A stabilized separation of variables method for the modified biharmonic equation

TL;DR

This paper introduces a stable separation of variables method for solving the modified biharmonic equation in polar coordinates, enabling efficient solutions in simple and complex geometries through new special functions and fast algorithms.

Contribution

It develops a stable separation of variables approach for the modified biharmonic equation using new special functions, applicable to interior and exterior disk problems.

Findings

Stable separation of variables representation derived

New class of special functions introduced for stability

Applicable to complex geometries with fast algorithms

Abstract

The modified biharmonic equation is encountered in a variety of application areas, including streamfunction formulations of the Navier-Stokes equations. We develop a separation of variables representation for this equation in polar coordinates, for either the interior or exterior of a disk, and derive a new class of special functions which makes the approach stable. We discuss how these functions can be used in conjunction with fast algorithms to accelerate the solution of the modified biharmonic equation or the "bi-Helmholtz" equation in more complex geometries.