## 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 language | English (US) |
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Pages (from-to) | A1789-A1808 |

Journal | SIAM Journal on Scientific Computing |

Volume | 42 |

Issue number | 3 |

DOIs | |

State | Published - 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.

## Keywords

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