A Collocation Technique via Pell-Lucas Polynomials to Solve Fractional Differential EquationModel for HIV/AIDS with Treatment Compartment

In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.

Legendre-Weighted Residual Methods for System of Fractional Order Differential Equations

The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.

Meta-Auto-Decoder:a Meta-Learning-Based Reduced Order Model for Solving Parametric Partial Differential Equations

Many important problems in science and engineering require solving the so-called parametric partial differential equations(PDEs),i.e.,PDEs with different physical parameters,boundary conditions,shapes of computational domains,etc.Typical reduced order modeling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the ofline stage.These methods often need a predefined mesh as well as a series of precomputed solution snapshots,and may struggle to balance between the efficiency and accuracy due to the limitation of the linear ansatz.Utilizing the nonlinear representation of neural networks(NNs),we propose the Meta-Auto-Decoder(MAD)to construct a nonlinear trial manifold,whose best possible performance is measured theoretically by the decoder width.Based on the meta-learning concept,the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage.Fast adaptation to new(possibly heterogeneous)PDE parameters is enabled by searching on this trial manifold,and optionally fine-tuning the trial manifold at the same time.Extensive numerical experiments show that the MAD method exhibits a faster convergence speed without losing the accuracy than other deep learning-based methods.

On Nonlinear Conformable Fractional Order Dynamical System via Differential Transform Method

This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution.Indeed,Schauder and Banach fixed point theorems are utilized to prove our claim.Further,an algorithm for the approximate analytical solution to the proposed problem has been established.In this regard,the conformable fractional differential transform(CFDT)technique is used to compute the required results in the form of a series.Using Matlab-16,we simulate the series solution to illustrate our results graphically.Finally,a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.

Fractional Order Environmental and Economic Model Investigations Using Artificial Neural Network

The motive of these investigations is to provide the importance and significance of the fractional order(FO)derivatives in the nonlinear environmental and economic(NEE)model,i.e.,FO-NEE model.The dynamics of the NEE model achieves more precise by using the form of the FO derivative.The investigations through the non-integer and nonlinear mathematical form to define the FO-NEE model are also provided in this study.The composition of the FO-NEEmodel is classified into three classes,execution cost of control,system competence of industrial elements and a new diagnostics technical exclusion cost.The mathematical FO-NEE system is numerically studied by using the artificial neural networks(ANNs)along with the Levenberg-Marquardt backpropagation method(ANNs-LMBM).Three different cases using the FO derivative have been examined to present the numerical performances of the FO-NEE model.The data is selected to solve the mathematical FO-NEE system is executed as 70%for training and 15%for both testing and certification.The exactness of the proposed ANNs-LMBM is observed through the comparison of the obtained and the Adams-Bashforth-Moulton database results.To ratify the aptitude,validity,constancy,exactness,and competence of the ANNs-LMBM,the numerical replications using the state transitions,regression,correlation,error histograms and mean square error are also described.

An Artificial Approach for the Fractional Order Rape and Its Control Model

The current investigations provide the solutions of the nonlinear fractional order mathematical rape and its controlmodel using the strength of artificial neural networks(ANNs)along with the Levenberg-Marquardt backpropagation approach(LMBA),i.e.,artificial neural networks-Levenberg-Marquardt backpropagation approach(ANNs-LMBA).The fractional order investigations have been presented to find more realistic results of the mathematical form of the rape and its control model.The differential mathematical form of the nonlinear fractional order mathematical rape and its control model has six classes:susceptible native girls,infected immature girls,susceptible knowledgeable girls,infected knowledgeable girls,susceptible rapist population and infective rapist population.The rape and its control differential system using three different fractional order values is authenticated to perform the correctness of ANNs-LMBA.The data is used to present the rape and its control differential system is designated as 70%for training,14%for authorization and 16%for testing.The obtained performances of the ANNs-LMBA are compared with the dataset of the Adams-Bashforth-Moulton scheme.To substantiate the consistency,aptitude,validity,exactness,and capability of the LMBA neural networks,the obtained numerical values are provided using the state transitions(STs),correlation,regression,mean square error(MSE)and error histograms(EHs).

Exact Solution to Nonlinear Differential Equations of Fractional Order via (<i>G’</i>/<i>G</i>)-Expansion Method

In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (G'/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Burgers’ Equation have been discussed.

Solving the Nonlinear Variable Order Fractional Differential Equations by Using Euler Wavelets

An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented.Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations.And the convergence of the Euler wavelets basis is given.The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.

