Approximate Controllability of Neutral Functional Differential Systems with State-Dependent Delay

This paper deals with the approximate controllability of semilinear neutral functional differential systems with state-dependent delay. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can apply to the systems involving derivatives of spatial variables. By methods of functional analysis and semigroup theory, sufficient conditions of approximate controllability are formulated and proved. Finally, an example is provided to illustrate the applications of the obtained results.     

拟等级网格下非线性延迟微分方程间断有限元法

本文利用间断有限元法求解非线性延迟微分方程,在拟等级网格下.给出非线性延迟微分方程间断有限元解的整体收敛阶和局部超收敛阶,数值实验验证了理论结果的正确性.     

Sobolev方程的连续时空有限元方法

本文研究Sobolev方程的连续时空有限元方法.首先建立Sobolev方程的连续时空有限元格式,然后证明了解的存在唯一性和稳定性并给出连续时空有限元解各种范数下的误差估计.最后给出数值算例来验证理论分析的正确性,并进一步说明本文所建立的格式关于时间可以得到比传统有限元方法更高的精度.     

带有非线性收获和妊娠时滞的微分代数捕食者-食饵系统的稳定性及Hopf分支

本文研究了一类同时带有非线性食饵收获和捕食者妊娠时滞的微分代数捕食者-食饵系统的稳定性及Hopf分支问题.利用了分支理论和稳定性理论,以捕食者妊娠时滞作为系统的分支参数,获得了所提出的新系统在正平衡点处系统稳定性的相关判据条件和Hopf分支的产生条件.推广了一般带有线性收获和时滞的微分代数捕食者-食饵系统的结论.     

An optimal asymptotic-numerical method for convection dominated systems having exponential boundary layers

The convection dominated diffusion problems are studied. Higher order accurate numerical methods are presented for problems in one and two dimensions. The underlying technique utilizes a superposition of given problem into two independent problems. The first one is the reduced problem that refers to the outer or smooth solution. Stretching transformation is used to obtain the second problem for inner layer solution. The method considered for outer or degenerate problems are based on higher order Runge–Kutta methods and upwind finite differences. However, inner problem is solved analytically or asymptotically. The schemes presented are proved to be consistent and stable. Possible extensions to delay differential equations and to nonlinear problems are outlined. Numerical results for several test examples are illustrated and a comparative analysis is presented. It is observed that the method presented is highly accurate and easy to implement. Moreover, the numerical results obtained are not only comparable with the exact solution but also in agreement with the theoretical estimates.     

Stepanov ergodic perturbations for some neutral partial functional differential equations

In this work, we give sufficient conditions for the existence and uniqueness of μ?pseudo almost periodic integral solutions for some neutral partial functional differential equations with Stepanov μ?pseudo almost periodic forcing functions. Our working tools are based on the variation of constant formula and the spectral decomposition of the phase space. To illustrate our main results, we give applications to a neutral model arising in physical systems, as well as an application to heat equations with discrete and continuous delay. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence and global exponential stability of almost periodic solution for delayed competitive neural networks with discontinuous activations

In this paper, we study a class of delayed competitive neural networks with discontinuous activations. Without assuming the boundedness and local Lipschizian on the activation functions, some new criteria ensuring the existence and global exponential stability of almost periodic solutions for the neural network model considered in this work are established by constructing some suitable Lyapunov functionals and employing the theory of nonsmooth analysis. Finally, we present some applications and numerical examples with simulations to show the effectiveness of our main results. Copyright © 2016 John Wiley & Sons, Ltd.     

Non‐relativistic continuous states in arbitrary dimension for a ring‐shaped pseudo‐Coulomb and energy‐dependent potentials

In this research, the non‐relativistic particle scattering has been investigated for an alternative pseudo‐Coulomb potential plus ring‐shaped and an energy‐dependent potentials in D‐dimensional space. The normalized wave functions of continuous states on the k/2π scale are expressed in terms of the hyper‐geometric series, and formula of phase shifts is presented. Analytical properties of the scattering amplitude and thermodynamics properties are discussed. Some of the numerical results of energy levels have been calculated too. Copyright © 2015 John Wiley & Sons, Ltd.     

Fitted Galerkin spectral method to solve delay partial differential equations

In this paper, we consider a class of parabolic partial differential equations with a time delay. The first model equation is the mixed problems for scalar generalized diffusion equation with a delay, whereas the second model equation is a delayed reaction‐diffusion equation. Both of these models have inherent complex nature because of which their analytical solutions are hardly obtainable, and therefore, one has to seek numerical treatments for their approximate solutions. To this end, we develop a fitted Galerkin spectral method for solving this problem. We derive optimal error estimates based on weak formulations for the fully discrete problems. Some numerical experiments are also provided at the end. Copyright © 2015 John Wiley & Sons, Ltd.     

Decomposition of Analysis Operators and Frame Ranges for Continuous Frames

In this article, the measure space associated with a continuous frame is supposed to be σ-finite and positive, and a frame range is the range of the analysis operator for a continuous frame. Gabardo and Han in 2003 asked whether two frame ranges can both be contained in another one. To solve this problem, we give two decompositions of analysis operators and frame ranges for continuous frames respectively, which essentially establish a relationship between continuous frames and Hilbert-Schmidt operator valued frames. As applications, it follows that only separable Hilbert space can have a continuous frame, that there exists a continuous frame of Riesz-type if and only if the associated measure space is purely atomic, and that the sum of two frame ranges is still a frame range when the sum is closed. Finally, we construct a counterexample which shows that the Gabardo-Han problem is not necessarily true in general.