一维弱噪声随机Burgers方程的奇摄动解

讨论了一类有界区域上具有有色噪声干扰的随机Burgers方程奇摄动解,其波动率服从弱噪声Ornstein-Uhlenbeck(O-U)过程.由波运动的转移概率密度函数满足的后向Kolmogorov方程,得到随机Burgers的期望所满足的后向Kolmogorov方程.由于期望满足的后向Kolmogorov方程的初边值问题条件涉及到一类确定性Burgers方程的解,因此该问题实际上是Burgers方程和Kolmogorov方程的联立形式.首先,应用奇摄动方法,对一类确定性Burgers方程进行了正则渐近展开,由Schauder估计、Ascoli-Arzela定理证明了非线性抛物方程渐近解的有界性与存在性,由Lax-Milgram定理证明了线性抛物方程渐近解的有界性与存在性,得到波速率的形式渐近解.其次,由奇摄动理论,对期望满足的方程进行了奇摄动渐近展开和边界层矫正,由二阶线性偏微分方程理论,得到边界层函数渐近解存在且有界.应用极值原理、De-Giorgi迭代技术分别证明了波速率和波期望渐近解的余项有界,得到渐近解的一致有效性.     

一类变系数非线性奇摄动方程的变形坐标法

利用变形坐标法,讨论了一类变系数的非线性奇摄动问题:(xn+εym)dy/dx+nxn-1y=1,y(1)=a>1,x∈[0,1],0<ε<<1,m,n为自然数,a为常数.通过与L-P方法的对比和对参数几种不同取值的分类探讨,得到了该变系数非线性奇摄动方程的一致有效的渐近解.并且通过数值模拟,证实了方程的精确解和用变形坐标法得到的渐近解的一致性,从而说明用变形坐标法解此类奇摄动方程的渐近解的有效性.     

非线性超抛物型方程广义渐近解的研究

讨论了一类超抛物型方程的非线性奇摄动问题.首先引入了相应问题的比较定理,然后利用奇摄动方法构造了问题的形式渐近解,最后利用比较定理,证明了问题广义解的存在性及其渐近性态.     

一类非线性奇摄动方程的匹配解法和解的精度估计

利用匹配渐近展开法,研究了一类非线性奇异摄动方程.在适当的条件下,得出了该类问题解的渐近展开式.并将结果应用于例子,对渐近解与精确解和用两变量方法求得的解进行比较,可知所得到的渐近解达到了较高精度.     

参数展开摄动法

本文提出一个参数展开摄动法,作为一个应用,讨论了非线性项上带有的参数不是很小时的一般Duffing方程的解。求得了解的渐近展开式。 本文还讨论了广义Duffing方程λ~2x+ex~3=O,这个方程不宜用寻常的摄动法求其渐近解,但用参数展开摄动法可以求其渐近解,文中构造了解的渐近形式,提出了二次近似与一次近似渐近解的稳定判据。     

非Fourier温度场分布的奇摄动解

应用非Fourier热传导定律构建了单层材料中温度场模型，即一类在无界域上带小参数的奇摄动双曲方程，通过奇摄动展开方法，得到了该问题的渐近解.首先应用奇摄动方法得到了该问题的外解和边界层矫正项，通过对内解和外解的最大模估计和关于时间导数的最大模估计以及线性抛物方程理论，得到了内外解的存在唯一性，从而得到了解的形式渐近展开式.通过余项估计，给出了渐近解的L2估计，得到了渐近解的一致有效性，从而得到了无界域上温度场的分布.通过奇摄动分析，给出了非Fourier 温度场与Fourier 温度场的关系，描述了非Fourier温度场的具体形态.     

边界和算子双摄动的高阶椭圆型方程一般边值问题解的渐近式

本文在文[1]和[2]的基础上研究边界和算子双摄动的高阶椭圆型方程一般边值问题的奇摄动,建立含两参数的渐近解表达式,导出求渐近解的迭代过程,给出余项估计,改进和拓广了前文的工作.     

一类拟线性椭圆型方程的奇摄动

本文讨论了一类二阶拟线性椭圆型方程的奇摄动问题,给出了外部解和边界层项的N阶递推方程,并对余项进行了估计,从而导得了解的渐近展开式和摄动问题解的存在唯一性.     

