EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED TYPE INTEGRAL BOUNDARY CONDITIONS

In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.

On a Discrete Fractional Boundary Value Problem with Nonlocal Fractional Boundary Conditions

In this paper, we investigate the nonlinear fractional difference equation with nonlocal fractional boundary conditions. We derive the Green's function for this problem and show that it satisfies certain properties. Some existence results are obtained by means of nonlinear alternative of Leray-Schauder type theorem and Krasnosel-skii's fixed point theorem.

Spline Fictitious Boundary Element Alternating Method for Edge Crack Problems with Mixed Boundary Conditions

The alternating method based on the fundamental solutions of the infinite domain containing a crack,namely Muskhelishvili’s solutions,divides the complex structure with a crack into a simple model without crack which can be solved by traditional numerical methods and an infinite domain with a crack which can be solved by Muskhelishvili’s solutions.However,this alternating method cannot be directly applied to the edge crack problems since partial crack surface of Muskhelishvili’s solutions is located outside the computational domain.In this paper,an improved alternating method,the spline fictitious boundary element alternating method(SFBEAM),based on infinite domain with the combination of spline fictitious boundary element method(SFBEM)and Muskhelishvili’s solutions is proposed to solve the edge crack problems.Since the SFBEM and Muskhelishvili’s solutions are obtained in the framework of infinite domain,no special treatment is needed for solving the problem of edge cracks.Different mixed boundary conditions edge crack problems with varies of computational parameters are given to certify the high precision,efficiency and applicability of the proposed method compared with other alternating methods and extend finite element method.

AMBARZUMYAN’S THEOREM WITH EIGENPARAMETER IN THE BOUNDARY CONDITIONS

In this paper, the classical Ambarzumyan’s theorem for the regular SturmLiouville problem is extended to the case in which the boundary conditions are eigenparameter dependent. Specifically, we show that if the spectrum of the operator D 2 ＋q with eigenparameter dependent boundary conditions is the same as the spectrum belonging to the zero potential, then the potential function q is actually zero.

Investigation of nanofluid flow in a vertical channel considering polynomial boundary conditions by Akbari-Ganji’s method

In this research,a vertical channel containing a laminar and fully developed nanofluid flow is investigated.The channel surface’s boundary conditions for temperature and volume fraction functions are considered qth-order polynomials.The equations related to this problem have been extracted and then solved by the AGM and validated through the Runge-Kutta numerical method and another similar study.In the study,the effect of parameters,including Grashof number,Brownian motion parameter,etc.,on the motion,velocity,temperature,and volume fraction of nanofluids have been analyzed.The results demonstrate that increasing the Gr number by 100%will increase the velocity profile function by 78%and decrease the temperature and fraction profiles by 20.87%and 120.75%.Moreover,rising the Brownian motion parameter in five different sizes(0.1,0.2,0.3,0.4,and 0.5)causes lesser velocity,about 24.3%at first and 4.35%at the last level,and a maximum 52.86%increase for temperature and a 24.32%rise for ψ occurs when N b rises from 0.1 to 0.2.For all N_(t) values,at least 55.44%,18.69%,for F(η),andΩ(η),and 20.23%rise for ψ(η)function is observed.Furthermore,enlarging the N r parameter from 0.25 to 0.1 leads F(η)to rise by 199.7%,fluid dimensionless temperature,and dimensional volume fraction to decrease by 18%and 92.3%.In the end,a greater value of q means a more powerful energy source,amplifying all velocity,temperature,and volume fraction functions.The main novelty of this research is the combined convection qth-order polynomials boundary condition applied to the channel walls.Moreover,The AMG semi-analytical method is used as a novel method to solve the governing equations.

