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APPLICATION OF PROJECTION ALGORITHMS TO DIFFERENTIAL EQUATIONS: BOUNDARY VALUE PROBLEMS

Part of: Boundary value problems Nonlinear operators and their properties

Published online by Cambridge University Press:  18 February 2019

BISHNU P. LAMICHHANE Open the ORCID record for BISHNU P. LAMICHHANE[Opens in a new window] ,
SCOTT B. LINDSTROM Open the ORCID record for SCOTT B. LINDSTROM[Opens in a new window]  and
BRAILEY SIMS Open the ORCID record for BRAILEY SIMS[Opens in a new window]
BISHNU P. LAMICHHANE
Affiliation:
Priority Research Centre for Computer-Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, New South Wales, Australia email bishnu.lamichhane@newcastle.edu.au, scott.lindstrom@uon.edu.au, brailey.sims@newcastle.edu.au
SCOTT B. LINDSTROM*
Affiliation:
Priority Research Centre for Computer-Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, New South Wales, Australia email bishnu.lamichhane@newcastle.edu.au, scott.lindstrom@uon.edu.au, brailey.sims@newcastle.edu.au
BRAILEY SIMS
Affiliation:
Priority Research Centre for Computer-Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, New South Wales, Australia email bishnu.lamichhane@newcastle.edu.au, scott.lindstrom@uon.edu.au, brailey.sims@newcastle.edu.au
*
✉scott.lindstrom@uon.edu.au
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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.

Keywords

boundary value problemDouglas–Rachford methodNewton’s methodhypersurface

MSC classification

Primary: 34B15: Nonlinear boundary value problems
Secondary: 47H10: Fixed-point theorems
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 1 , January 2019 , pp. 23 - 46
Copyright
© 2019 Australian Mathematical Society 

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