Fundamental frequency analysis of microtubules under different boundary conditions using differential quadrature method

Microtubules (MTs) are a central part of the cytoskeleton in eukaryotic cells. The dynamic behaviors of MTs are of great interest in biomechanics. Many researchers have studied the vibration analysis of MTs by modeling them as an orthotropic cylindrical elastic shell and the exact solution to its displacements is investigated under simply supported boundary conditions. Other boundary conditions lead to some coupled equations, which there are no exact solution to them. Considering various boundary conditions requires implementing semi-analytic or numerical methods. In this study, the differential quadrature method (DQM) has been used to solve the nonlinear problem of seeking fundamental frequency. At first to verify the DQM results, this method has been applied to the equations of MTs under simply supported boundary condition. The coincidence of the exact solution results and the results of DQM shows the effectiveness and precision of this method. After verification, DQM has been used for the other boundary conditions. These boundary conditions are including clamped–clamped (CC), clamped–simply (CS), clamped–free (CF) and free–free (FF) constraints. Finally, the effect of edges boundary condition, radius of MTs and half wave numbers on the vibration behavior of MTs is considered.     

半线性抛物方程初边值问题解的Blow up

本文考虑了一类半线性抛物型方程的第三类非线性初边值问题,在某些假设条件下,证明了其解在有限时间内Blow up.     

具非线性边界条件的非线性发展方程解的Blow up

讨论一类非线性发展方程具有非线性边界条件的初边值问题.在某些假设条件下,利用抛物型方程的最大值原理和凸性方法证明了其解在有限时间内具有爆破性质.     

非线性边界条件的非线性发展方程解的爆破

讨论一类非线性发展方程具有非线性边界条件的初边值问题.在某些假设条件下,利用抛物型方程的最大值原理和凸性方法证明了其解在有限时间内具有爆破性质.     

Large time behavior of the solutions to the Boltzmann equation with specular reflective boundary condition

In this paper a half space problem for the one-dimensional Boltzmann equation with specular reflective boundary condition is investigated. It is shown that the solution of the Boltzmann equation time-asymptotically converges to a global Maxwellian under some initial conditions. Furthermore, a time-decay rate is also obtained.     

On the nonuniqueness of the solution of a mixed boundary value problem for the Laplace equation

We study the solvability of the mixed boundary value problem for the Laplace equation with three distinct boundary conditions, two of which include two directional derivatives with distinct tilt angles and the remaining one is the first boundary condition. An example of a nontrivial solution of the homogeneous problem is given, and conditions under which the problem has a unique solution are established. The solvability of the problem with a nonhomogeneous first boundary condition is studied.     

Persistency of wellposedness of Ventcel’s boundary value problem under shape deformations

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to well-posed variational problems under a sign condition of the coefficient. This is achieved when physical situations are considered. Nevertheless, situations where this condition is violated appeared in several recent works where absorbing boundary conditions or equivalent boundary conditions on rough surfaces are sought for numerical purposes. The well-posedness of such problems was recently investigated: up to a countable set of parameters, existence and uniqueness of the solution for the Ventcel boundary value problem holds without the sign condition. However, the values to be avoided depend on the domain where the boundary value problem is set. In this work, we address the question of the persistency of the solvability of the boundary value problem under domain deformation.     

Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions

This paper is concerned with some dynamical property of a reaction-diffusion equation with nonlocal boundary condition. Under some conditions on the kernel in the boundary condition and suitable conditions on the reaction function, the asymptotic behavior of the time-dependent solution is characterized in relation to a finite or an infinite set of constant steady-state solutions. This characterization is determined solely by the initial function and it leads to the stability and instability of the various steady-state solutions. In the case of finite constant steady-state solutions, the time-dependent solution blows up in finite time when the initial function in greater than the largest constant solution. Also discussed is the decay property of the solution when the kernel function in the boundary condition prossesses alternating sign in its domain.     

On a class of parabolic equations with nonlocal boundary conditions

In this paper we study a class of parabolic equations subject to a nonlocal boundary condition. The problem is a generalized model for a theory of ion-diffusion in channels. By using energy method, we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution. Moreover, under some assumptions on the nonlinear boundary condition, it is shown that the solution blows up in finite time. Finally, the long-time behavior of solution to a linear problem is also studied in the paper.     

PROPERTIES OF POSITIVE SOLUTIONS FOR A NONLOCAL NONLINEAR DIFFUSION EQUATION WITH NONLOCAL NONLINEAR BOUNDARY CONDITION

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.     

