Boundedness for the Singular Integral with Variable Kernel and Fractional Differentiation on Weighted Morrey Spaces

Let T be the singular integral operator with variable kernel, T＊ be the adjoint of T and T# be the pseudo-adjoint of T. Let TIT2 be the product of T1 and T2, T1 o T2 be the pseudo product of T1 and T2. In this paper, we establish the boundedness for commutators of these operators and the fractional differentiation operator D^γ on the weighted Morrey spaces.

Ground Nephogram Enhancement Algorithm Based on Improved Adaptive Fractional Differentiation

The texture of ground-based nephogram is abundant and multiplicity.Many cloud textures are not as clear as artificial textures.A nephogram enhancement algorithm based on Adaptive Fractional Differential is established to extract the natural texture of visible ground-based cloud image.Grunwald-Lentikov(G-L)and Grunwald-Lentikov(R-L)fractional differential operators are applied to the enhancement algorithm of ground-based nephogram.An operator mask based on adaptive differential order is designed.The corresponding mask template is used to process each pixel.The results show that this method can extract image texture and edge details and simplify the process of differential order selection.

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.

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress inten- sity factors （DSIFs） for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractional wave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation （ODE）. The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional dif- ferential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.

Existence of Positive Solutions to Boundary Value Problem for a Nonlinear Fractional Differential Equation

In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.

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.

EXISTENCE OF SOLUTION FOR BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.

EXISTENCE, UNIQUENESS AND STABILITY OF RANDOM IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS

In this article, we study the existence, uniqueness, stability through continuous dependence on initial conditions and Hyers-Ulam-Rassias stability results for random impulsive fractional differential systems by relaxing the linear growth conditions. Finally, we give examples to illustrate its applications.

MIXED VARIATIONAL INEQUALITIES DRIVEN BY FRACTIONAL EVOLUTIONARY EQUATIONS

The goal of the present paper is to investigate an abstract system, called fractional differential variational inequality, which consists of a mixed variational inequality combined with a fractional evolution equation in the framework of Banach spaces. Using discrete approximation approach, an existence theorem of solutions for the inequality is established under some suitable assumptions.

NEW EXISTENCE RESULTS OF POSITIVE SOLUTION FOR A CLASS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term without any monotone requirement. Our main method to the problem is the method of upper and lower solutions and Schauder fixed point theorem. Finally, we give an example to illuminate our results.

Sliding Mode Control for Fractional Differential Systems with State-delay

The problem of sliding mode control for fractional differential systems with statedelay is considered.A novel sliding surface is proposed and a controller is designed correspondingly,such that the state starting from any initial value will move toward the switching surface and reach the sliding surface in finite time and the state variables on the sliding surface will converge to equilibrium point.And the stability of the proposed control design is discussed.

Asymptotical stability analysis of linear fractional differential systems

It has been recently found that many models were established with the aid of fractional derivatives, such as viscoelastic systems, colored noise, electrode-electrolyte polarization, dielectric polarization, boundary layer effects in ducts, electromagnetic waves, quantitative finance, quantum evolution of complex systems, and fractional kinetics. In this paper, the asymptotical stability of higher-dimensional linear fractional differential systems with the Riemann-Liouville fractional order and Caputo fractional order were studied. The asymptotical stability theorems were also derived.

A Novel Analytical Technique of the Fractional Bagley-Torvik Equations for Motion of a Rigid Plate in Newtonian Fluids

The current paper is concerned with a modified Homotopy perturbation technique.This modification allows achieving an exact solution of an initial value problem of the fractional differential equation.The approach is powerful,effective,and promising in analyzing some classes of fractional differential equations for heat conduction problems and other dynamical systems.To crystallize the new approach,some illustrated examples are introduced.

FIXED POINTS OF α-TYPE F-CONTRACTIVE MAPPINGS WITH AN APPLICATION TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

In this paper, we introduce new concepts of a-type F-contractive mappings which are essentially weaker than the class of F-contractive mappings given in [21, 22] and different from a-GF-contractions given in [8]. Then, sufficient conditions for the existence and uniqueness of fixed point are established for these new types of contractive mappings, in the setting of complete metric space. Consequently, the obtained results encompass various generalizations of the Banach contraction principle. Moreover, some examples and an application to nonlinear fractional differential equation are given to illustrate the usability of the new theory.

FRACTIONAL INTEGRAL INEQUALITIES AND THEIR APPLICATIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, first we obtain some new fractional integral inequalities. Then using these inequalities and fixed point theorems, we prove the existence of solutions for two different classes of functional fractional differential equations.

Fractional Analysis of Thin Film Flow of Non-Newtonian Fluid

Modeling and analysis of thin film flow with respect to magneto hydro dynamical effect has been an important theme in the field of fluid dynamics,due to its vast industrial applications.The analysis involves studying the behavior and response of governing equations on the basis of various parameters such as thickness of the film,film surface profile,shear stress,liquid velocity,volumetric flux,vorticity,gravity,viscosity among others,along with different boundary conditions.In this article,we extend this analysis in fractional space using a homotopy based scheme,considering the case of a Non-Newtonian Pseudo-Plastic fluid for lifting and drainage on a vertical wall.After applying similarity transformations,the given problems are reduced to highly non-linear and inhomogeneous ordinary differential equations.Moreover,fractional differential equations are obtained using basic definitions of fractional calculus.The Homotopy Perturbation Method(HPM),along with fractional calculus is used for obtaining approximate solutions.Physical quantities such as the velocity profile,volume flux and average velocity respectively for lift and drainage cases have been calculated.To the best of our knowledge,the given problems have not been attempted before in fractional space.Validity and convergence of the obtained solutions are confirmed by finding residual errors.From a physical perspective,a comprehensive study of the effects of various parameters on the velocity profile is also performed.Study reveals that Stokes number St,non-Newtonian parameterand magnetic parameter M have inverse relationship with fluid velocity in lifting case.In the drainage case,Stokes number St and non-Newtonian parameterhave direct relationship with fluid velocity,but magnetic parameter M shows inverse relationship with velocity.The investigation also shows that the fractional parameterhas direct relationship with the fluid velocity in lifting case,while it has inverse relationship with velocity in the drainage case.

PARAMETER IDENTIFICATION IN FRACTIONAL DIFFERENTIAL EQUATIONS

This article investigates the fractional derivative order identification, the coefficient identification, and the source identification in the fractional diffusion problems. If 1 〈 α〈 2, we prove the unique determination of the fractional derivative order and the dif- fusion coefficient p（x） by fo u（0, s）ds, 0 〈 t 〈 T for one-dimensional fractional diffusion-wave equations. Besides, if 0 〈 α 〈 1, we show the unique determination of the source term f（x, y） by U（0, 0, t）, 0 〈 t 〈 T for two-dimensional fractional diffusion equations. Here, a denotes the fractional derivative order over t.