一类强非线性非自治方程的奇摄动Robin边值问题

本文研究一类强非线性非自治方程的奇摄动Robin边值问题.利用奇异摄动的定性、稳定性理论和方法讨论两端边界值对解的渐近性态.并得到相应解具有边界层、内部层解的结果.     

关于钱氏摄动法的高阶解的计算机求解和收敛性的研究

本文借助于中心受集中载荷圆板小挠度问题的积分方程,获得了摄动参数为中心挠度的任意n阶摄动解的解析式.于是,任意次摄动解的所有待定系数能用计算机求解.因此,获得了相当高阶的摄动解.在此基础上,讨论了钱氏摄动法的渐近性和适用区.     

两参数非线性反应扩散积分微分方程的内层激波渐近解

研究了一类两参数非线性反应扩散积分微分奇摄动问题.利用奇摄动方法,构造了问题的外部解、内部激波层、边界层及初始层校正项,由此得到了问题解的形式渐近展开式.最后利用积分微分方程的比较定理证明了该问题解的渐近展开式的一致有效性.     

奇摄动高阶微分方程边值问题的套层解

研究一类高阶微分方程两点边值问题的奇摄动,揭示了其解分别在两端点呈现双重边界层的性质.根据不同的层次,引进不同量级的伸长变量,分别构造了不同量级的边界层校正项,并利用微分不等式理论,得到了解的一致有效的渐近展开式.     

The singular perturbation of boundary value problem for third-order nonlinear vector integro-differential equation and its application

In this paper, the singular perturbation of boundary value problem to a class of third-order nonlinear vector integro-differential equation is studied. Using the method of differential inequalities, under certain conditions, the existence of perturbed solution is proved, the uniformly valid asymptotic expansion for arbitrary order and the estimation of remainder term are given. Finally, the results are applied to study singularly perturbed boundary value problem to a nonlinear vector fourth-order differential equation. The existence of solution and its asymptotic estimation can be obtained conveniently.     

Large solutions for quasilinear elliptic equation with nonlinear gradient term

We study the existence, asymptotic behavior near the boundary and uniqueness of large solutions for a class of quasilinear elliptic equation with a nonlinear gradient term. By constructing the suitable blow-up upper and lower solutions, we obtain the existence and the asymptotic behavior of radial large solutions of the problem in balls and then derive the existence of solutions in a general domain by a comparison argument. By using a perturbation method and constructing comparison functions, we show the exact asymptotic behavior of any nonnegative solution of it near the boundary. The uniqueness is shown by a standard argument.     

Singular Perturbation for a Boundary Value Problem of Fourth Order Nonlinear Differential Equation

The singular perturbation for a boundary value problem of fourth order nonlinear differential equation is studied. Under suitable assumptions using differential inequalities the author finds a solution of the original problem and obtains the uniformly valid asymptotic expansions.     

A Note on a Moving Boundary Problem Arising in the American Put Option

We consider an American put option, under the Black–Scholes model. This corresponds to a moving boundary problem for a PDE. We convert the problem to a nonlinear integral equation for the moving boundary, which corresponds to the optimal exercise of the option. We use singular perturbation methods to compute the moving boundary, as well as the full solution to the PDE, in various asymptotic limits. We consider times close to the expiration date, as well as systems where the interest rate is large or small, relative to the volatility of the asset for which the option is sold.     

一类具有边界摄动的非线性泛函椭圆型方程奇摄动问题

研究了具有边界摄动的非线性泛函椭圆型方程奇摄动边值问题．在适当的条件下．利用伸长变量、微分不等式理论，讨论了问题解的渐近性态和原问题解的存在唯一性。     

A CLASS OF NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR ELLIPTIC EQUATION WITH BOUNDARY PERTURBATION

A class of nonlinear nonlocal singularly perturbed boundary value problems for elliptic equation with boundary perturbation is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the stretched variable, the composing expansion method and the expanding theory of power series, the boundary layer is constructed, finally, using the theory of differential inequalities the asymptotic behavior of solution for the boundary value problems is studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed.     

Layer potential techniques for the narrow escape problem

The narrow escape problem consists in deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. The asymptotic formula explicitly exhibits the nonlinear interaction of many small absorbing targets. Based on the asymptotic theory for eigenvalue problems developed in Ammari et al. (2009) [3], we also construct high-order asymptotic formulas for the perturbation of eigenvalues of the Laplace and the drifted Laplace operators for mixed boundary conditions on large and small pieces of the boundary.