Contact Classification of Linear Ordinary Differential Equations: I

It is known that a linear ordinary differential equation of order n3 can be transformed to the Laguerre–Forsyth form y ⁽ⁿ⁾= _i=3 ⁿ a _n–i (x)y ^(n–i) by a point transformation of variables. The classification of equations of this form in a neighborhood of a regular point up to a contact transformation is given.     

Uniform l1 behaviour for time discretization of a Volterra equation with completely monotonic kernel: I. stability

This paper is the first of two papers on the time discretizationof the equation u_t + ^t ₀ ß (t — s) Au (s) ds= 0, t > 0, u (0) = u₀, where A is a self-adjoint denselydefined linear operator on a Hilbert space H with a completeeigensystem {_m, _m}_{m = 1}, and ß (t) is completely monotonicand locally integrable, but not constant. The equation is discretizedin time using first-order differences in combination with order-oneconvolution quadrature. The stability properties of the timediscretization are derived in the l¹_t (0, ; H) norm.     

Liouville Theorems for Generalized Harmonic Functions

Each nonzero solution of the stationary Schrödinger equation u(x)–c(r)u(x)=0 in R ⁿ with a nonnegative radial potential c(r) must have certain minimal growth at infinity. If r ² c(r)=O(1), r, then a solution having power growth at infinity, is a generalized harmonic polynomial.     

一类二阶渐近周期微分方程的正同宿轨道

§ 1　IntroductionIn this note we are concerned with the asymptotically periodic second order equation-u″+α( x) u =β( x) uq +γ( x) up,　 x∈ R,( 1 )where1 相似文献   

具有时滞的非线性强迫微分方程振动的充分必要条件

对于一类特殊的具有时滞的二阶次线性强迫微分方程x"(t) +a (t)|x(τ(t))|γsgnx(τ(t)) =g(t)(0＜τ＜1),给出了它的所有解振动的一个充要条件.     

The Design and Implementation of Usable ODE Software

In the last decade it has become standard for students and researchers to be introduced to state-of-the-art numerical software through a problem solving environment (PSE) rather than through the use of scientific libraries callable from a high level language such as Fortran or C. In this paper we will identify the constraints and implications that this imposes on the ODE software we investigate and develop. In particular, the way a numerical solution is displayed and viewed by a user dictates that new measures of performance and quality must be adopted. We will use the MATLAB environment and ODE software for initial value problems, boundary value problems and delay problems to illustrate the issues that arise and the progress that has been made. One of the major implications is the expectation that accurate approximations at off-mesh points must be provided. Traditional numerical methods for ODEs have produced approximations to the underlying solution on an associated discrete, adaptively chosen mesh. In recent years it has become common for the ODE software to also deliver approximations at off-mesh values of the independent variable. Such a feature can be extremely valuable in applications and leads to new measures of quality and performance which are more meaningful to users and more consistently interpreted and implemented in contemporary ODE software. Numerical examples of the robust and reliable behaviour of such software will be presented and the cost/reliability trade-offs that arise will be quantified.     

On a semilinear Schrödinger equation with critical Sobolev exponent

We consider the semilinear Schrödinger equation , , where , are periodic in for , 0$">, is of subcritical growth and 0 is in a gap of the spectrum of . We show that under suitable hypotheses this equation has a solution . In particular, such a solution exists if and .

     

A criterion for correct solvability of the Sturm-Liouville equation in the space

We consider an equation

where , and By a solution of equation (1), we mean any function such that and equality (1) holds almost everywhere on In this paper, we obtain a criterion for the correct solvability of (1) in ,