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IMA Journal of Applied Mathematics Advance Access originally published online on September 4, 2008
IMA Journal of Applied Mathematics 2009 74(1):128-148; doi:10.1093/imamat/hxn025
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Generalized Calderón–Ryaben'kii's potentials

Sergei V. Utyuzhnikov{dagger}

School of Mechanical Aerospace and Civil Engineering, University of Manchester, PO Box 88, Manchester M60 1QD, UK

{dagger} Email: s.utyuzhnikov{at}manchester.ac.uk

Received on December 5, 2007; Accepted on July 23, 2008

Calderón–Ryaben'kii potentials provide the foundation for the difference potential method, which is an efficient way for solving boundary-value problems (BVPs) in arbitrary domains. This method allows us to reduce a uniquely solvable and well-posed BVP to a pseudo-differential boundary equation. The general theory of Calderón–Ryaben'kii potentials is considered via the theory of distributions. The definition of Calderón–Ryaben'kii potentials is based on the notion of a clear trace. The criterion of the clear trace is formulated. Partial differential equations of the first order and the second order are considered as particular examples. On the basis of the Calderón–Ryaben'kii potential theory, a solution of the active sound control problem is obtained in a general formulation. For the first time, the solution of the problem takes into account the feedback of the active shielding sources on the input (measurement) data. The exact transfer of the boundary conditions from the original boundary to an artificial boundary is also considered.

Keywords: potential; difference potential method; pseudo-differential equation; active noise shielding; artificial boundary condition; clear trace.


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