High accuracy least-squares solutions of nonlinear differential equations

This study shows how to obtain least-squares solutions to initial and boundary value problems of ordinary nonlinear differential equations. The proposed method begins using an approximate solution obtained by any existing integrator. Then, a least-squares fitting of this approximate solution is obtained using a constrained expression, derived from Theory of Connections. In this expression, the differential equation constraints are embedded and are always satisfied. The resulting constrained expression is then used as an initial guess in a Newton iterative process that increases the solution accuracy to machine error level in no more than two iterations for most of the problems considered. An analysis of speed and accuracy has been conducted for this method using two nonlinear differential equations. For non-smooth solutions or for long integration times, a piecewise approach is proposed. The highly accurate value estimated at the final time is then used as the new initial guess for the next time range, and this process is repeated for subsequent time ranges. This approach has been applied and validated solving the Duffing oscillator obtaining a final solution error on the order of 10^-12. To complete the study, a final numerical test is provided for a boundary value problem with a known solution.