A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems


We present a unified boundary integral approach for the stable numerical solution of the ill-posed Cauchy problem for the heat and wave equation. The method is based on a transformation in time (semi-discretisation) using either the method of Rothe or the Laguerre transform, to generate a Cauchy problem for a sequence of inhomogenous elliptic equations; the total entity of sequences is termed an elliptic system. For this stationary system, following a recent integral approach for the Cauchy problem for the Laplace equation, the solution is represented as a sequence of single-layer potentials invoking what is known as a fundamental sequence of the elliptic system thereby avoiding the use of volume potentials and domain discretisation. Matching the given data, a system of boundary integral equations is obtained for finding a sequence of layer densities. Full discretisation is obtained via a Nyström method together with the use of Tikhonov regularization for the obtained linear systems. Numerical results are included both for the heat and wave equation confirming the practical usefulness, in terms of accuracy and resourceful use of computational effort, of the proposed approach.

Publication DOI: https://doi.org/10.1016/j.apnum.2018.03.004
Divisions: College of Engineering & Physical Sciences > School of Informatics and Digital Engineering > Mathematics
College of Engineering & Physical Sciences > Systems analytics research institute (SARI)
College of Engineering & Physical Sciences
Additional Information: © 2018, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/
Uncontrolled Keywords: Cauchy parabolic and hyperbolic problems,System of elliptic equations,Single layer potentials,Boundary integral equations,Nyström method,Tikhonov regularization
Publication ISSN: 0168-9274
Full Text Link: http://linkingh ... 168927418300667
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PURE Output Type: Article
Published Date: 2018-07
Published Online Date: 2018-03-08
Accepted Date: 2018-03-02
Authors: Chapko, Roman
Johansson, B. Tomas (ORCID Profile 0000-0001-9066-7922)

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