A Stabilized Separation of Variables Method for the Modified Biharmonic Equation

T. Askham

doi:10.1007/s10915-018-0679-9

A Stabilized Separation of Variables Method for the Modified Biharmonic Equation

T. Askham

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

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.

Original language	English (US)
Pages (from-to)	1674-1697
Number of pages	24
Journal	Journal of Scientific Computing
Volume	76
Issue number	3
DOIs	https://doi.org/10.1007/s10915-018-0679-9
State	Published - Sep 1 2018
Externally published	Yes

All Science Journal Classification (ASJC) codes

Software
Theoretical Computer Science
Numerical Analysis
Engineering(all)
Computational Theory and Mathematics
Computational Mathematics
Applied Mathematics

Keywords

Fast multipole method
Integral equation
Modified biharmonic equation
Separation of variables
Special functions

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10.1007/s10915-018-0679-9

Cite this

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title = "A Stabilized Separation of Variables Method for the Modified Biharmonic Equation",

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.",

keywords = "Fast multipole method, Integral equation, Modified biharmonic equation, Separation of variables, Special functions",

author = "T. Askham",

note = "Funding Information: T. Askham{\textquoteright}s work was supported by the Office of the Assistant Secretary of Defense for Research and Engineering and AFOSR under NSSEFF Program Award FA9550-10-1-0180 and the Office of the Assistant Secretary of Defense for Research and Engineering and AFOSR under Award FA9550-15-1-0385. Publisher Copyright: {\textcopyright} 2018, Springer Science+Business Media, LLC, part of Springer Nature.",

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N2 - 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.

AB - 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.

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KW - Integral equation

KW - Modified biharmonic equation

KW - Separation of variables

KW - Special functions

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A Stabilized Separation of Variables Method for the Modified Biharmonic Equation

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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