分数阶微分方程边值问题研究简介

分数阶微积分是一个古老而又新颖的课题,近30年来,由于在包括分形现象在内的物理、工程等诸多应用学科领域应用的拓展,激发了科研人员对分数阶微积分的巨大热情。分数阶微分方程现在已应用于分数物理学、混沌与湍流、粘弹性力学与非牛顿流体力学、高分子材料的解链、自动控制理论、化学物理、随机过程和反常扩散等许多科学领域。分数阶微分方程边值问题是非线性常微分方程理论研究中一个活跃而成果丰硕的领域。本文讨论了分数阶微分方程边值问题的一些理论,介绍了作者的著作《分数阶微分方程边值问题理论及应用》的基本内容。     

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.     

Generalized Newtonian fractional model for the vertical motion of a particle

Based on the Riemann-Liouville (R-L) fractional derivative and the generalized Newtonian law of gravitation, the nonlinear fractional differential equation describing the vertical motion of a particle is solved. Such solution is investigated to obtain the escape velocity (EV) following the fractional Newtonian mechanics. It is well known that the EV from the Earth’s gravitational field is about 11.18 km/s within the paradigm of the classical Newtonian mechanics using integer derivatives, but its value has not been yet determined in the scope of fractional calculus. Therefore, we can pose the question: Is the classical value of the EV identical when analyzed under the light of the fractional mechanics? The paper answers this question for the first time. It is found that the fractional escape velocity (FEV) depends on the non-integer order α and a parameter σ with dimension of seconds. The general relation between σ and α is established. The results reveal that the values of the FEV approaches the classical one when α → 1 and σ ≈ 5 × 10³ seconds.     

Necessary optimality conditions for fractional action‐like integrals of variational calculus with Riemann–Liouville derivatives of order (α, β)

We derive Euler–Lagrange‐type equations for fractional action‐like integrals of the calculus of variations which depend on the Riemann–Liouville derivatives of order (α, β), α>0, β>0, recently introduced by Cresson. Some interesting consequences are obtained and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Martingale solutions of stochastic fractional nonlinear Schrödinger equation on a bounded interval*

This paper is concerned with stochastic fractional nonlinear Schrödinger equation, which plays a very important role in fractional nonrelativistic quantum mechanics. Due to disturbing and interacting of the fractional Laplacian operator on a bounded interval with white noise, the stochastic fractional nonlinear Schrödinger equation is too complicated to be understood. This paper would explore and analyze this stochastic fractional system. Using a suitable weighted space with some fractional operator skills, it overcame the difficulties coming from the fractional Laplacian operator on a bounded interval. Applying the tightness instead of the common compactness, and combining Prokhorov theorem with Skorokhod embedding theorem, it solved the convergence problem in the case of white noise. It finally established the existence of martingale solutions for the stochastic fractional nonlinear Schrödinger equation on a bounded interval.     

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.     

Analysis of projectile motion in view of fractional calculus

The projectile motion is examined by means of the fractional calculus. The fractional differential equations of the projectile motion are introduced by generalizing Newton’s second law and Caputo’s fractional derivative is considered to use the physical initial conditions. In the absence of air resistance it is found that at certain conditions, the range and the maximum height of the projectile obtained by using the fractional calculus give the same results of the classical calculus. It is also found that, unlike the classical projectile motion, the launching angle that maximizes the horizontal range is a function of the arbitrary order of the fractional derivative α. Moreover, in a resistant medium, the parametric equations are expressed in terms of Mittag-Leffler function and the results agree with those of the classical projectile as α → 2. Moreover, the trajectories of the projectile are discussed in graphs and compared with those of the classical calculus. In order to explore the validity of modelling the projectile motion by the fractional approach, we compared our results with the experimental data of mortar.     

On solvability of differential equations with the Riesz fractional derivative

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy-type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well-known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory.     

Properties of fractional operators with fixed memory length

In this present paper, we discuss some properties of fractional operators with fixed memory length (Riemann–Liouville fractional integral, Riemann–Liouville and Caputo fractional derivatives). Some observations and examples are discussed during the article, in order to make the results well defined and clear. Furthermore, we consider the fundamental theorem of calculus for fractional operators with fixed memory length.     

