论文标题
带有奇异系数的普通微分方程：一种固有的配方
Ordinary differential equations with singular coefficients: an intrinsic formulation with applications to the Euler-Bernoulli beam equation
论文作者
论文摘要
我们研究具有分布系数的一类线性普通微分方程（ODE）。这些方程是使用schwartz分布的{\ it固有}的乘积产物来定义的，这是分布的hörmander产品的扩展，并具有非相邻的单数支持[L.霍曼德（Hörmander），线性部分差异\ - 级算子I的分析，Springer-verlag，1983年]。我们为这些ODE提供了正则化程序，并证明了其解决方案的存在和独特定理。我们还确定了解决方案是规则和分布的条件。这些结果用于研究Euler-Bernoulli束方程，并具有不连续和奇异系数。过去使用内在产品（在某些限制性条件下）和哥伦布形式主义（在一般情况下）解决了这个问题。在这里，我们提出了一种更简单，更通用的新内在公式。作为一种应用，讨论了表现出结构裂纹的不均匀静态光束的情况。
We study a class of linear ordinary differential equations (ODE)s with distributional coefficients. These equations are defined using an {\it intrinsic} multiplicative product of Schwartz distributions which is an extension of the Hörmander product of distributions with non-intersecting singular supports [L. Hörmander, The Analysis of Linear Partial Diffe\-rential Operators I, Springer-Verlag, 1983]. We provide a regularization procedure for these ODEs and prove an existence and uniqueness theorem for their solutions. We also determine the conditions for which the solutions are regular and distributional. These results are used to study the Euler-Bernoulli beam equation with discontinuous and singular coefficients. This problem was addressed in the past using intrinsic products (under some restrictive conditions) and the Colombeau formalism (in the general case). Here we present a new intrinsic formulation that is simpler and more general. As an application, the case of a non-uniform static beam displaying structural cracks is discussed in some detail.
