A stabilized separation of variables method for the modified biharmonic equation
This work provides a stable analytical approach for solving the modified biharmonic equation, which is relevant for applications such as fluid dynamics, but the method is domain-specific and incremental.
The authors develop a stabilized separation of variables method for the modified biharmonic equation in polar coordinates, introducing a new class of special functions to ensure stability. The method can be combined with fast algorithms to accelerate solutions in complex geometries.
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.