown through graphs and error tables. The convergence analysis of the introduced scheme has been explained by some useful theorems and lemmas. Finally, it has been observed that the approximate solution consistently converges to the exact solution with less computational overhead. The proposed method would be very effective in solving various fractional partial differential equation models.

Meshfree numerical approach for some time-space dependent order partial differential equations in porous media

<abstract><p>In this article, the meshfree radial basis function method based on the Gaussian function is proposed for some time-space dependent fractional order partial differential equation (PDE) models. These PDE models have significant applications in chemical engineering and physical science. Some main advantages of the proposed method are that it is easy to implement, and the output response is quick and highly accurate, especially in the higher dimension. In this method, the time-dependent derivative terms are treated by Caputo fractional derivative while space-dependent derivative terms are treated by Riesz, Riemann-Liouville, and Grünwald-Letnikov derivatives. The proposed method is tested on some numerical examples and the accuracy is analyzed by $ \|L\|_\infty $.</p></abstract>

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.

Football Offense Training Strategy Based on Fractional Differential Mathematical Modeling

Abstract This paper establishes the parameters of the ball trajectory based on the fractional differential equation model. At the same time, this paper quantifies the shooting power of direct free kicks. In this paper, the resistance effect of air on the motion of the center of mass is considered. And the dynamic equation of the football arc motion is established. In this way, we obtain the analytical solution of the differential equation of order. The research shows that the correlation between the fitting equation of the football flight trajectory and the prediction equation is 96.55%. The algorithm model in this paper can effectively predict the flight path of football. The research theory of this paper provides a theoretical basis for football offensive training strategies.

The application of an optimized fractional order accumulated grey model with variable parameters in the total energy consumption of Jiangsu Province and the consumption level of Chinese residents

<abstract> <p>Fractional order imply the idea of "in between", the grey model generated by fractional accumulation has better prediction and adaptability than that generated by first-order accumulation. General grey model of the differential equation of the left is a cumulative function derivative of time, in order to improve the adaptability of the model and prediction ability, general fractional order differential equation model is presented. In this paper, on the basis of the derivation of time $ t $ extensions to the derivation of $ {t^u} $, added a variable coefficient, and through the integral differential equation and tectonic background value. We establish an optimized fractional order cumulative grey model with variable parameters, i.e., optimized fractional order accumulated grey model (FOGM (1, 1)). By using the Particle swarm optimization (PSO) algorithm, we search for the order and variable parameters of the optimal fractional order. Then we apply the proposed model to predict the total energy consumption of Jiangsu province and the consumption level of Chinese residents. The results indicate that the proposed model has high fitting and prediction accuracy compared to other classical grey prediction models, such as grey model (GM (1, 1)), non-homogeneous grey model (NGM (1, 1)) and fractional order accumulated grey model (FGM (1, 1)). It also validates that the proposed model is a practical and promising model for forecasting the energy consumption as well as the consumption level of Chinese residents.</p> </abstract>

Modelling of natural growth with memory effect in economics: An application of adomian decomposition and variational iteration methods

The power-law memory effect is taken into consideration in a generalisation of the economic model of natural growth. The memory effect refers to a process's reliance on its current state and its history of previous changes. However, the study that focuses on natural growth in economics considering the memory effect with fractional order-linear differential equation model is still limited. The current investigation seeks to solve the natural growth with memory effect in the economics model and decide the best model using fractional differential equation (FDE), namely Adomian Decomposition and Variational Iteration Methods. Also, this study assumes the level of consumer loss memory during a certain time interval denoted by a parameter (α). This study showed the model of loss memory effect with 0 < α ≤ 1 given a slowdown in output growth compared to a model without memory effect. Besides that, this study also found that output Y(t) is growing faster with the Variational Iteration method compared to the Adomian decomposition method. Also, using graphical simulation, this study found the output Y(t) is closer to the exact solution with α=0.4 and α=0.9. In conclusion, this study successfully solved natural growth with memory effect in economics and decided the best model between FDE, namely Adomian decomposition and Variational iterative methods using numerical analysis.

Numerical surfaces of fractional Zika virus model with diffusion effect of mosquito‐borne and sexually transmitted disease

This paper analyzes the dynamics of fractional partial differential equation (FPDE) model of Zika virus that incorporates diffusion using Atangana–Baleanu (AB) fractional derivative. Zika virus disease is an infection transmitted predominantly by the bite of an infected Aedes species mosquito and may be a severe epidemic if not contained in its premature stages. The q‐homotopy analysis transform method is employed to analyze and compute the solutions for this nonlinear partial differential model, and the fractional derivative is defined in Atangana–Baleanu sense. We determine some new approximate numerical results for different values of parameters of alpha. Numerical models focused on various distributions of the population help to explain how the spread of humans and mosquitoes influences the disease's transmission. With the utilization fixed‐point hypothesis, the existence and uniqueness of the solutions obtained for the proposed model are presented.

