Fractional regularisation of the Cauchy problem for Laplace’s equation and application in some free boundary value problems
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- by Barbara Kaltenbacher and William Rundell;
- Math. Comp. 94 (2025), 647-679
- DOI: https://doi.org/10.1090/mcom/3974
- Published electronically: June 12, 2024
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Abstract:
In this paper we revisit the classical Cauchy problem for Laplace’s equation as well as two further related problems in the light of regularisation of this highly ill-conditioned problem by replacing integer derivatives with fractional ones. We do so in the spirit of quasi-reversibility, replacing a classically severely ill-posed partial differential equations problem by a nearby well-posed or only mildly ill-posed one. In order to be able to make use of the known stabilising effect of one-dimensional fractional derivatives of Abel type we work in a particular rectangular (in higher space dimensions cylindrical) geometry. We start with the plain Cauchy problem of reconstructing the values of a harmonic function inside this domain from its Dirichlet and Neumann trace on part of the boundary (the cylinder base) and explore three options for doing this with fractional operators. The two other related problems are the recovery of a free boundary and then this together with simultaneous recovery of the impedance function in the boundary condition. Our main technique here will be Newton’s method, combined with fractional regularisation of the plain Cauchy problem. The paper contains numerical reconstructions and convergence results for the devised methods.References
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Bibliographic Information
- Barbara Kaltenbacher
- Affiliation: Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Austria
- MR Author ID: 616341
- ORCID: 0000-0002-3295-6977
- Email: barbara.kaltenbacher@aau.at
- William Rundell
- Affiliation: Department of Mathematics, Texas A&M University, Texas 77843
- MR Author ID: 213241
- ORCID: 0000-0001-9579-2183
- Email: rundell@tamu.edu
- Received by editor(s): September 22, 2023
- Received by editor(s) in revised form: February 26, 2024
- Published electronically: June 12, 2024
- Additional Notes: The work of the first author was supported by the Austrian Science Fund (FWF) [10.55776/P36318]; the second author was supported in part by the National Science Foundation through award DMS-2111020.
- © Copyright 2024 American Mathematical Society
- Journal: Math. Comp. 94 (2025), 647-679
- MSC (2020): Primary 35J25, 35R11, 35R30, 35R35, 65J20
- DOI: https://doi.org/10.1090/mcom/3974
- MathSciNet review: 4841483