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.

Pseudo S-Asymptotically(ω,c)-Periodic Solutions to Fractional Differential Equations of Sobolev Type

In this paper,we firstly recall some basic results on pseudo S-asymptotically(ω,c)-periodic functions and Sobolev type fractional differential equation.We secondly investigate some existence of pseudo S-asymptotically(ω,c)-periodic solutions for a semilinear fractional differential equations of Sobolev type.We finally present a simple example.

Multiple Solutions for a Class of Singular Boundary Value Problems of Hadamard Fractional Differential Systems with p-Laplacian Operator

This paper discusses the existence and multiplicity of positive solutions for a class of singular boundary value problems of Hadamard fractional differential systems involving the p-Laplacian operator. First, for the sake of overcoming the singularity, sequences of approximate solutions to the boundary value problem are obtained by applying the fixed point index theory on the cone. Next, it is demonstrated that these sequences of approximate solutions are uniformly bounded and equicontinuous. The main results are then established through the Ascoli-Arzelà theorem. Ultimately, an instance is worked out to test and verify the validity of the main results.

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.

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.

On the partial stability of nonlinear impulsive Caputo fractional systems

In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptotic uniform stability and Mittag Leffler stability.The approach presented is based on the specially introduced piecewise continuous Lyapunov functions.Furthermore,some numerical examples are given to show the effectiveness of our obtained theoretical results.

On Time Fractional Partial Differential Equations and Their Solution by Certain Formable Transform Decomposition Method

This paper aims to investigate a new efficient method for solving time fractional partial differential equations.In this orientation,a reliable formable transform decomposition method has been designed and developed,which is a novel combination of the formable integral transform and the decomposition method.Basically,certain accurate solutions for time-fractional partial differential equations have been presented.Themethod under concern demandsmore simple calculations and fewer efforts compared to the existingmethods.Besides,the posed formable transformdecompositionmethod has been utilized to yield a series solution for given fractional partial differential equations.Moreover,several interesting formulas relevant to the formable integral transform are applied to fractional operators which are performed as an excellent application to the existing theory.Furthermore,the formable transform decomposition method has been employed for finding a series solution to a time-fractional Klein-Gordon equation.Over and above,some numerical simulations are also provided to ensure reliability and accuracy of the new approach.

Solving Fractional Differential Equations via Fixed Points of Chatterjea Maps

In this paper,we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric spaces.Furthermore,we study fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space using the Krasnoselskii-Ishikawa iteration method associated withSλand consider some applications of our results to prove the existence of solutions for nonlinear integral and nonlinear fractional differential equations.We also establish certain interesting examples to illustrate the usability of our results.

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

On Fractional Differential Inclusion for an EpidemicModel via L-Fuzzy Fixed Point Results

The real world is filled with uncertainty,vagueness,and imprecision.The concepts we meet in everyday life are vague rather than precise.In real-world situations,if a model requires that conclusions drawn from it have some bearings on reality,then two major problems immediately arise,viz.real situations are not usually crisp and deterministic;complete descriptions of real systems often require more comprehensive data than human beings could recognize simultaneously,process and understand.Conventional mathematical tools which require all inferences to be exact,are not always efficient to handle imprecisions in a wide variety of practical situations.Following the latter development,a lot of attention has been paid to examining novel L-fuzzy analogues of conventional functional equations and their various applications.In this paper,new coincidence point results for single-valued mappings and an L-fuzzy set-valued map in metric spaces are proposed.Regarding novelty and generality,the obtained invariant point notions are compared with some well-known related concepts via non-trivial examples.It is observed that our principal results subsume and refine some important ones in the corresponding domains.As an application,one of our results is utilized to discussmore general existence conditions for realizing the solutions of a non-integer order inclusion model for COVID-19.

Image Enhancement Using Adaptive Fractional Order Filter

Image enhancement is an important preprocessing task as the contrast is low in most of the medical images,Therefore,enhancement becomes the mandatory process before actual image processing should start.This research article proposes an enhancement of the model-based differential operator for the images in general and Echocardiographic images,the proposed operators are based on Grunwald-Letnikov(G-L),Riemann-Liouville(R-L)and Caputo(Li&Xie),which are the definitions of fractional order calculus.In this fractional-order,differentiation is well focused on the enhancement of echocardiographic images.This provoked for developing a non-linear filter mask for image enhancement.The designed filter is simple and effective in terms of improving the contrast of the input low contrast images and preserving the textural features,particularly in smooth areas.The novelty of the proposed method involves a procedure of partitioning the image into homogenous regions,details,and edges.Thereafter,a fractional differential mask is appropriately chosen adaptively for enhancing the partitioned pixels present in the image.It is also incorporated into the Hessian matrix with is a second-order derivative for every pixel and the parameters such as average gradient and entropy are used for qualitative analysis.The wide range of existing state-of-the-art techniques such as fixed order fractional differential filter for enhancement,histogram equalization,integer-order differential methods have been used.The proposed algorithm resulted in the enhancement of the input images with an increased value of average gradient as well as entropy in comparison to the previous methods.The values obtained are very close(almost equal to 99.9%)to the original values of the average gradient and entropy of the images.The results of the simulation validate the effectiveness of the proposed algorithm.

