ON THE GROWTH OF SOLUTIONS OF A CLASS OF HIGHER ORDER DIFFERENTIAL EQUATION

In this paper, a class of higher order linear differential equation is investigated. The order and the hyper-order of the solutions of the equation are exactly estimated under some certain conditions.     

SOLUTION OF SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS BY ADOMIAN DECOMPOSITION METHOD

The aim of this paper is to apply the relatively new Adomian decomposition method to solving the system of linear fractional, in the sense of Riemann-Liouville and Caputo respectively, differential equations. The solutions are expressed in terms of Mittag-Leffler functions of matric argument. The Adomian decomposition method is straightforward, applicable for broader problems and avoids the difficulties in applying integral transforms. As the order is 1, the result here is simplified to that of first order differential equation.     

STABILITY OF GENERAL LINEAR METHODS FOR SYSTEMS OF FUNCTIONAL-DIFFERENTIAL AND FUNCTIONAL EQUATIONS

This paper is concerned with the numerical solution of functional-differential and func-tional equations which include functional-differential equations of neutral type as special cases. The adaptation of general linear methods is considered. It is proved that A-stable general linear methods can inherit the asymptotic stability of underlying linear systems.Some general results of numerical stability are also given.     

H-STABILITY OF RUNGE-KUTTA METHODS WITH VARIABLE STEPSIZE FOR SYSTEM OF PANTOGRAPH EQUATIONS

This paper deals with H-stability of Runge-Kutta methods with variable stepsize for the system of pantograph equations. It is shown that both Runge-Kutta methods with nonsingular matrix coefficient A and stiffly accurate Runge-Kutta methods are H-stable if and only if the modulus of stability function at infinity is less than 1.     

SLIDING MODE CONTROL OF A CLASS OF ITO^ TYPE DISTRIBUTED PARAMETER SYSTEMS WITH DELAY

Sliding mode control problem of a class of Ito type partial differential equations with delay is probed. The variable structure controller is designed. The existence of motion of sliding mode is shown. And the character of invariance of sliding control system about uncertainty on the sliding switching surface and stability are analyzed.     

THE UNIQUENESS OF LIMIT CYCLE AND THE STRUCTURE OF CRITICAL POINT AT INFINITY FOR A CLASS OF CUBIC SYSTEM

A class of cubic system, which is an accompany system of a quadratic differential one, is studied. It is proved that the system has at most one limit cycle, and the critical point at infinity is a higher order one. The structure and algebraic character of the critical point at infinity are obtained.     

OSCILLATION OF A CLASS OF HILLE EQUATION

This paper studies a class of Hille equation. A formula for solutions of a class of Hille equation is given. Under some suitable conditions the oscillation and nonoscillation of a class of Hille equation are established. Our results generalize the known Hille's ones.     

BEHAVIOR OF SOLUTIONS FOR A CLASS OF DIFFERENCE SYSTEMS

In this paper, the discrete-time neural network model of two neurons with piecewise constant argument is considered. Some sufficient conditions under which every solution is either periodic or convergent are obtained.     

ASYMPTOTIC BEHAVIOR OF SOLUTION FOR A CLASS OF REACTION DIFFUSION EQUATIONS

A class of initial boundary value problems for the reaction diffusion equations are considered. The asymptotic behavior of solution for the problem is obtained using the theory of differential inequality.     

ON THE HYPER-ORDER OF MEROMORPHIC SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

In this paper the order and the hyper-order of the solutions of higher-order homoge-neous linear differential equations is investigated.     

Dual characterization of fractional capacity via solution of fractional p-Laplace equation

In terms of weak solutions of the fractional p-Laplace equation with measure data, this paper offers a dual characterization for the fractional Sobolev capacity on bounded domain. In addition, two further results are given: one is an equivalent estimate for the fractional Sobolev capacity; the other is the removability of sets of zero capacity and its relation to solutions of the fractional p-Laplace equation.     

