具有非分离边界条件的非线性分数阶微分包含弱解的存在性（英文）

许多物理、航天科学、生态科学、工程中的实际问题都需要用分数阶微分方程来描述,因此对于分数阶微分方程的研究有着十分重要的理论意义和实践价值.本文在Pettis可积性假设条件下讨论了一类带有非分离边值条件的非线性分数阶微分包含弱解的存在性.微分算子是Caputo导算子,并且非线性项具有弱序列闭图像.本文的理论分析基于Monch不动点定理和弱非紧性测度的技巧,并举例论证了结论的有效性.     

分数阶时滞微分方程边值问题解的存在性与唯一性

时滞因素对分数阶微分系统的解有重要影响,系统解的变化不仅取决于现在状态,而且受到过去状态的约束,因此在分数阶微分系统中考虑时滞效应具有重要的意义.本文主要研究一类分数阶时滞微分方程边值问题解的存在性与唯一性问题.首先通过构建Green函数并利用分数阶微积分的相关性质给出该类分数阶时滞微分方程的等价方程.然后将此等价方程的求解问题转换为Banach空间中的不动点问题.再利用Banach压缩映像原理和Schauder不动点定理分别给出了保证分数阶时滞微分方程边值问题解的唯一性与存在性的充分性条件.最后,通过两个例子验证了定理结论的有效性.在考虑该类边值问题解的存在唯一性时,本文利用Banach空间中一个特殊的范数,得到系统解的存在唯一性充分性条件较以往的研究结果更为简单.这种方法是新颖的,在后续的研究过程中将尝试利用这种方法讨论带有时滞的分数阶Langevin方程边值问题的正解存在唯一性问题。     

分数阶微分方程边值问题解的存在性

应用Green函数将分数阶微分方程边值问题可转化为等价的积分方程.近来此方法被应用于讨论非线性分数阶微分方程边值问题解的存在性.本文讨论非线性分数阶微分方程边值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Carathéodory条件,利用非紧性测度的性质和M(o)nch,s不动点定理证明解的存在...     

分数阶脉冲微分方程初值问题解的存在性

利用Green函数可以将分数阶微分方程初值问题转化为等价的积分方程.近来此方法被应用于讨论非线性分数阶微分方程初值问题解的存在性.本文讨论菲线性分数阶脉冲微分方程初值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Carathéodory条件,利用非紧性测度的性质和M(o)nch,8不动点定理证明解的存在性.     

高阶分数阶微分方程系统的解的注记

分数阶导数在描述不同物质的记忆与遗传性质方面提供了有力的工具.在科学和工程的不同领域,都用分数阶微分方程组来描述动力系统.本文主要探讨分数阶微分方程系统初值问题局部解的存在性与唯一性.对于线性系统,运用Schur分解定理,给出其局部解的存在性与唯一性,并通过举例说明该方法是有效的.对于非线性系统,利用Schauder不动点定理,给出了解的存在性;运用Banach不动点定理,给出了解的唯一性.     

分数阶微分方程耦合系统边值问题解的存在性

应用Green函数将微分方程边值问题可转化为等价的积分方程.近来此方法被应用于讨论分数阶微分方程边值问题正解的存在性.本文讨论非线性分数阶微分方程耦合系统的两点边值问题,应用Green函数,将其转化为等价的积分方程耦合系统,并设非线性项在无穷远处有增长条件,应用Schauder不动点定理证明解而非限于正解的存在性.     

奇异分数阶微分方程边值问题正解的唯一性(英文)

应用Green函数将分数阶微分方程边值问题可转化为等价的积分方程.近来此方法被应用于讨论非线性分数阶微分方程边值问题解的存在性.本文讨论奇异非线性分数阶微分方程边值问题正解的唯一性.应用Green函数将其转化为等价的积分方程,利用偏序集上的不动点定理证明正解的唯一性.     

一类四阶两点边值问题多个正解的存在性

两端简单支撑弹性梁的形变可以用四阶常微分方程两点边值问题来描述。由于其在物理中的重要性,已有许多人研究了该类问题解的存在性,但在实际应用中该类问题正解以及多个正解的存在性更为重要。本文应用锥上的不动点定理,研究了该类四阶常微分方程两点边值问题多个正解的存在性,给出了该类问题多个正解存在的充分条件,本文结果推广和改进了一些已知结果。最后给出一例作为所获结果的应用。     

具有分数阶形状记忆合金系统的时滞反馈控制与稳定性分析

将分数阶引入到形状记忆合金振子模型中,针对分数阶对其系统动力学的影响进行了研究。首先基于分数阶微分方程理论,构建了分数阶形状记忆合金系统,并给出了该系统的稳定性和 Hopf 分岔存在性条件。其次,设计了时滞反馈控制器,用来控制分数阶形状记忆合金系统的稳定性。研究结果表明,形状记忆合金系统中的时滞和分数阶对系统的动力学性质有着重要的调控作用。     

非线性分数阶微分方程奇异两点边值问题的解(英文)

本文讨论了一类非线性分数阶微分方程奇异有界边值问题解的存在性.微分算子是Riemann-Liouville导算子,并且非线性项依赖于低阶分数阶导数.本文的理论分析基于Schauder不动点定理,并举例论证了结论的有效性.     

