Finding a sparse solution of a class of linear differential equations by solving a nonlinear system

Abstract

In this paper, a new numerical method is proposed for finding a sparse solution of a class of linear differential equations with highly oscillatory coefficients. In contrast to spectral methods and finite element methods: 1) The numerical solution is represented by a linear combination of undetermined basis functions instead of a linear combination of predetermined basis functions; 2) A small nonlinear system is obtained rather than a large linear one and, the nonlinear system can be efficiently solved by Prony method; 3) The amount of computational work of our new method is independent of the parameter in the oscillatory coefficient. Some numerical examples are given to show that our new method is promising.