Application of projection algorithms to differential equations: boundary value problems

Bishnu Lamichhane, Scott Boivin Lindstrom, Brailey Sims

Abstract


The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails.

doi:10.1017/S1446181118000391

Keywords


boundary value problem, Douglas–Rachford method, Newton’s method, hypersurface.



DOI: https://doi.org/10.21914/anziamj.v61i0.12165



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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.