Existence of fractional differential chains and factorizations based on transformations

In this work, we deal with the existence of the fractional integrable equations involving two generalized symmetries compatible with nonlinear systems. The method used is based on the Bä cklund transformation or B‐transformation. Furthermore, we shall factorize the fractional heat operator in order to yield the fractional Riccati equation. This is done by utilizing matrix transform Miura type and matrix operators, that is, matrices whose entries are differential operators of fractional order. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. Copyright © 2014 John Wiley & Sons, Ltd.     

关于分数k-因子临界图与分数k-可扩图的若干结果

一个简单图G,如果对于V(G)的任意k元子集S,子图G-S都包含分数完美匹配,那么称G为分数后-因子临界图.如果图G的每个k-匹配M都包含在一个分数完美匹配中,那么称图G为分数k-可扩图.给出一个图是分数k-因子临界图和分数k-可扩图的充分条件,并给出一个图是分数k-因子临界图的充分必要条件.     

The effect of fractional calculus on the formation of quantum-mechanical operators

In this paper, the deformation of the ordinary quantum mechanics is formulated based on the idea of conformable fractional calculus. Some properties of fractional calculus and fractional elementary functions are investigated. The fractional wave equation in 1 + 1 dimension and fractional version of the Lorentz transformation are discussed. Finally, the fractional quantum mechanics is formulated; infinite potential well problem, density of states for the ideal gas, and quantum harmonic oscillator problem are discussed.     

A fractional partial differential equation based multiscale denoising model for texture image

Traditional integer‐order partial differential equation based image denoising approach can easily lead edge and complex texture detail blur, thus its denoising effect for texture image is always not well. To solve the problem, we propose to implement a fractional partial differential equation (FPDE) based denoising model for texture image by applying a novel mathematical method—fractional calculus to image processing from the view of system evolution. Previous studies show that fractional calculus has some unique properties that it can nonlinearly enhance complex texture detail in digital image processing, which is obvious different with integer‐order differential calculus. The goal of the modeling is to overcome the problems of the existed denoising approaches by utilizing the aforementioned properties of fractional differential calculus. Using classic definition and property of fractional differential calculus, we extend integer‐order steepest descent approach to fractional field to implement fractional steepest descent approach. Then, based on the earlier fractional formulas, a FPDE based multiscale denoising model for texture image is proposed and further analyze optimal parameters value for FPDE based denoising model. The experimental results prove that the ability for preserving high‐frequency edge and complex texture information of the proposed fractional denoising model are obviously superior to traditional integral based algorithms, as for texture detail rich images. Copyright © 2013 John Wiley & Sons, Ltd.     

分数k-因子临界图的条件

设G是-个连通简单无向图,如果删去G的任意k个项点后的图有分数完美匹配,则称G是分数k-因子临界图．给出了G是分数k-因子临界图的韧度充分条件与度和充分条件,这些条件中的界是可达的,并给出G是分数k-因子临界图的一个关于分数匹配数的充分必要条件．     

Solution of fractional order heat equation via triple Laplace transform in 2 dimensions

In this article, we introduce the triple Laplace transform for the solution of a class of fractional order partial differential equations. As a consequence, fractional order homogeneous heat equation in 2 dimensions is investigated in detail. The corresponding solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. Numerical plots to the concerned solutions are provided to demonstrate our results.     

Caputo–Hadamard Fractional Derivatives of Variable Order

In this article, we present three types of Caputo–Hadamard derivatives of variable fractional order and study the relations between them. An approximation formula for each fractional operator, using integer-order derivatives only, is obtained and an estimation for the error is given. At the end, we compare the exact fractional derivative of a concrete example with some numerical approximations.     

分数阶微分方程的比较定理

本文给出了非线性Riemann—Liouville分数阶微分方程和Caputo分数阶微分方程与相应的非线性Volterra积分方程的等价性,并在此基础上建立了分数阶微分方程的比较定理．     

Optimality conditions for fractional variational problems with dependence on a combined Caputo derivative of variable order

We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.