条件方程

Spectral element methods for partial differencial equation is introduced in this study from viewpoint of the collocation approximation of Chebyshev polynomial. Wave Equation and its space discretization are deduced. Two time integral methods, central difference method and implicit Newmark method, are introduced, and their stability and applicability are also discussed in some details. The significance of absorbing boundary conditions in spectral element methods for Aeroacoustics is explained, and Clayton-Engquist-Majda absorbing boundary conditions is emphasized and introduced, then the discrete scheme of this boundary conditions is deduced and applied to spectral element methods for wave equation.

本文从Chebyshev多项式逼近理论出发，详细介绍了谱元方法求解偏微分方程的过程；推导了流体中的声波动方程并在空间上对其进行了谱元离散；详细讨论了两种时间积分方法──中心差分法和Newmark方法，分析了它们的稳定性条件，并从理论上对比了两种方法的优缺点和适用范围；将吸收边界条件推广应用于谱元方法求解气动声学问题中，重点介绍了Clayton-Engquist-Majda吸收边界条件的原理和公式，推导了该吸收边界条件的变分形式，并将其引入波动方程的离散形式中。

Weak formulation of equilibrium equations including boundary conditions of laminated cylindrical shell are presented, and thermal stresses mixed state equation for axisymmetric problem of closed cantilever cylindrical shell is established.

导出层合柱壳轴对称问题的平衡方程和边界条件的弱形式，提供了方程和边界条件放在一起的算子形式，建立了悬臂柱壳轴对称问题的热应力混合方程，给出了正交异性层合悬臂柱壳在热荷载和机械荷载作用下的弱形式解·本文提出的方法弱化了求解方程和边界条件，化解了问题，具有一般性并便于推

In Chapters 3 and 4, we investigate the existence of periodic solutions of a class of second order functional differential equation and a class of third order functional differential equation, respectively, and obtain new sufficient conditions of their existence result of periodic solutions under growth conditions and linear growth condition, respectively.

第三章与第四章分别研究了一类二阶泛函微分方程及一类三阶泛函微分方程的周期解存在性，分别在增长条件及线性增长条件下，获得了这两类方程周期解存在性的新的充分条件，与文献中已知结果比较，我们所讨论的方程更为一般，所得结果较简洁且易于验证。

In Chapters 3 and 4, we investigate the existence of periodic solutions of a class of second order functional differential equation and a class of third order functional differential equation, respectively, and obtain new sufficient conditions of their existence result of periodic solutions under growth conditions and linear growth condition, respectively. Compared with some known results in the literature, the equations discussed by us are more general, and our results are concise and easy to be verified.

第三章与第四章分别研究了一类二阶泛函微分方程及一类三阶泛函微分方程的周期解存在性，分别在增长条件及线性增长条件下，获得了这两类方程周期解存在性的新的充分条件，与文献中已知结果比较，我们所讨论的方程更为一般，所得结果较简洁且易于验证。

In chapter two, under non-Lipschitz condition, the existence and uniqueness of the solution of the second kind of BSDE is researched, based on it, the stability of the solution is proved; In chapter three, under non-Lipschitz condition, the comparison theorem of the solution of the second kind of BSDE is proved and using the monotone iterative technique , the existence of minimal and maximal solution is constructively proved; in chapter four, on the base of above results, we get some results of the second kind of BSDE which partly decouple with SDE, which include that the solution of the BSDE is continuous in the initial value of SDE and the application to optimal control and dynamic programming. At the end of this section, the character of the corresponding utility function has been discussed, e.g monotonicity, concavity and risk aversion; in chapter 5, for the first land of BSDE ,using the monotone iterative technique , the existence of minimal and maximal solution is proved and other characters and applications to utility function are studied.

首先，第二章在非Lipschitz条件下，研究了第二类方程的解的存在唯一性问题，在此基础上，又证明了解的稳定性；第三章在非Lipschitz条件下，证明了第二类BSDE解的比较定理，并在此基础上，利用单调迭代的方法，构造性证明了最大、最小解的存在性；第四章在以上的一些理论基础之上，得到了相应的与第二类倒向随机微分方程耦合的正倒向随机微分方程系统的一些结果，主要包括倒向随机微分方程的解关于正向随机微分方程的初值是具有连续性的，得到了最优控制和动态规划的一些结果，在这一章的最后还讨论了相应的效用函数的性质，如，效用函数的单调性、凹性以及风险规避性等；第五章，针对第一类倒向随机微分方程，运用单调迭代方法，证明了最大和最小解的存在性，并研究了解的其它性质及在效用函数上的应用。

Secondly, based on the different structure characteristics and additional conditions, we study several kinds of inverse problems of pseudoparabolic equations. One is a kind of pseudoparabolic inverse problem of identifying a constant coefficient solved by combining the formal solution of the problem and the additional condition properly. The second is the pseudoparabolic inverse problems of identifying an unknown boundary function and an unknown source term solved by using the Riemann function method to get the formal solution of the problem and then using the additional condition to transform the problem into a Volterra integral equation of the second kind. The third is a kind of backward heat flow problem of nonlinear pseudoparabolic equation solved by combining the Riemann function method and the fixed point theory properly.

