EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A CLASS OF NONLINEAR FRACTIONAL ORDER DIFFERENTIAL EQUATIONS WITH DELAY

A class of nonlinear fractional order differential equations with delay is investigated in this paper. Using Leray-Schauder fixed point theorem and the contraction mapping theorem, we obtain some sufficient conditions for the existence and uniqueness of solutions to the fractional order differential equations.

An Exploration on Adaptive Iterative Learning Control for a Class of Commensurate High-order Uncertain Nonlinear Fractional Order Systems

This paper explores the adaptive iterative learning control method in the control of fractional order systems for the first time. An adaptive iterative learning control(AILC) scheme is presented for a class of commensurate high-order uncertain nonlinear fractional order systems in the presence of disturbance.To facilitate the controller design, a sliding mode surface of tracking errors is designed by using sufficient conditions of linear fractional order systems. To relax the assumption of the identical initial condition in iterative learning control(ILC), a new boundary layer function is proposed by employing MittagLeffler function. The uncertainty in the system is compensated for by utilizing radial basis function neural network. Fractional order differential type updating laws and difference type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. The hyperbolic tangent function and a convergent series sequence are used to design robust control term for neural network approximation error and bounded disturbance, simultaneously guaranteeing the learning convergence along iteration. The system output is proved to converge to a small neighborhood of the desired trajectory by constructing Lyapnov-like composite energy function(CEF)containing new integral type Lyapunov function, while keeping all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.

A New Technique for Solving Fractional Order Systems: Hermite Collocation Method

In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, this method converts the mentioned system into a matrix equation which corresponds to a system of linear equations with unknown Hermite polynomial coefficients. Construction of the method on the aforementioned type of equations has been presented and tested on some numerical examples. Results related to the effectiveness and reliability of the method have been illustrated.

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.

Higher Order Fractional Differentiability for the Stationary Stokes System

This paper focuses on the higher order fractional differentiability of weak solution pairs to the following nonlinear stationary Stokes system{div A(x-Du)-■π=divF,inΩdivu=0,inΩ.In terms of the difference quotient method,our first result reveals that if F∈B_(p,q.loc)^(β)(Ω,R^(n))for p=2 and 1≤q≤2n/n-2β,then such extra Besov regularity can transfer to the symmetric gradient Du and its pressureπwith no losses under a suitable fractional differentiability assumption on x■A(x,ξ).Furthermore,when the vector field A(x,Du)is simplified to the full gradient■u,we improve the aforementioned Besov regularity for all integrability exponents p and q by establishing a new Campanato-type decay estimates for(■u,π).

EXISTENCE OF MILD SOLUTIONS TO SEMILINEAR FRACTIONAL ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper,we use the analytic semigroup theory of linear operators and fixed point method to prove the existence of mild solutions to a semilinear fractional order functional differential equations in a Banach space.

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO NONLINEAR DELAY FRACTIONAL DIFFERENTIAL EQUATIONS

Using a fixed point theorem,this paper discusses the existence and uniqueness of positive solutions to a system of nonlinear delay fractional differential equations and obtains some new results.

A Novel Approach for Enhancement of Geometric and Contrast Resolution Properties of Low Contrast Images

The present work encompasses a new image enhancement algorithm using newly constructed Chebyshev fractional order differentiator. We have used Chebyshev polynomials to design Chebyshev fractional order differentiator. We have generated the high pass filter corresponding to it. The designed filters are applied for decomposing the input image into four bands and low-low(L-L) sub-band is updated using correction coefficients. Reconstructed image with updated L-L sub-band provides the enhanced image. The visual results obtained are encouraging for image enhancement. The applicability of the developed algorithm is illustrated on three different test images.The effects of order of differentiation on the edges of images have also been presented and discussed.