Properties of positive solutions for a nonlocal nonlinear diffusion equation with nonlocal nonlinear boundary condition

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.     

The iterative solutions of 2nth-order nonlinear multi-point boundary value problems

The aim of this paper is to investigate the existence of iterative solutions for a class of 2nth-order nonlinear multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as three- or four-point boundary condition, (n + 2)-point boundary condition and 2(n − m)-point boundary condition. The existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique. A monotone iteration is developed so that the iterative sequence converges monotonically to a maximal solution or a minimal solution, depending on whether the initial iteration is an upper solution or a lower solution. Two examples are presented to illustrate the results.     

Mixed initial–boundary value problem for equations of motion of Kelvin–Voigt fluids

The initial–boundary value problem for equations of motion of Kelvin–Voigt fluids with mixed boundary conditions is studied. The no-slip condition is used on some portion of the boundary, while the impermeability condition and the tangential component of the surface force field are specified on the rest of the boundary. The global-in-time existence of a weak solution is proved. It is shown that the solution is unique and depends continuously on the field of external forces, the field of surface forces, and initial data.     

一类具有保护区域的Leslie-Gower捕食-食饵模型的分歧分析

研究了一类Neumann边界条件下带有保护区域的Leslie-Gower捕食-食饵模型,分析稳态系统从半平凡解处发生分歧的条件,得到了分歧方向及分歧值的唯一性,得到了在确定参数范围内,从半平凡解出发的分支解曲线的稳定性.     

Global existence of solutions to a parabolic–elliptic chemotaxis system with critical degenerate diffusion

This paper is devoted to the analysis of nonnegative solutions for a degenerate parabolic–elliptic Patlak–Keller–Segel system with critical nonlinear diffusion in a bounded domain with homogeneous Neumann boundary conditions. Our aim is to prove the existence of a global weak solution under a smallness condition on the mass of the initial data, thereby completing previous results on finite blow-up for large masses. Under some higher regularity condition on solutions, the uniqueness of solutions is proved by using a classical duality technique.     

Discrete absorbing boundary conditions for Schrödinger-type equations. Practical implementation

Recently, some absorbing boundary conditions for Schrödinger-type equations have been studied by Fevens, Jiang and Alonso-Mallo, and Reguera. These conditions make it possible to obtain a very high absorption at the boundary avoiding the nonlocality of transparent boundary conditions. However, the implementations used in the literature, where the boundary condition is chosen in a manual way in accordance with the solution or fixed independently of the solution, are not practical because of the small absorption. In this paper, a new practical adaptive implementation is developed that allows us to obtain automatically a very high absorption.

     

Numerical-Analytic Methods for Differential-Algebraic Systems

The numerical-analytic method is useful to show that a differential equation with a boundary condition has a solution. In this paper we use this method to differential-algebraic systems with boundary conditions. Some existence results are formulated under assumptions that the corresponding functions satisfy Lipschitz conditions in matrix notation.     

守恒方程的Chebychev-Legendre拟谱粘性方法

给出了非线性守恒方程初边值问题的Chebychev-Legendre拟谱粘性法(CLSV). 文中,用补偿方法处理边界条件,而对高频部分使用粘性法,以恢复精度. 最后证明了在适当条件下,CLSV解收敛于唯一的熵解.     

Non-constant positive steady-states of a diffusive predator-prey system in homogeneous environment

In this paper, we investigate the existence and non-existence of non-constant positive steady-states of a diffusive predator-prey interaction system under homogeneous Neumann boundary condition. In homogeneous environment, we show that the predator-prey model with Leslie-Gower functional response has no non-constant positive solution, but the system with a general functional response may have at least one non-constant positive steady-state under some conditions.     

A non‐linear and non‐local boundary condition for a diffusion equation in petroleum engineering

This article deals with a boundary value problem for Laplace equation with a non‐linear and non‐local boundary condition. This problem comes from petroleum engineering and is used to obtain an estimation of well productivity. The non‐linear and non‐local boundary condition is written on the well boundary. On the outer reservoir boundaries, we have both Dirichlet and Neumann conditions. In this paper, we prove the existence and uniqueness of a solution to this problem. The existence is proved by Schauder theorem and the uniqueness is obtained under more restricted conditions, when the involved operator is a contraction. Copyright © 2005 John Wiley & Sons, Ltd.