An approach to differential geometry of fractional order via modified Riemann-Liouville derivative

In order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann-Liouville definition of fractional derivative, one (Jumarie) has proposed recently an alternative referred to as (local) modified Riemann-Liouville definition, which directly, provides a Taylor’s series of fractional order for non differentiable functions. We examine here in which way this calculus can be used as a framework for a differential geometry of fractional order. One will examine successively implicit function, manifold, length of curves, radius of curvature, Christoffel coefficients, velocity, acceleration. One outlines the application of this framework to Lagrange optimization in mechanics, and one concludes with some considerations on a possible fractional extension of the pseudo-geodesic of thespecial relativity and of the Lorentz transformation.     

The minimax approach for a class of variable order fractional differential equation

This paper introduces an approximate solution for Liouville‐Caputo variable order fractional differential equations with order 0 < α(t) ≤ 1 . The solution is adapted using a family of fractional‐order Chebyshev functions with unknown coefficients. These coefficients have been obtained by using an optimization approach based on minimax technique and the least pth optimization function. Several linear and nonlinear fractional‐order differential equations are discussed using the proposed technique for fixed and variable order fractional‐order derivatives. Moreover, the response of RC charging circuit with variable order fractional capacitor is studied for different cases. Several comparisons with related published techniques have been added to illustrate the accuracy of the proposed approach.     

Stochastic calculus of variations

A theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variations to stochastic processes. Generalizations of the Euler equation and Noether's theorem are obtained and several conservation laws are discussed. An application to Nelson's probabilistic framework of quantum mechanics is also given.     

一个分形函数的分数阶微积分函数

Based on the combination of fractional calculus with fractal functions, a new type of is introduced； the definition, graph, property and dimension of this function are discussed.     

Space-time fractional Schrödinger equation with time-independent potentials

We develop a space-time fractional Schrödinger equation containing Caputo fractional derivative and the quantum Riesz fractional operator from a space fractional Schrödinger equation in this paper. By use of the new equation we study the time evolution behaviors of the space-time fractional quantum system in the time-independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively. The space-time fractional Schrödinger equation with time-independent potentials is divided into a space equation and a time one. A general solution, which is composed of oscillatory terms and decay ones, is obtained. We investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative, but also on the sign (positive or negative) of the eigenvalues of the space equation. We also find that the limit value of the total probability can be greater or less than one, which means the space-time fractional Schrödinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials. Additionally, the non-Markovian time evolution laws of the space-time fractional quantum system are discussed. The formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space-time fractional quantum system. We also get a Mittag-Leffler type of time evolution operator of wave functions and then establish a Heisenberg equation containing fractional operators.     

Modeling with fractional difference equations

In this paper, we develop some basics of discrete fractional calculus such as Leibniz rule and summation by parts formula. We define simplest discrete fractional calculus of variations problem and derive Euler-Lagrange equation. We introduce and solve Gompertz fractional difference equation for tumor growth models.     

分数外微分在三维空间中的应用

首先介绍了分数微积分和分数微分形式。讨论了在原点处对曲线坐标的分数外微分变换,并且获得了从三维卡氏坐标到球面坐标和柱面坐标的两个分数微分变换。特别地,当v=m=1时,这两个分数微分变换约化的结果与通过外微积分获得的结果是一致的。     

A Functional Model for Quantum Mechanics: Unbounded Operators

We extend the recently developed Riesz–Clifford monogenic functional calculus (based on Clifford analysis) for a set of unbounded non-commuting operators. Connections with quantum mechanics are discussed. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Unique continuation property for anomalous slow diffusion equation

A Carleman estimate and the unique continuation of solutions for an anomalous diffusion equation with fractional time derivative of order 0 < α <1 are given. The estimate is derived through some subelliptic estimates for an operator associated to the anomalous diffusion equation using calculus of pseudo-differential operators.     

Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach

In the present paper, the wave propagation in one-dimensional elastic continua, characterized by nonlocal interactions modeled by fractional calculus, is investigated. Spatial derivatives of non-integer order 1 < α < 2 are involved in the governing equation, which is solved by fractional finite differences. The influence of long-range interactions is then analyzed as α varies: the resonant frequencies and the standing waves of a nonlocal bar are evaluated and the deviations from the classical (local) ones, recovered by imposing α = 2, are discussed.