User online consumption behaviour based on fractional differential equation

Abstract User consumption behaviour is a subject worthy of study. Because consumers’ consumption behaviours are dynamic and with individual differences, various factors need to be considered when establishing a fractional differential equation model of users’ online consumption behaviour. The two elements, namely advertising and price are more evident in influencing consumer behaviour. Therefore, the paper establishes a product diffusion fractional differential equation model of price and advertising presence or absence to study the impact of these two factors on consumer behaviour. It turns out that ignoring the advertisement and the cost of the product is related to the characteristics of the consumer network. When there are advertising and price factors, product awareness is related to price constraints.

Positive solutions and stability of fuzzy Atangana–Baleanu variable fractional differential equation model for a novel coronavirus (COVID-19)

This work provides a new fuzzy variable fractional COVID-19 model and uses a variable fractional operator, namely, the fuzzy variable Atangana–Baleanu fractional derivatives in the Caputo sense. Next, we explore the proposed fuzzy variable fractional COVID-19 model using the fixed point theory approach and determine the solution’s existence and uniqueness conditions. We choose an appropriate mapping and with the help of the upper/lower solutions method. We prove the existence of a positive solution for the proposed fuzzy variable fractional COVID-19 model and also obtain the result on the existence of a unique positive solution. Moreover, we discuss the generalized Hyers–Ulam stability and generalized Hyers–Ulam–Rassias stability. Further, we investigate the results on maximum and minimum solutions for the fuzzy variable fractional COVID-19 model.

Industrial Service Skill Development in the Service Regions Based on the Nonlinear Fractional Differential Equation Model

Based on a questionnaire survey of economics students at school, 290 students majoring in economics were selected as the survey sample, and a nonlinear fractional differential equation model was constructed to comprehensively score the innovation ability of school students. The study found that the innovative spirit of economics students is more active, but the innovative skills are relatively scarce. In this regard, it is proposed to promote the reform of the education system, improve the implementation and evaluation mechanism, focus on theory and practice, promote the application of innovative abilities, improve the quality of teaching, cultivate a team of learning teachers, cultivate students’ independent thinking ability, strengthen students’ logical thinking ability, and promote “industry-university-research” Suggestions on base training model.

Numerical Solution of the Multiterm Time-Fractional Model for Heat Conductivity by Local Meshless Technique

Fractional partial differential equation models are frequently used to several physical phenomena. Despite the ability to express many complex phenomena in different disciplines, researchers have found that multiterm time-fractional PDEs improve the modeling accuracy for describing diffusion processes in contrast to the results of a single term. Nowadays, it attracts the attention of the active researchers. The aim of this work is concerned with the approximate numerical solutions of the three-term time-fractional Sobolev model equation using computationally attractive and reliable technique, known as a local meshless method. Because of the meshless character and the simple application in higher dimensions, there is a growing interest in meshless techniques. To assess the reliability and accuracy of the proposed method, three test problems and two types of irregular domains are taken into account.

An Investigation on Analytical Properties of Delayed Fractional Order HIV Model: A Case Study

In this manuscript, we design a fractional order delay differential equation model for HIV transmission with the implementation of three distinct therapies for three different infectious stages. We investigate the positivity of solutions, analyze the stability properties, followed by Hopf bifurcation analysis. To probe the parameters that expedite the spread of infection, uncertainty and sensitivity analysis were performed. The numerical review was carried out to substantiate our theoretical results. Our proposed model parameters have been calibrated to fit yearly data from Afghanistan, Australia, France, Italy, Netherlands and New Zealand.

A fractional order differential equation model for Hepatitis B virus with saturated incidence

This manuscript presents a nonlinear fractional order Hepatitis B virus (HBV) model with saturated incidence. Model governing equations are defined using Caputo fractional derivatives. We use the Adams–Bashforth-Moulton technique to compute numerical solutions of the presented epidemic model. Generally, fractional derivatives are used to model real-world phenomena that contains nonlocal effects, history and/or memory. The main objective is to develop and explore the dynamics of an HBV fractional derivative epidemic model via the Caputo definition. Firstly, we discuss the model basic properties which include positivity of solutions and the invariant regions where the solution set is bounded using the generalized fractional mean value theorem. The model dynamics are described based on the reproduction number R0. Thereafter, we use the fractional Routh-Hurwitz stability criterion as well as simple matrix algebra to determine sufficient conditions for the stability of the equilibrium points. Lastly, we present graphical representation of numerical results. The results indicate the importance of fractional order in modeling HBV transmission dynamics and ensures that by including the memory effects, the model is more appropriate and insightful for such a study.

