A framework for rigorous computational methods using Haar wavelets for differential equations

Abstract

This work presents a framework for a-posteriori error-estimating algorithms for differential equations which combines the radii polynomial approach with Haar wavelets. By using Haar wavelets, we obtain recursive structures for the matrix representations of the differential operators and quadratic nonlinearities, which can be exploited for the radii polynomial method in order to get error estimates in the $L^2$ sense. This allows the method to be applicable when the system or solution is not continuous, which is a limitation of other radii-polynomial-based methods. Numerical examples show how the method is implemented in practice.