Local Solvability and Propagation of Singularities for a Class of Psendodifferential Operators with Irregular Singularity

Consider a class of pseudodifferential operators which satisfy the conditions 1—4 (or 4').By microlocal analysis, we can reduce the operators to $\[P = {t^m}\frac{\partial }{{\partial t}} - B(x,t,{D_x},{D_t})\]$ ($\[B \in L_C^0,m > 1\]$ is integer), which we call non-Fuchsian operators. Then, we give an explicit construction for the microlocal right and left parametrices of these operators near multi-characteristics and compute wave front sets of those parametrices. Finally, we study the local solvability and the propagation of singularities for the equation corresponding to the non-Fuchsian operator. In order to obtain the previous results, the following are noteworthy. 1 The singularity of the operators $\[{t^m}\frac{\partial }{{\partial t}} - B\]$ is concentrated on $\[{t^m}\frac{\partial }{{\partial t}}\]$, So,for simplicity, we may suppose that B depends only on x and D_x. Obviously, it will lead to considerable simplification of working process. 2 It's necessary to distinguish the odd integer m from even, however, we can study these different cases in the same way. In this paper, we study only that m is odd; i. e. $\[P = {t^{2N + 1}}\frac{\partial }{{\partial t}} - 2NB(x,{D_x})\]$ (N>=1 integer) (2) 3 We have to make some hypothesis about B to obtain local solvability. Here we assume $\[{\mathop{\rm Re}\nolimits} ({b_0}(x,\xi )) < 0\]$ near characteristic point, (3) where $\[{b_0}(x,\xi )\]$ is the principal symbol of B(x, D_x). By the discussion on this subject we prove that the operator (2) is, under the assumption (3), $\[{C^\infty } - locally\]$ solvable near the multi-characteristic point (x_0, 0); and obtain the following result for the propagation of singularities: Assume the multi- oharacteristio point (x_0,0,\xi_0,0) of the operator (2) doesn't belong to WF(pu). Let v denote the null bicharaoteristio strip of symbol to through (x_0, 0,\xi_0,0). Then, if $\[WF(u) \cap v\backslash \{ ({x_0},0,{\xi _0},0)\} = \phi \]$, we have $\[({x_0},0,{\xi _0},0) \notin WF(u)\]$.     

On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity

We prove the existence of continuously differentiable solutions with required asymptotic properties as t +0 and determine the number of solutions of the following Cauchy problem for a functional differential equation:

线性微分方程的微分算子级数解法

介绍了微分算子级数法及其求解线性常微分方程通解、特解的原理、方法和实例.这个方法和其它解法的差别,在于不借助其它学科知识的启示,直接通过方程中微分算子的运算求出方程的特解或通解.     

Periodic Solutions of a Second-Order Functional Differential Equation with State-Dependent Argument

In this paper, we use Schauder and Banach fixed point theorem to study the existence, uniqueness and stability of periodic solutions of a class of iterative differential equation

$$\begin{aligned} c_0x''(t)+c_1x'(t)+c_2x(t)=x(p(t)+bx(t))+h(t). \end{aligned}$$

     

Exact Solutions of a Second-Order Nonlinear Partial Differential Equation

The equation ?²u/?t?x + u^p?u/?x = u^q describing a nonstationary process in semiconductors, with parameters p and q that are a nonnegative integer and a positive integer, respectively, and satisfy p + q ≥ 2, is considered in the half-plane (x, t) ∈ ? × (0,∞). All in all, fourteen families of its exact solutions are constructed for various parameter values, and qualitative properties of these solutions are noted. One of these families is defined for all parameter values indicated above.     

A Delay Second-Order Set-Valued Differential Equation with Hukuhara Derivatives

In this article, we study a second-order differential equation with three-point boundary conditions with the notion of Hukuhara derivatives. The existence and uniqueness of a solution is given under a Lipschitz condition on the right-hand side in the second and third variables.     

一类含参数的二阶两点边值问题的解和正解

本是研究一类含参数的二阶两点边值问题的解和正解的存在性，基本工具是Leray-Schauder不动点定理。主要结论的条件是局部的，即通过考察非线性项在其定义域的有界集上的“高度”即可决定解的存在性。     

用无限阶Toeplitz矩阵求常系数微分方程的级数解

无限阶Toeplitz矩阵的属于0的特征向量可递推地求得,可表示常系数齐次微分方程的解.用它的逆可求得常系数非齐次微分方程的特解.     

Generalized CR Equation with a Singularity

This article deals with the solvability of the CR equation rw = (A₀(θ) ? O(r^α))w ? (B₀(θ) ? O(r^α)), where z = r e^iθ and where A₀, B₀ are continuous and 2π-periodic. The solutions of this equation are shown to be similar to those of the model equation rw = A₀(θ)w ? B₀(θ). The solutions of the model equation are completely characterized by using dynamical systems and Fourier techniques.     

On the Stability of the Zero Solution of a Second-Order Differential Equation under a Periodic Perturbation of the Center

Small periodic perturbations of the oscillator \(\ddot x + {x^{2n}}\) sgn x = Y(t, x, \(\dot x\)) are considered, where n < 1 is a positive integer and the right-hand side is a small perturbation periodic in t, which is an analytic function in \(\dot x\) and x in a neighborhood of the origin. New Lyapunov-type periodic functions are introduced and used to investigate the stability of the equilibrium position of the given equation. Sufficient conditions for asymptotic stability and instability are given.     

Numerical Solution of the Cauchy Problem for a Second-Order Integro-Differential Equation

Differential Equations - In a finite-dimensional Hilbert space, we consider the Cauchy problem for a second-order integro-differential evolution equation with memory where the integrand is the...     

一类二阶微分差分方程边值问题的奇摄动解

本文利用两变量展开直接构造边界层项的方法,讨论了一类二阶微分差分方程边值问题的奇摄动解,构造了形式渐近解,作出了余项估计,从而证明了解的存在性.     

形态学上一个非线性偏微分方程的正定性及渐近解

One reaction-diffusion equation φ_tx+εφ_tφ_xx-φ_xxx=γ(φ_x) has been presented in the study of Morphogenesis. In this paper, reasonable de-finite conditions of the equation are proposed and the asymptotic form of its solution is obtained by using perturbation method. So the existence of solution of this probiem is solved.     

Second-Order Elliptic Equation of Divergence Form Having a Compactly Supported Solution

An equation of the form -div (gu) = u has a solution u of class C ₀ , where g is a real positive definite matrix-valued function belonging to the Holder classes with exponent less than 1. From the spectral point of view, this means that there exists a Schrodinger operator with periodic metric and the spectrum of this operator contains an eigenvalue of infinite multiplicity. Bibliography: 9 titles.     

二阶中立型泛函微分方程的周期解

考虑如下一类二阶中立型泛函微分方程的周期解:u″(t)-cu″(t-δ)+a(t)u(t)=λf(t,u(t-τ(t))),其中,λ>0为参数,c和δ为常数.通过应用Krasnoselskii锥不动点定理及一些分析技巧给出了这类方程周期正解的存在性非存在性和多解性.     

一类n阶中立型泛函数微分方程的周期解

利用Fouier级数理论和不动点原理研究下列方程：d^n/dt^n(x(t)-cs(t-τ))=n/∑/j=1ajx^(n-j)(t) n/∑/j=1bjx(n-j)t-τ) f(t,xt,x′t,…,x^(n-1)t)的周期解问题,得到了解的存在性和唯一性。