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Iterated Neumann problem for the higher order Poisson equation
Author(s) -
Begehr H.,
Vanegas C. J.
Publication year - 2006
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310344
Subject(s) - mathematics , neumann boundary condition , laplace operator , poisson's equation , partial differential equation , neumann series , von neumann architecture , mathematical analysis , boundary value problem , pure mathematics
Rewriting the higher order Poisson equation Δ n u = f in a plane domain as a system of Poisson equations it is immediately clear what boundary conditions may be prescribed in order to get (unique) solutions. Neumann conditions for the Poisson equation lead to higher‐order Neumann (Neumann‐ n ) problems for Δ n u = f . Extending the concept of Neumann functions for the Laplacian to Neumann functions for powers of the Laplacian leads to an explicit representation of the solution to the Neumann‐ n problem for Δ n u = f . The representation formula provides the tool to treat more general partial differential equations with leading term Δ n u in reducing them into some singular integral equations. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)