非线性分数阶耦合泛函微分方程组边值问题的可解性

由于运动速度是有限的，因此在信号传输等过程中时滞现象往往是不可避免的。分数阶泛函微分方程是研究时滞系统运动规律的重要模型，当系统中具有两个或多个状态变量且这些状态变量相互作用时，常常运用耦合微分方程组来刻画。对一类具有 Riemann-Liouville 分数阶导数的非线性时滞耦合泛函微分方程组边值问题正解的存在唯一性进行了研究。首先，根据方程与边界条件的特点，建立了比较定理，构造了上解与下解的单调序列，并确定了上下解的关系。运用上下解的方法建立并证明了边值问题正解的存在性定理，同时得到了正解的取值范围。然后，利用迭代技术建立并证明了边值问题正解的存在唯一性定理。最后，给出了具体例子用于说明所得主要结论的适应性与广泛性。     

p-Laplacian分数微分方程的周期边界问题

本文利用重合度理论，研究了一类具周期边界条件的p-Laplacian分数微分方程解的存在条件。本文的结果改进了已有的结论。     

带分布时滞分数阶微分方程非振动解的存在性北大核心CSCD

研究了一类中立型部分带有分布时滞且具有正负系数的分数阶微分方程,利用Banach压缩映像原理,通过克服算子构造和不等式放缩技巧,得到了方程有界的非振动解存在的充分条件,将其系数的适用范围拓展成不等于1和-1的实数,并通过算例验证了相关结果。     

Multiple positive solutions of integral BVPs for high-order nonlinear fractional differential equations with impulses and distributed delays

In this article, we study the existence of multiple solutions of the integral boundary value problems for high-order nonlinear fractional differential equations with impulses and distributed delays. Some sufficient criteria will be established by the fixed point index theorem in cones. As application, one example is given to demonstrate the validity of our main results.     

几类边值问题的正解

本文利用存在性定理,考察了二阶常微分方程两点、三点以及m-点边值问题正解的存在性.在较弱的条件下,给出了几类边值问题至少有一个正解存在的充分性条件.所得结果改进和推广了文献中的相应结论.     

Application of fractional derivative models in linear viscoelastic problems

Appropriate knowledge of viscoelastic properties of polymers and elastomers is of fundamental importance for a correct modelization and analysis of structures where such materials are present, especially when dealing with dynamic and vibration problems. In this paper experimental results of a series of compression and tension tests on specimens of styrene-butadiene rubber and polypropylene plastic are presented; tests consist of creep and relaxation tests, as well as cyclic loading at different frequencies. Experimental data are then used to calibrate some linear viscoelastic models; besides the classical approach based on a combination in series or parallel of standard mechanical elements as springs and dashpots, particular emphasis is given to the application of models whose constitutive equations are based on differential equations of fractional order (Fractional Derivative Model). The two approaches are compared analyzing their capability to reproduce all the experimental data for given materials; also, the main computational issues related with these models are addressed, and the advantage of using a limited number of parameters is demonstrated.     

Linear dynamical analysis of fractionally damped beams and rods

The aim of this study is to develop a general model for beams and rods with fractional derivatives. Fractional time derivatives can represent the damping term in dynamical models of continuous systems. Linear differential operators with spatial derivatives make it possible to generalize a wide range of problems. The method of multiple scales is directly applied to equations of motion. For the approximate solution, the amplitude and phase modulation equations are obtained in terms of the operators. Stability boundaries are derived from the solvability condition. It is shown that a fractional derivative influences the stability boundaries, natural frequencies, and amplitudes of vibrations. The solution procedure may be applied to many problems with linear vibrations of continuous systems.     

On the numerical solution of fractional differential equations under white noise processes

The aim of this paper is to elucidate some relevant aspects concerning the numerical solution of stochastic differential equations in structural and mechanical applications. Specifically, the attention is focused on those differential problems involving fractional operators to model the viscoelastic behavior of the structural/mechanical components and involving a white noise process as stochastic input. Starting from the consideration that the Grünwald–Letnikov based integration scheme, that is a step-by-step procedure often invoked in literature to discretize and integrate the aforementioned differential equations, is not properly employed due to the discontinuous nature of the input, an alternative numerical integration scheme is proposed. The latter is based on the Riemann–Liouville fractional integral and relies on the parabolic piecewise approximation of the response function to be integrated, leading to a more effective and more advantageous solution than that provided by the Grünwald–Letnikov based integration scheme. This is demonstrated analyzing the case study of a fractional Euler–Bernoulli beam and comparing the numerical results with those obtained by an analytical solution available in literature.     

The Investigation of the Fractional-View Dynamics of Helmholtz Equations Within Caputo Operator

It is eminent that partial differential equations are extensively meaningful in physics, mathematics and engineering. Natural phenomena are formulated with partial differential equations and are solved analytically or numerically to interrogate the system’s dynamical behavior. In the present research, mathematical modeling is extended and the modeling solutions Helmholtz equations are discussed in the fractional view of derivatives. First, the Helmholtz equations are presented in Caputo’s fractional derivative. Then Natural transformation, along with the decomposition method, is used to attain the series form solutions of the suggested problems. For justification of the proposed technique, it is applied to several numerical examples. The graphical representation of the solutions shows that the suggested technique is an accurate and effective technique with a high convergence rate than other methods. The less calculation and higher rate of convergence have confirmed the present technique’s reliability and applicability to solve partial differential equations and their systems in a fractional framework.