Hydraulic cylinder control of injection molding machine based on differential evolution fractional order PID

Injection molding machine,hydraulic elevator,speed actuators belong to variable speed pump control cylinder system.Because variable speed pump control cylinder system is a nonlinear hydraulic system,it has some problems such as response lag and poor steady-state accuracy.To solve these problems,for the hydraulic cylinder of injection molding machine driven by the servo motor,a fractional order proportion-integration-diferentiation(FOPID)control strategy is proposed to realize the speed tracking control.Combined with the adaptive differential evolution algorithm,FOPID control strategy is used to determine the parameters of controller on line based on the test on the servo-motor-driven gear-pump-controlled hydraulic cylinder injection molding machine.Then the slef-adaptive differential evolution fractional order PID controller(SADE-FOPID)model of variable speed pump-controlled hydraulic cylinder is established in the test system with simulated loading.The simulation results show that compared with the classical PID control,the FOPID has better steady-state accuracy and fast response when the control parameters are optimized by the adaptive differential evolution algorithm.Experimental results show that SADE-FOPID control strategy is effective and feasible,and has good anti-load disturbance performance.

Existence and Uniqueness of Solution for a Fractional Order Integro-Differential Equation with Non-Local and Global Boundary Conditions

In this paper, we prove an important existence and uniqueness theorem for a fractional order Fredholm – Volterra integro-differential equation with non-local and global boundary conditions by converting it to the corresponding well known Fredholm integral equation of second kind. The considered in this paper has been solved already numerically in [1].

EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES

We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.

Controllability of Fractional Order Stochastic Differential Inclusions with Fractional Brownian Motion in Finite Dimensional Space

In this paper,sufficient conditions are formulated for controllability of fractional order stochastic differential inclusions with fractional Brownian motion(f Bm) via fixed point theorems,namely the Bohnenblust-Karlin fixed point theorem for the convex case and the Covitz-Nadler fixed point theorem for the nonconvex case.The controllability Grammian matrix is defined by using Mittag-Leffler matrix function.Finally,a numerical example is presented to illustrate the efficiency of the obtained theoretical results.

Fixed Point Theorem and Fractional Differential Equations with Multiple Delays Related with Chaos Neuron Models

In this paper, we show a fixed point theorem which deduces to both of Lou’s fixed point theorem and de Pascale and de Pascale’s fixed point theorem. Moreover, our result can be applied to show the existence and uniqueness of solutions for fractional differential equations with multiple delays. Using the theorem, we discuss the fractional chaos neuron model.

Application of Fractional Calculus to a Linear Third Order (Nonhomogeneous and Homogeneous) Ordinary Differential Equation

In this paper we apply fractional calculus to solve the 3rd order ordinary differential equation of the following form: (z-a)(z-b)(z-c)φ 3+(βz 2+γz+D)φ 2+(α(2β-3α-3)z+αγ+α(α+1)(a+b+c))φ 1+α(α-1)(β-2α-2)φ=f.

Dynamics of Fractional Differential Model for Schistosomiasis Disease

In the present study,a design of a fractional order mathematical model is presented based on the schistosomiasis disease.To observe more accurate performances of the results,the use of fractional order derivatives in the mathematical model is introduce based on the schistosomiasis disease is executed.The preliminary design of the fractional order mathematical model focused on schistosomiasis disease is classified as follows:uninfected with schistosomiasis,infected with schistosomiasis,recovered from infection,susceptible snail unafflicted with schistosomiasis disease and susceptible snail afflicted with this disease.The solutions to the proposed system of the fractional order mathematical model will be presented using stochastic artificial neural network(ANN)techniques in conjunction with the LevenbergMarquardt backpropagation(LMBP),referred to as ANN-LMBP.To illustrate the preciseness of the ANN-LMBP method,mathematical presentations of three different values focused on fractional order will be performed.These statics performances are taken in these investigations are 78%and 11%for both learning and certification.The accuracy of the ANN-LMBP method is determined by comparing the values obtained by the database Adams-Bash forth-Moulton scheme.The simulation-based error histograms(EHs),MSE,recurrence,and state transitions(STs)will be offered to achieve the capability,accuracy,steadiness,abilities,and finesse of the ANN-LMBP method.

Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.