Exact Time Domain Solutions of 1-D Transient Dynamic Piezoelectric Problems with Nonlinear Damper Boundary Conditions

Novel exact solutions of one-dimensional transient dynamic piezoelectric problems for thickness polarized layers and disks, or length polarized rods, are obtained. The solutions are derived using a time-domain Green’s function method that leads to an exact analytical recursive procedure which is applicable for a wide variety of boundary conditions including nonlinear cases. A nonlinear damper boundary condition is considered in more detail. The corresponding nonlinear relationship between stresses and velocities at a current time moment is used in the recursive procedure. In addition to the exact recursive procedure that is effective for calculations, some new practically important explicit exact solutions are presented. Several examples of the time behavior of the output electric potential difference are given to illustrate the effectiveness of the proposed exact approach.

Construction of Wave-free Potentials and Multipoles in a Two-layer Fluid Having Free-surface Boundary Condition with Higher-order Derivatives

There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems associated with the Laplace’s equation （or Helmholtz equation） arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the multipole expansion method. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace’s equation or two-dimensional Helmholz equation. Construction of these wave-free potentials and multipoles are presented here in a systematic manner for a number of situations such as two-dimensional non-oblique and oblique waves, three dimensional waves in two-layer fluid with free surface condition with higher order partial derivative are considered. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.

Analytical Solution of Boundary Integral Equations for 2-D Steady Linear Wave Problems

Based on the Fourier transform, the analytical solution of boundary integral equations formulated for the complex velocity of a 2-D steady linear surface flow is derived. It has been found that before the radiation condition is imposed,free waves appear both far upstream and downstream. In order to cancel the free waves in far upstream regions, the eigensolution of a specific eigenvalue, which satisfies the homogeneous boundary integral equation, is found and superposed to the analytical solution. An example, a submerged vortex, is used to demonstrate the derived analytical solution. Furthermore,an analytical approach to imposing the radiation condition in the numerical solution of boundary integral equations for 2-D steady linear wave problems is proposed.

Availability of the Boundary Condition for Vorticity

By applying a boundary condition for vorticity [1] in addition to that for velocity, a velocity distribution on a flat plate set in a parallel homogeneous flow has been numerically obtained through a one-way calculation from surface to infinity, without the “matching” procedure between an analysis from surface to infinity and that from infinity to surface. The numerical results obtained were in excellent agreement with those by Howarth [2]. The usage of the boundary condition for vorticity has raised the accuracy of velocity distribution near a plate’s surface and made it possible to realize the one-way calculation from surface to infinity.

解耦N-S/DSMC方法计算推力器真空羽流的边界条件研究

准确模拟羽流流场是合理评估空间推力器真空羽流效应的基础。因羽流流场同时包含连续和稀薄两种流动机理,通常采用解耦方式的Navier-Stokes(N-S)方程和直接模拟蒙特卡罗(DSMC)混合方法对其进行模拟。为保证解耦N-S/DSMC方法应用于羽流计算时的准确性并尽量提高其计算效率,对计算中的DSMC入口和喷管壁面等边界条件设置问题开展了研究。通过与文献中低总压工况的羽流试验数据比较,确定了合理设置DSMC入口边界位置、壁面反射类型、壁温等边界条件的方法。针对将此方法应用于实际推力器工况计算时效率过低的问题,通过多种数值试验,表明从喷管出口处开始沿KnGL为0.05的等值线作为DSMC入口界面可同时保证仿真精度和较高的计算效率;并证明实际工况下壁温设置对羽流场仿真结果影响不大。

K-S方程的精确边界控制

运用Fourier基函数的展开以及Fourier变换的方法研究带有周期边界条件的Kuramoto-Sivashinsky方程在有限时间区间[0,T]上的精确控制.首先研究线性化K-S方程的精确控制,运用Reimann-Lebesgue收敛定理以及R iese基函数的性质证明了在给定的时间T>0,对于两个任意给定的函数u0(x),u1(x)属于一定的Sobolev空间,总能找到一个控制函数使得线性化K-S方程有一个存在于某一合适的空间的解u(x,t)使其满足u(x,0)=u0(x),u(x,T)=u1(x).然后结合线性化K-S方程的精确控制,再通过定义Fredholm算子并应用此算子的一些理论可以找到K-S方程的控制函数,使其达到精确控制.

