On the Numerical Solution of Fourth-Order Linear Two-Point Boundary Value Problems

TL;DR

This paper presents a fast, stable numerical algorithm for solving fourth-order linear boundary value problems by reformulating them into integral equations, enabling efficient and accurate solutions suitable for physics and signal processing applications.

Contribution

The paper introduces a novel integral equation-based method with deferred corrections for stable, efficient, and accurate solutions of fourth-order boundary value problems.

Findings

Method achieves high accuracy with linear computational cost.

Algorithm is stable for a variety of boundary value problems.

Numerical examples demonstrate effectiveness and efficiency.

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 requires 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.