An SVE Approach for the Numerical Solution of Ordinary Differential Equations

The derivative operator is reformulated as a Volterra integral operator of the first kind. So, the singular value expansion (SVE) of the kernel of such integral operator can be used to obtain new numerical methods to solve differential equations. We present such ideas in the solution of initial value problems for ordinary differential equations of first order. In particular, we develop an iterative scheme where global error in the solution of this problem is gradually reduced at each step. The global error is approximated by using the system of the singular functions in the aforementioned SVE. Some experiments are used to show the performances of the proposed numerical method.