On the numerical solution of fourth-order linear two-point boundary value problems

William Leeb, Vladimir Rokhlin

Research output: Contribution to journalArticlepeer-review

Abstract

This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method reformulates the equation as a collection of second-kind integral equations defined on local subdomains. Each such equation can be stably discretized and solved. The boundary values of these local solutions are matched by solving a banded linear system. The method of deferred corrections is then used to increase the accuracy of the scheme. Deferred corrections require applying the integral operator to a function on the entire domain, for which we provide an algorithm with linear cost. We illustrate the performance of our method on several numerical examples.

Original languageEnglish (US)
Pages (from-to)A1789-A1808
JournalSIAM Journal on Scientific Computing
Volume42
Issue number3
DOIs
StatePublished - 2020

Bibliographical note

Funding Information:
\ast Submitted to the journal's Methods and Algorithms for Scientific Computing section September 18, 2018; accepted for publication (in revised form) April 8, 2020; published electronically June 23, 2020. https://doi.org/10.1137/18M1214810 Funding: The work of the first author was supported by a postdoctoral fellowship from the Simons Collaboration on Algorithms and Geometry, by the NSF through grant IIS-1837992, and by the BSF through grant 2018230. The work of the second author was supported by the Office of Naval Research under grant N00014-16-1-2123 and by the Air Force Office of Scientific Research under grant FA9550-16-1-0175. \dagger School of Mathematics, University of Minnesota, Minneapolis, MN 55455 (wleeb@umn.edu). \ddagger Department of Computer Science and Program in Applied Mathematics, Yale University, New Haven, CT 06511 (vladimir.rokhlin@yale.edu).

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

Keywords

  • Deferred corrections
  • Fourth-order boundary value problem
  • Second-kind integral equation

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