Numerical methods for differential equations with distributional derivatives

Abstract

A recently developed numerical method, the discrete singular convolution (DSC) method, for solving high order differential equations with distributional derivatives is presented in the different framework of distribution theory and sampling theory, respectively. In order to use this method to solve a class of differential equations with the delta distribution and its distributional derivatives, the classical approximations to the delta distribution and their convergence rates in Sobolev space $H^\alpha$ of negative order $\alpha$ are studied in details. As an example, the model of Euler-Bernoulli beam with jump discontinuities is used to test the efficiency of some delta sequences and the DSC method.