Dynamical Analysis of the Stochastic COVID-19 Model Using Piecewise Differential Equation Technique

Various data sets showing the prevalence of numerous viral diseases have demonstrated that the transmission is not truly homogeneous.Two examples are the spread of Spanish flu and COVID-19.The aimof this research is to develop a comprehensive nonlinear stochastic model having six cohorts relying on ordinary differential equations via piecewise fractional differential operators.Firstly,the strength number of the deterministic case is carried out.Then,for the stochastic model,we show that there is a critical number RS0 that can predict virus persistence and infection eradication.Because of the peculiarity of this notion,an interesting way to ensure the existence and uniqueness of the global positive solution characterized by the stochastic COVID-19 model is established by creating a sequence of appropriate Lyapunov candidates.Adetailed ergodic stationary distribution for the stochastic COVID-19 model is provided.Our findings demonstrate a piecewise numerical technique to generate simulation studies for these frameworks.The collected outcomes leave no doubt that this conception is a revolutionary doorway that will assist mankind in good perspective nature.

An Intelligence Computational Approach for the Fractional 4D Chaotic Financial Model

The main purpose of the study is to present a numerical approach to investigate the numerical performances of the fractional 4-D chaotic financial system using a stochastic procedure.The stochastic procedures mainly depend on the combination of the artificial neural network(ANNs)along with the Levenberg-Marquardt Backpropagation(LMB)i.e.,ANNs-LMB technique.The fractional-order term is defined in the Caputo sense and three cases are solved using the proposed technique for different values of the fractional orderα.The values of the fractional order derivatives to solve the fractional 4-D chaotic financial system are used between 0 and 1.The data proportion is applied as 73%,15%,and 12%for training,testing,and certification to solve the chaotic fractional system.The acquired results are verified through the comparison of the reference solution,which indicates the proposed technique is efficient and robust.The 4-D chaotic model is numerically solved by using the ANNs-LMB technique to reduce the mean square error(MSE).To authenticate the exactness,and consistency of the technique,the obtained performances are plotted in the figures of correlation measures,error histograms,and regressions.From these figures,it can be witnessed that the provided technique is effective for solving such models to give some new insight into the physical behavior of the model.

A Neural Study of the Fractional Heroin Epidemic Model

This works intends to provide numerical solutions based on the nonlinear fractional order derivatives of the classical White and Comiskey model(NFD-WCM).The fractional order derivatives have provided authentic and accurate solutions for the NDF-WCM.The solutions of the fractional NFD-WCM are provided using the stochastic computing supervised algorithm named Levenberg-Marquard Backpropagation(LMB)based on neural networks(NNs).This regression approach combines gradient descent and Gauss-Newton iterative methods,which means finding a solution through the sequences of different calculations.WCM is used to demonstrate the heroin epidemics.Heroin has been on-growth world wide,mainly in Asia,Europe,and the USA.It is the fourth foremost cause of death due to taking an overdose in the USA.The nonlinear mathematical system NFD-WCM discusses the overall circumstance of different drug users,such as suspected groups,drug users without treatment,and drug users with treatment.The numerical results of NFD-WCM via LMB-NNs have been substantiated through training,testing,and validation measures.The stability and accuracy are then checked through the statistical tool,such asmean square error(MSE),error histogram,and fitness curves.The suggested methodology’s strength is demonstrated by the high convergence between the reference solutions and the solutions generated by adding the efficacy of a constructed solver LMB-NNs,with accuracy levels ranging from 10?9 to 10?10.

Stochastic Investigations for the Fractional Vector-Host Diseased Based Saturated Function of Treatment Model

The goal of this research is to introduce the simulation studies of the vector-host disease nonlinear system(VHDNS)along with the numerical treatment of artificial neural networks(ANNs)techniques supported by Levenberg-Marquardt backpropagation(LMQBP),known as ANNs-LMQBP.This mechanism is physically appropriate,where the number of infected people is increasing along with the limited health services.Furthermore,the biological effects have fadingmemories and exhibit transition behavior.Initially,the model is developed by considering the two and three categories for the humans and the vector species.The VHDNS is constructed with five classes,susceptible humans Sh(t),infected humans Ih(t),recovered humans Rh(t),infected vectors Iv(t),and susceptible vector Sv(t)based system of the fractional-order nonlinear ordinary differential equations.To solve the number of variations of the VHDNS,the numerical simulations are performed using the stochastic ANNs-LMQBP.The achieved numerical solutions for solving the VHDNS using the stochastic ANNs-LMQBP have been described for training,verifying,and testing data to decrease the mean square error(MSE).An extensive analysis is provided using the correlation studies,MSE,error histograms(EHs),state transitions(STs),and regression to observe the accuracy,efficiency,expertise,and aptitude of the computing ANNs-LMQBP.