其次，根据不同模型的结构特点和附加条件，研究了几类伪抛物型方程的反问题：一是利用问题的形式解并结合附加条件，解决了一类伪抛物型方程常数系数的反问题；二是利用Riemann函数方法获得问题的形式解，利用附加条件将问题转化成求解第二类Volterra积分方程问题，解决了一类伪抛物型方程未知边界值的反问题和未知源项的反问题；三是将Riemann函数方法和不动点定理相结合，解决了一类非线性伪抛物型方程的后向热流问题。

To avoid the trouble to construct the basis functions, we introduce a Lagrange multiplier into the variational formulation to replace the constraint condition for the basis.

本文采用同样的思路用Galerkin 边界元方法求解带有约束条件的与第一类Fredholm 间接边界积分方程等价的变分方程，为避免构造基函数时约束条件的限制，我们采用Lagrange 乘子法，把约束条件放入变分方程中去，形成扩展的变分方程，用线形单元离散后求解。

And by using the initial conditions as well as the end conditions, the dynamic problem is then transferred to a second kind Volterra integral equation about the function of the axial strain with respect to time which can also be solved successfully by the interpolation method. For piezoelectric and pyroelectric hollow cylinders, by following the solving procedure for elastic hollow cylinder and by using the electric boundary conditions, the dynamic problems are transferred to two Volterra integral equations about two functions of time, one is axial strain and the other is related to electric displacement, which can also be solved efficiently and quickly by employing interpolation method. The elastodynamic solutions of hollow spheres, which are made of elastic, piezoelectric and pyroelectric materials, respectively, for spherically symmetric problems are also obtained.

对于弹性空心圆柱，通过引入一特定函数将非齐次边界条件化为齐次边界条件，然后利用正交展开技术，导出关于时间函数的方程，再结合初始条件和端部边界条件，将原问题转化为关于一个时间函数的第二类Volterra积分方程，运用插值法可给出此积分方程的解；对于压电和热释电空心圆柱，利用求解弹性空心圆柱相似的方法，再结合电学边界条件，原问题转化为关于两个时间函数（轴向应变和与电位移有关的函数）的第二类Volterra积分方程组，同样可用插值法来构造相应的递推公式高效地求解此积分方程组。

In this project, we study the theory of higher order differential equations in Banach spaces and related topics. We solve an open problem put forward by two American Mathematicians and two Italian Mathematicians concerning wave equations with generalized Weztzell boundary conditions, introduce an existence family of operators from a Banach space $Y$ to $X$ for the Cauchy problem for higher order differential equations in a Banach space $X$, establish a sufficient and necessary condition ensuring $ACP_n$ possesses an exponentially bounded existence family, as well as some basic results in a quite general setting about the existence and continuous dependence on initial data of the solutions of $ACP_n$ and $IACP_n$. We set up quite a few multiplicative and additive perturbation theorems for existence families governing a wide class of higher order differential equations, regularized cosine operator families, regularized semigroups, and solution operators of Volterra integral equations, obtain classical and strict solutions having optimal regularity for the inhomogeneous nonautonomous heat equations with generalized Wentzell boundary conditions, gain novel existence and uniqueness theorems,which extend essentially the existing results, for mild and classical solutions of nonlocal Cauchy problems for semilinear evolution equations, present a new theorem with regard to the boundary feedback stabilization of a hybrid system composed of a viscoelastic thin plate with one part of its edge clamped and the rest-free part attached to a visocelastic rigid body. Also we obtain many other research results.

在本研究中，我们对Banach空间中的高阶算子微分方程的理论以及相关理论进行了深入研究，解决了由美国和意大利的四位数学家联合提出的一个关于广义Wentzell边界条件下的波动方程适定性的公开问题，恰当地定义了Banach空间中的高阶算子微分方程Cauchy问题的算子存在族及唯一族，建立了齐次和非齐次高阶算子微分方程Cauchy问题适定性的判别定理，获得了关于高阶退化算子微分方程的算子存在族、正则余弦算子族、正则算子半群、Volterra积分方程解算子族的乘积扰动和混合扰动定理，得到了关于以依赖于时间的二阶微分算子为系数的一大类非自治热方程非齐次情形下的时变广义Wentzell动力边值问题的古典解、严格解的最大正则性结果，获得了半线性发展方程非局部Cauchy问题广义解和经典解存在唯一的判别条件，从实质上推广了现有的相关结果；得到了一部分边缘固定而另一部分附在一粘弹性刚体上的薄板构成的混合粘弹性系统的边界反馈稳定化的新稳定化定理，还建立了一系列其他研究结果。

The transmission of heat differential equation for the heat conduction process of thermoelectric module was derived And the solution with the first kind boundary conditions under the influence of Thomson effect and the solution with the third kind boundary conditions without the influence of Thomson effect was obtained.

在假设热电制冷的热端温度固定的条件下，得出热电制冷器在不同工况下的最佳工作参数并进一步求得在三类边界条件下的最佳工作参数；导出了电偶臂在电与热的偶合作用下传热微分方程，并对其进行解析求解：在考虑汤姆逊效应的影响下得出一类跟三类边界条件下传热方程的解析解，在忽略汤姆逊效应的影响下得出二类跟三类相的边界条件下传热方程的解析解。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。