Least-squares solutions of nonlinear differential equations
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© 2017, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved. This study shows how to obtain least-squares solutions to initial and boundary value problems of 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, introduced in Ref. . This expression has embedded the differential equations constraints. The resulting 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. 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 validated for the simple oscillator and Duffing oscillator for 10, 000 subsequent time ranges obtaining a 10−12-level of final accuracy. A final numerical test is provided for a boundary value problem with a known solution, requiring 18 iterations to reach machine error accuracy.
author list (cited authors)
Mortari, D., Johnston, H., & Smith, L.