Kalman Filter-Based Real-Time Implementable Optimization of the Fuel Efficiency of Solid Oxide Fuel Cells

The electric power characteristic of solid oxide fuel cells (SOFCs) depends on numerous influencing factors. These are the mass flow of supplied hydrogen, the temperature distribution in the interior of the fuel cell stack, the temperatures of the supplied reaction media at the anode and cathode, and—most importantly—the electric current. Describing all of these dependencies by means of analytic system models is almost impossible. Therefore, it is reasonable to identify these dependencies by means of stochastic filter techniques. One possible option is the use of Kalman filters to find locally valid approximations of the power characteristics. These can then be employed for numerous online purposes of dynamically operated fuel cells such as maximum power point tracking or the maximization of the fuel efficiency. In the latter case, it has to be ensured that the fuel cell operation is restricted to the regime of Ohmic polarization. This aspect is crucial to avoid fuel starvation phenomena which may not only lead to an inefficient system operation but also to accelerated degradation. In this paper, a Kalman filter-based, real-time implementable optimization of the fuel efficiency is proposed for SOFCs which accounts for the aforementioned feasibility constraints. Essentially, the proposed strategy consists of two phases. First, the parameters of an approximation of the electric power characteristic are estimated. The measurable arguments of this function are the hydrogen mass flow and the electric stack current. In a second stage, these inputs are optimized so that a desired stack power is attained in an optimal way. Simulation results are presented which show the robustness of the proposed technique against inaccuracies in the a-priori knowledge about the power characteristics. For a numerical validation, three different models of the electric power characteristic are considered: (i) a static neural network input/output model, (ii) a first-order dynamic system representation and (iii) the combination of a static neural network model with a low-order fractional differential equation model representing transient phases during changes between different electric operating points.

Explicit tight frames for simulating a new system of fractional nonlinear partial differential equation model of Alzheimer disease

This paper is devoted to develop a new mathematical model for Alzheimer disease based on a system of fractional-order partial differential equations. The system of Alzheimer disease includes neurons, astrocytes, microglias and peripheral macrophages, as well as amyloid β aggregation and hyperphosphorylated tau proteins. We consider the Caputo fractional derivative definition to analyze the formulated system by simulating the effect of drugs that either failed or currently in clinical trials. To simulate the model, we use tight frame (framelet) systems generated using the unitaryand oblique extension principle. According to the simulation results, and based on using such new direction of fractional modeling, the progression of Alzheimer disease and its consequences will be slowing down. Which may give clinical insights on intervention measures against the disease and its effective therapies.

An Anomalous Diffusion Approach for Speckle Noise Reduction in Medical Ultrasound Images

Medical ultrasound images are usually degraded by a specific type of noise, called speckle. The presence of speckle noise in medical ultrasound images will reduce the image quality and affect the effective information, which can potentially cause a misdiagnosis. Therefore, medical image enhancement processing has been extensively studied and several denoising approaches have been introduced and developed. In the current work, a robust fractional partial differential equation (FPDE) model based on the anomalous diffusion theory is proposed and used for medical ultrasound image enhancement. An efficient computational approach based on a combination of a time integration scheme and localized meshless method in a domain decomposition framework is performed to deal with the model. {In order to evaluate the performance of the proposed de-speckling approach, it is used for speckle noise reduction of a synthetic ultrasound image degraded by different levels of speckle noise. The results indicate the superiority of the proposed approach in comparison with classical anisotropic diffusion denoising model (Catt$acute{e}$'s pde model).}

The behavior of solution function of the fractional differential equations using modified homotopy perturbation method

The discourse regarding fractional calculus, in particular those related to fractional differential equation, is still continue to attract researcher attention. Previous studies have elaborated on the variation of fractional differential equation models. This study aims to uncover the problem of convergence of the solution function sequence related to the order of fractional differential equation. Firstly, this study presents how to find a solution for the model by using a Modified Homotopy Perturbation Method as the improvement of Homotopy Perturbation Method. Furthermore, the solution function with the sequence of fractional order is drawn by Maple. Using the geometrical analysis, the result of this study shows that if fractional order sequence is convergent to α, then sequence of its solution function will be convergent to a solution function of fractional differential equation with order α.