A transform method for the biharmonic equation in multiply connected circular domains

Abstract

A new transform approach for solving mixed boundary value problems for the biharmonic equation in simply and multiply connected circular domains is presented. This work is a sequel to Crowdy (2015, IMA J. Appl. Math., 80, 1902–1931) where new transform techniques were developed for boundary value problems for Laplace’s equation in circular domains. A circular domain is defined to be a domain, which can be simply or multiply connected, having boundaries that are a union of circular arc segments. The method provides a flexible approach to finding quasi-analytical solutions to a wide range of problems in fluid dynamics and plane elasticity. Three example problems involving slow viscous flows are solved in detail to illustrate how to apply the method; these concern flow towards a semicircular ridge, a translating and rotating cylinder near a wall as well as in a channel geometry.

The unified transform for mixed boundary condition problems in unbounded domains

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“…where Λ is a domain containing all λ such that the integral converges (which depends on how D behaves at infinity if it is unbounded, and the conditions satisfied by q at infinity). The global relation (9) may be used directly, however to simplify it in the case of a polygonal domain where ∂D consists of M straight sides, we let q j and q j n denote the Dirichlet and Neumann boundary values a Whilst here we discuss convex polygons, for extensions to non-convex polygons one should consult Colbrook et al 9 The unified transform can also be used for circular domains [10][11][12] and non-polygonal domains with general curved edges. 13 on the j th side which connects corners z j and z j+1 .…”

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Lightning Stokes solver

“…The Goursat representation has been applied for numerical solution of biharmonic problems by Luca and coauthors [23,24] and Kazakova and Petrov [19], among others. However, it is used far less than more general tools such as the finite element method and integral equations.…”