Numerical manifold method for steady-state nonlinear heat conduction using Kirchhoff transformation

The numerical manifold method(NMM)introduces the mathematical and physical cover to solve both continuum and discontinuum problems in a unified manner.In this study,the NMM for solving steady-state nonlinear heat conduction problems is presented,and heat conduction problems consider both convection and radiation boundary conditions.First,the nonlinear governing equation of thermal conductivity,which is dependent on temperature,is transformed into the Laplace equation by introducing the Kirchhoff transformation.The transformation reserves linearity of both the Dirichlet and the Neumann boundary conditions,but the Robin and radiation boundary conditions remain nonlinear.Second,the NMM is employed to solve the Laplace equation using a simple iteration procedure because the nonlinearity focuses on parts of the problem domain boundaries.Finally,the temperature field is retrieved through the inverse Kirchhoff transformation.Typical examples are analyzed,demonstrating the advantages of the Kirchhoff transformation over the direct solution of nonlinear equations using the NewtonRaphson method.This study provides a new method for calculating nonlinear heat conduction.

MATLAB PDE工具箱在Stokes第一问题中的应用

Stokes第一问题在工业生产领域是常见的一类问题,运用传统的相似解法解决此类问题比较抽象,难以理解。运用Matlab PDE工具箱,可以形象地描绘Stokes第一问题,可输出所研究问题的具体数值并形象地显示出某一面上的运动状态。Matlab PDE工具箱为研究管道外流体介质的粘性流动提供一种直观、快速、准确、形象的数值求解方法。

具动力边界条件的半线性Kirchhoff方程整体解的不存在性(英文)

利用凸性方法得到了具动力边界条件的半线性Kirchhoff方程整体解的不存在性.

一类带Neumann边值问题的p(x)-Kirchhoff型问题解的存在性

利用Ricceri's三临界点定理,研究了一类带Neumann边值条件的p(x)-Kirchhoff型问题解的存在性与多解性.

带有Neumann边界的Kirchhoff问题无穷多径向解的存在性

研究了有界区域上带有Neumann边界的Kirchhoff方程解的存在性.在非线性项次临界的条件下,利用喷泉定理,得到了Kirchhoff方程有无穷多个径向解.

带非局部边值条件的周期Fisher's模型正解的渐近性态

应用bootstrap技巧,讨论了带非局部边值条件的周期Fisher's模型正周期解的存在性和一般时变解的渐近性态.

一类混合边条件下的右定Sturm-Liouville问题的渐近分析

利用函数论的方法,结合Green-Liouville变换,讨论了一类带有混合边条件的右定Sturm-Liouville问题特征值的渐近估计,给出了在混合边条件下的右定Sturm-Liouville问题特征值的渐近表示.

RVSP弹性波数值模拟及传播特征分析

基于纵波、横波解耦的弹性波高阶有限差分方程和PML吸收边界条件,实现了RVSP观测统中弹性波数值模拟。采用完全弹性波波动方程进行数值模拟,可以得到纵波和横波的混合波场,且波场丰富,符合实际地震波的传播规律。该正演模拟方法纵波、横波自然解耦,产生全波场、纯纵波和纯横波模拟记录。通过对层状介质模型、凹陷模型,以及实际复杂介质模型的RVSP弹性波进行数值模拟,得到各模型不同分量下全波、纯纵波和纯横波的波场,并对弹性波传播特征进行了分析,为下一步的RVSP地震资料处理和解释工作奠定了理论基础。

基于Hopf-Cole变换的Burgers方程初边值问题的Sinc-Galerkin法

利用Sinc-Galerkin法数值求解Burgers方程的初边值问题。首先,用Hopf-Cole变换将二阶非线性的Burgers方程变换为二阶线性方程,同时把第一类边界条件变为第二类边界条件。时间上的导数采用θ加权格式离散,空间导数采用Sinc-Galerkin法离散,端点处分别引入权函数处理变换后的第二类边界条件。最后,通过数值算例验证了Sinc-Galerkin法的指数收敛性,与精确解相比,本文构造的数值格式精度高,能够有效捕捉激波等物理现象。