The Fractional Investigation of Fornberg-Whitham Equation Using an Efficient Technique

In the last few decades,it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences.For this reason,fractional partial differential equations(FPDEs)are of more importance to model the different physical processes in nature more accurately.Therefore,the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution.In the current work,the idea of fractional calculus has been used,and fractional FornbergWhithamequation(FFWE)is represented in its fractional view analysis.Awell-knownmethod which is residual power series method(RPSM),is then implemented to solve FFWE.TheRPSMresults are discussed through graphs and tables which conform to the higher accuracy of the proposed technique.The solutions at different fractional orders are obtained and shown to be convergent toward an integer-order solution.Because the RPSM procedure is simple and straightforward,it can be extended to solve other FPDEs and their systems.

The Fractional Investigation of Some Nonlinear Partial Differential Equations by Using an Efficient Procedure

The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.

Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay

We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay.By using new conformable delayed matrix functions and the method of variation,we obtain a representation of their solutions.As an application,we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayedmatrix functions.The obtained results are new,and they extend and improve some existing ones.Finally,an example is presented to illustrate the validity of our theoretical results.

Numerical Procedure for Fractional HBV Infection with Impact of Antibody Immune

The current investigations are presented to solve the fractional order HBV differential infection system(FO-HBV-DIS)with the response of antibody immune using the optimization based stochastic schemes of the Levenberg-Marquardt backpropagation(LMB)neural networks(NNs),i.e.,LMBNNs.The FO-HBV-DIS with the response of antibody immune is categorized into five dynamics,healthy hepatocytes(H),capsids(D),infected hepatocytes(I),free virus(V)and antibodies(W).The investigations for three different FO variants have been tested numerically to solve the nonlinear FO-HBV-DIS.The data magnitudes are implemented 75%for training,10%for certification and 15%for testing to solve the FO-HBV-DIS with the response of antibody immune.The numerical observations are achieved using the stochastic LMBNNs procedures for soling the FO-HBV-DIS with the response of antibody immune and comparison of the results is presented through the database Adams-Bashforth-Moulton approach.To authenticate the validity,competence,consistency,capability and exactness of the LMBNNs,the numerical presentations using the mean square error(MSE),error histograms(EHs),state transitions(STs),correlation and regression are accomplished.

Existence Results for Systems of Nonlinear Caputo Fractional Differential Equations

We aim, in this work, to demonstrate the existence of minimal and maximal coupled quasi-solutions for nonlinear Caputo fractional differential systems with order q ∈ (1,2). Our approach is based on mixed monotone iterative techniques developed under the concept of lower and upper quasi-solutions. Our results extend those obtained for ordinary differential equations and fractional ones.

Existence and Stability of Solutions for a Class of Fractional Impulsive Differential Equations with Atangana-Baleanu-Caputo Derivative

In this paper, we are concerned with the existence of solutions to a class of Atangana-Baleanu-Caputo impulsive fractional differential equation. The existence and uniqueness of the solution of the fractional differential equation are obtained by Banach and Krasnoselakii fixed point theorems, and sufficient conditions for the existence and uniqueness of the solution are also developed. In addition, the Hyers-Ulam stability of the solution is considered. At last, an example is given to illustrate the main results.

Exact Traveling Wave Solutions of the Generalized Fractional Differential mBBM Equation

By using the fractional complex transform and the bifurcation theory to the generalized fractional differential mBBM equation, we first transform this fractional equation into a plane dynamic system, and then find its equilibrium points and first integral. Based on this, the phase portraits of the corresponding plane dynamic system are given. According to the phase diagram characteristics of the dynamic system, the periodic solution corresponds to the limit cycle or periodic closed orbit. Therefore, according to the phase portraits and the properties of elliptic functions, we obtain exact explicit parametric expressions of smooth periodic wave solutions. This method can also be applied to other fractional equations.

Discussion on the Homology Theory of Lie Algebras

Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent manifolds. So we start with the differential sequence of Lie algebras. The Lie algebra g has the differential sequence E0,E1,⋯,Es⋯, which leads to the chain complex Es0→Δs0Ess→Δs1⋯→ΔsiEs(i+1)s→Δsi+1⋯of Esby discussing the chain complex E10→Δ10E11→Δ11⋯→Δ1r−1E1r→Δ1r⋯of E1and proves that Es+1i≅Hi(Es)=KerΔsi+1/ImΔsiand therefore Es+1≅H(Es)by the chain complex of Es(see Theorem 2).

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED TYPE INTEGRAL BOUNDARY CONDITIONS

In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.